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icml26/a3c6aa1c-cfec-4174-aab8-a4cb5af0d892/appendix_chunks.jsonl

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+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0105", "section": "A. Motivating example 2", "page_start": 11, "page_end": 11, "type": "Text", "text": "In this section we report the posterior correlation plot obtained from an agent that learned from ML-DS as described in the second motivating example in Section 1. After observing a prediction that did not match expectation, the agent revises their beliefs, entertaining multiple hypotheses on how their beliefs should change to account for the situation. In this case, either the treatment effect is lower than anticipated and the dosage higher than assumed, or the other way around.", "source": "marker_v2", "marker_block_id": "/page/10/Text/2"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0106", "section": "A. Motivating example 2", "page_start": 11, "page_end": 11, "type": "FigureGroup", "text": "Figure 4. Plot showing the correlation of variables N E and µ A in the posterior beliefs of the agent after update with ML-DS information. The agent entertains multiple options", "source": "marker_v2", "marker_block_id": "/page/10/FigureGroup/364"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0107", "section": "B. Sanity checks on the Bayesian model", "page_start": 11, "page_end": 11, "type": "Text", "text": "We have implemented a series of experiments to make sure that the framework is functioning as intended.", "source": "marker_v2", "marker_block_id": "/page/10/Text/6"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0108", "section": "B. Sanity checks on the Bayesian model", "page_start": 11, "page_end": 11, "type": "ListGroup", "text": "1. We compare the posterior distributions with the explicit plate and with the sufficient statistics formulation given the same observations. We check if the posteriors of the two agents end up the same. This is repeated for different sizes of plate to control for potential effects of size. 2. We run an update with a single observation X o = 200 that is much higher than the prior mean of 80 (with variance 10). We verify that after the update the posterior of µ X has moved in the right direction. We also check if this creates correlations of µ X with other parameters. This is repeated for different plate sizes. 3. We run an update with many (1000) observations drawn from the real distribution of X and check that the (incorrect) prior of the agent about µ X converges to the real parameter in the generating process. 4. We set one observation with X equal to the mean of the agent's prior, with the priors on A and N E very peaked, but with P red o set much higher than the value Y = coeff ∗ X + N E ∗ A. We also setthe coeff of X in the structural equation of A to zero. We observe in the posterior that there is now a correlation between the parameters of N E and A.", "source": "marker_v2", "marker_block_id": "/page/10/ListGroup/365"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0109", "section": "C. Results obtained when varying prior µ Y", "page_start": 11, "page_end": 11, "type": "Text", "text": "In Figure 5 we report the CATE and Y plots obtained when varying the belief of the agent concerning the mean of the outcome in the historical population, µ Y .", "source": "marker_v2", "marker_block_id": "/page/10/Text/12"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0110", "section": "D. Zoomed-in versions of the CATE plots", "page_start": 11, "page_end": 11, "type": "Text", "text": "The plots in Figure 3 encompass a wide sweep of options for the priors of the agent, which revealed interesting dynamics of CATE with ML-DS. In Figure 6 we 'zoom-in' on what happens in a more realistic range of priors within ± 3 std from the true SCM parameter.", "source": "marker_v2", "marker_block_id": "/page/10/Text/14"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0111", "section": "D. Zoomed-in versions of the CATE plots", "page_start": 12, "page_end": 12, "type": "FigureGroup", "text": "Figure 5. Figures displaying the effects of different agent's priors when using ML-DS (in orange) and without it (in blue). In all plots, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameter for µ Y . The left plot displays the agent's prior and posterior (predictive) belief of CATE, while the right one shows the impact of the agent's decisions on the downstream outcome Y . In a rather wide band is the posterior within 1 std of the true mean (gray area).", "source": "marker_v2", "marker_block_id": "/page/11/FigureGroup/341"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0112", "section": "D. Zoomed-in versions of the CATE plots", "page_start": 13, "page_end": 13, "type": "FigureGroup", "text": "Figure 6. Figures displaying the effects of different agent's priors on the posterior (predictive) belief of CATE when using ML-DS (in orange) and without it (in blue). In all plots except those for NE, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameters. The plots pertain to the belief on treatment effect, NE, the belief about past treatment policy (µA), the belief about past outcomes (µ Y ) and the belief about past covariate distribution (µX). The gray areas represents the portions of the belief in which the true CATE is within one std from the CATE learned from ML-DS (orange band.", "source": "marker_v2", "marker_block_id": "/page/12/FigureGroup/509"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0113", "section": "E. Derivations for plate model", "page_start": 14, "page_end": 14, "type": "Text", "text": "We elaborate here on the strategy to reduce the plate with the help of sufficient statistics", "source": "marker_v2", "marker_block_id": "/page/13/Text/2"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0114", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Text", "text": "In this section we will show how an observation of the model parameter ϕ from the agent's internal model can be generated. This is needed to estimate the posterior distributions of the agent's beliefs using a sampling approach. As running inference through a plate-model can be both computationally expensive and numerically unstable, we chose to reformulate the plate using low-dimensional auxiliary variables.", "source": "marker_v2", "marker_block_id": "/page/13/Text/5"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0115", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Text", "text": "We start from the SCM (fig. 2) that defines the relation between the variables and we are interested in finding the lowdimensional auxiliary variables that define the regression coefficient ϕ of the model M. As we are using PyMC for inference – which does not have an implementation for the non-central χ 2 -distribution – all random variables (RVs) will be reformulated using zero-centered normally distributed RVs (standard normals where needed) and (central) χ 2 distributed RVs. Some of these RVs will be dependent and that dependence will be dealt with in section E.2.", "source": "marker_v2", "marker_block_id": "/page/13/Text/6"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0116", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Text", "text": "We are specifically looking for a low-dimensional representation of the plate and treat a sample of the plate parameters {αXµ, αXσ, αAµ, αAσ, αY µ, αY σ, NE} as given. The relevant parts of the SCM, reproduced here for the reader's convenience, are defined as", "source": "marker_v2", "marker_block_id": "/page/13/Text/7"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0117", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Equation", "text": "Y_{i} = a + bX_{i} + N_{E}A_{i} + N_{Y_{i}} A_{i} = dX_{i} + N_{A_{i}} X_{i} = N_{X_{i}} \\forall i \\in [1, ..., n] (1)", "source": "marker_v2", "marker_block_id": "/page/13/Equation/8"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0118", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Text", "text": "where n is the plate size, a, b, and d are constants and", "source": "marker_v2", "marker_block_id": "/page/13/Text/9"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0119", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Equation", "text": "N_{X_i} \\sim \\mathcal{N}(\\alpha_{X\\mu}, \\alpha_{X\\sigma}^2) N_{A_i} \\sim \\mathcal{N}(\\alpha_{A\\mu}, \\alpha_{A\\sigma}^2) N_{Y_i} \\sim \\mathcal{N}(\\alpha_{Y\\mu}, \\alpha_{Y\\sigma}^2). (2)", "source": "marker_v2", "marker_block_id": "/page/13/Equation/10"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0120", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Text", "text": "We are modeling Y i without intercept as", "source": "marker_v2", "marker_block_id": "/page/13/Text/11"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0121", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Equation", "text": "Y_i \\approx \\phi X_i \\tag{3}", "source": "marker_v2", "marker_block_id": "/page/13/Equation/12"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0122", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Text", "text": "and looking for the ordinary least-squares estimate of ϕ, that is", "source": "marker_v2", "marker_block_id": "/page/13/Text/13"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0123", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Equation", "text": "\\min_{\\phi} \\left( \\sum_{i} (Y_i - \\phi X_i)^2 \\right). \\tag{4}", "source": "marker_v2", "marker_block_id": "/page/13/Equation/14"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0124", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Text", "text": "The least-squares estimate for this case has a well-known closed-form solution given by", "source": "marker_v2", "marker_block_id": "/page/13/Text/15"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0125", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Equation", "text": "\\phi = \\frac{\\sum_{i} X_i Y_i}{\\sum_{i} X_i^2}. (5)", "source": "marker_v2", "marker_block_id": "/page/13/Equation/16"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0126", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Text", "text": "Our goal is to express this equation in terms of RVs that summarize the sums thus removing the need to sample the full plate n times. The denominator of this equation can be expressed as", "source": "marker_v2", "marker_block_id": "/page/13/Text/17"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0127", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Equation", "text": "\\sum_{i} X_i^2 = \\sum_{i} N_{X_i}^2 \\tag{6}", "source": "marker_v2", "marker_block_id": "/page/13/Equation/18"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0128", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Text", "text": "⇔", "source": "marker_v2", "marker_block_id": "/page/13/Text/19"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0129", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Equation", "text": "\\sum_{i} X_i^2 = \\sum_{i} \\alpha_{X\\mu}^2 + 2\\alpha_{X\\mu}\\alpha_{X\\sigma}\\epsilon_{X_i} + \\alpha_{X\\sigma}^2\\epsilon_{X_i}^2 (7)", "source": "marker_v2", "marker_block_id": "/page/13/Equation/20"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0130", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Text", "text": "where", "source": "marker_v2", "marker_block_id": "/page/13/Text/21"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0131", "section": "E.1. Plate model, agent's internal model", "page_start": 14, "page_end": 14, "type": "Equation", "text": "\\epsilon_{X_i} \\sim \\mathcal{N}(0, 1). (8)", "source": "marker_v2", "marker_block_id": "/page/13/Equation/22"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0132", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Text", "text": "The denominator of (5) rewritten in (7) consists of three parts:", "source": "marker_v2", "marker_block_id": "/page/14/Text/1"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0133", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_1 = \\sum_{i} \\alpha_{X\\mu}^2 \\tag{9}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/2"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0134", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_2 = \\sum_{i} 2\\alpha_{X\\mu}\\alpha_{X\\sigma}\\epsilon_{X_i} \\tag{10}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/3"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0135", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_3 = \\sum_i \\alpha_{X\\sigma}^2 \\epsilon_{X_i}^2 \\tag{11}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/4"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0136", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Text", "text": "where", "source": "marker_v2", "marker_block_id": "/page/14/Text/5"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0137", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_1 = n\\alpha_{X\\mu}^2 \\tag{12}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/6"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0138", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_2 = 2\\alpha_{X\\mu}\\alpha_{X\\sigma} \\sum_{i} \\epsilon_{X_i} \\tag{13}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/7"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0139", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_3 = \\alpha_{X\\sigma}^2 \\sum_i \\epsilon_{X_i}^2. \\tag{14}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/8"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0140", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Text", "text": "Given one set of parameters for the equations in (2), we see that S_1 is a constant, S_2 = 2\\alpha_{X\\mu}\\alpha_{X\\sigma}S_X where S_X \\sim \\mathcal{N}(0,n) , and S_3 \\sim \\alpha_{X\\sigma}^2 Z_{S_3} where Z_{S_3} \\sim \\chi^2(n) . Here S_2 is the first low-dimensional auxiliary variable we need. Note that S_3 is dependent on S_2 and this has to be dealt with separately (see section E.2).", "source": "marker_v2", "marker_block_id": "/page/14/Text/9"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0141", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Text", "text": "Next we decompose the numerator of equation (5) in a similar way. First we replace the RVs by the definitions from (1)", "source": "marker_v2", "marker_block_id": "/page/14/Text/10"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0142", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "\\sum_{i} X_{i} Y_{i} = \\sum_{i} \\left[ a N_{X_{i}} + b N_{X_{i}}^{2} + d N_{E} N_{X_{i}}^{2} + N_{E} N_{A_{i}} N_{X_{i}} + N_{Y_{i}} N_{X_{i}} \\right]. (15)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/11"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0143", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Text", "text": "As we are actually interested in equation (5), we can divide this expression by \\sum_i N_{X_i}^2 (equation (6)) to cancel out some factors and get", "source": "marker_v2", "marker_block_id": "/page/14/Text/12"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0144", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "\\phi = b + dN_E + \\frac{a\\sum_{i} N_{X_i} + N_E \\sum_{i} N_{X_i} N_{A_i} + \\sum_{i} N_{X_i} N_{Y_i}}{\\sum_{i} X_i^2} (16)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/13"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0145", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Text", "text": "Remember that a, b, and d are constants. The denominator of the last term is equivalent to equation (7). We currently lack suitable expressions only for the numerator of the last term. The numerator consists of the three parts", "source": "marker_v2", "marker_block_id": "/page/14/Text/14"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0146", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_4 = a \\sum_i N_{X_i} \\tag{17}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/15"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0147", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_5 = N_E \\sum_{i} N_{X_i} N_{A_i} \\tag{18}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/16"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0148", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_6 = \\sum_i N_{X_i} N_{Y_i}. \\tag{19}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/17"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0149", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Text", "text": "S_4 can be expressed similarly to S_2 using S_X , i.e. S_4 = an\\alpha_{X\\mu} + a\\alpha_{X\\sigma}S_X . Both S_5 and S_6 are RVs following normal product distributions. We decompose S_6 as follows:", "source": "marker_v2", "marker_block_id": "/page/14/Text/18"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0150", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_6 = \\sum_{i} \\left[ (\\alpha_{X\\mu} + \\alpha_{X\\sigma} \\epsilon_{X_i})(\\alpha_{Y\\mu} + \\alpha_{Y\\sigma} \\epsilon_{Y_i}) \\right] (20)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/19"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0151", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_6 = \\sum_{i} \\left[ \\alpha_{X\\mu} \\alpha_{Y\\mu} + \\alpha_{X\\mu} \\alpha_{Y\\sigma} \\epsilon_{Y_i} + \\alpha_{Y\\mu} \\alpha_{X\\sigma} \\epsilon_{X_i} + \\alpha_{X\\sigma} \\alpha_{Y\\sigma} \\epsilon_{X_i} \\epsilon_{Y_i} \\right] (21)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/21"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0152", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Text", "text": "\\Leftrightarrow", "source": "marker_v2", "marker_block_id": "/page/14/Text/22"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0153", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Equation", "text": "S_6 = n\\alpha_{X\\mu}\\alpha_{Y\\mu} + \\alpha_{X\\mu}\\alpha_{Y\\sigma}S_Y + \\alpha_{Y\\mu}\\alpha_{X\\sigma}S_X + \\alpha_{X\\sigma}\\alpha_{Y\\sigma}\\sum_i \\epsilon_{X_i}\\epsilon_{Y_i} (22)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/23"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0154", "section": "E.1. Plate model, agent's internal model", "page_start": 15, "page_end": 15, "type": "Text", "text": "where \\epsilon_{Y_i} \\sim \\mathcal{N}(0,1) and S_Y \\sim \\mathcal{N}(0,n) . Note that S_X and S_Y are independent. We know that \\sum_i \\epsilon_{X_i} \\epsilon_{Y_i} follows a normal product distribution that is dependent on both S_X and S_Y and deal with this dependence in section E.2 below.", "source": "marker_v2", "marker_block_id": "/page/14/Text/24"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0155", "section": "E.1. Plate model, agent's internal model", "page_start": 16, "page_end": 16, "type": "Text", "text": "Following an almost identical derivation, we express", "source": "marker_v2", "marker_block_id": "/page/15/Text/1"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0156", "section": "E.1. Plate model, agent's internal model", "page_start": 16, "page_end": 16, "type": "Equation", "text": "S_5 = N_E \\left[ n\\alpha_{X\\mu}\\alpha_{A\\mu} + \\alpha_{X\\mu}\\alpha_{A\\sigma}S_A + \\alpha_{A\\mu}\\alpha_{X\\sigma}S_X + \\alpha_{X\\sigma}\\alpha_{A\\sigma}\\sum_i \\epsilon_{X_i}\\epsilon_{A_i} \\right] (23)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/3"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0157", "section": "E.1. Plate model, agent's internal model", "page_start": 16, "page_end": 16, "type": "Text", "text": "where", "source": "marker_v2", "marker_block_id": "/page/15/Text/5"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0158", "section": "E.1. Plate model, agent's internal model", "page_start": 16, "page_end": 16, "type": "Text", "text": "E.2. Separating dependent and independent parts", "source": "marker_v2", "marker_block_id": "/page/15/Text/46"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0159", "section": "E.1. Plate model, agent's internal model", "page_start": 16, "page_end": 16, "type": "Text", "text": "It can be shown that", "source": "marker_v2", "marker_block_id": "/page/15/Text/47"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0160", "section": "E.1. Plate model, agent's internal model", "page_start": 16, "page_end": 16, "type": "Text", "text": "and sums of squares.", "source": "marker_v2", "marker_block_id": "/page/15/Text/85"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0161", "section": "E.1. Plate model, agent's internal model", "page_start": 16, "page_end": 16, "type": "Text", "text": "It can be shown that", "source": "marker_v2", "marker_block_id": "/page/15/Text/87"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0162", "section": "E.1. Plate model, agent's internal model", "page_start": 16, "page_end": 16, "type": "Text", "text": "where U_{XA} \\sim \\chi^2(n-1) and V_{XA} \\sim \\chi^2(n-1) .", "source": "marker_v2", "marker_block_id": "/page/15/Text/64"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0163", "section": "E.1. Plate model, agent's internal model", "page_start": 16, "page_end": 16, "type": "Text", "text": "\\epsilon_{A_i} \\sim \\mathcal{N}(0,1)", "source": "marker_v2", "marker_block_id": "/page/15/Text/65"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0164", "section": "E.1. Plate model, agent's internal model", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\epsilon_{A_i} \\sim \\mathcal{N}(0, 1) (24) S_A \\sim \\mathcal{N}(0, n) . (25)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/66"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0165", "section": "S_7 = \\sum_{i} \\epsilon_{X_i}^2 = \\sum_{i} (\\epsilon_{X_i} - \\bar{\\epsilon_X})^2 - \\frac{(\\sum_{i} \\epsilon_{X_i})^2}{n} (26)", "page_start": 16, "page_end": 16, "type": "Text", "text": "where \\epsilon_{X_i} \\sim \\mathcal{N}(0,1) and \\epsilon_X^- is the sample average, and that \\sum_i (\\epsilon_{X_i} - \\epsilon_X^-)^2 is independent of \\frac{(\\sum_i \\epsilon_{X_i})^2}{n} . The first term is a sum of squares of a normal and follows a \\chi^2 -distribution with n-1 degrees of freedom, i.e. \\sum_i (\\epsilon_{X_i} - \\epsilon_X^-)^2 = Z_{XX} where Z_{XX} \\sim \\chi^2(n-1) . The second term is the square of the sum, where the sum is over an RV that is normally distributed. This second term in equation (26) is \\frac{\\left(\\sum_{i} \\epsilon_{X_{i}}\\right)^{2}}{n} = \\frac{S_{X}^{2}}{n} .", "source": "marker_v2", "marker_block_id": "/page/15/Text/71"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0166", "section": "E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS", "page_start": 16, "page_end": 16, "type": "Text", "text": "E.2.1. DEPENDENCE OF SUM OF SQUARES AND (SQUARE OF) SUM", "source": "marker_v2", "marker_block_id": "/page/15/Text/86"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0167", "section": "E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS", "page_start": 16, "page_end": 16, "type": "Equation", "text": "S_8 = \\sum_i \\epsilon_{X_i} \\epsilon_{Y_i} = n \\bar{\\epsilon_X} \\bar{\\epsilon_Y} + \\sum_i (\\epsilon_{X_i} - \\bar{\\epsilon_X}) (\\epsilon_{Y_i} - \\bar{\\epsilon_Y}) (27)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/74"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0168", "section": "E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS", "page_start": 16, "page_end": 16, "type": "Text", "text": "where bar denotes sample average so that \\bar{\\epsilon_X} = \\frac{\\sum_i \\epsilon_{X_i}}{n} and \\bar{\\epsilon_Y} = \\frac{\\sum_i \\epsilon_{Y_i}}{n} .", "source": "marker_v2", "marker_block_id": "/page/15/Text/75"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0169", "section": "E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS", "page_start": 16, "page_end": 16, "type": "Text", "text": "The first term of equation (27) can be written with familiar quantities as \\bar{\\epsilon_X} = \\frac{S_X}{n} and \\bar{\\epsilon_Y} = \\frac{S_Y}{n} resulting in n\\bar{\\epsilon_X}\\bar{\\epsilon_Y} = \\frac{S_XS_Y}{n} . The second part requires more deliberation. \\sum_i (\\epsilon_{X_i} - \\bar{\\epsilon_X})(\\epsilon_{Y_i} - \\bar{\\epsilon_Y}) is clearly a sum of an RV that follows a normal product distribution. To facilitate sampling, we rewrite this as", "source": "marker_v2", "marker_block_id": "/page/15/Text/76"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0170", "section": "E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\sum (\\epsilon_{X_i} - \\bar{\\epsilon_X})(\\epsilon_{Y_i} - \\bar{\\epsilon_Y}) = \\frac{1}{4} \\sum (\\epsilon_{X_i} - \\bar{\\epsilon_X} + \\epsilon_{Y_i} - \\bar{\\epsilon_Y})^2 + \\frac{1}{4} \\sum (\\epsilon_{X_i} - \\bar{\\epsilon_X} - \\epsilon_{Y_i} + \\bar{\\epsilon_Y})^2. (28)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/77"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0171", "section": "E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS", "page_start": 16, "page_end": 16, "type": "Text", "text": "Both terms (sums of squares) in the right-hand side of this equation are sums of squared normals which are \\chi^2 -distributed with n-1 degrees of freedom. Hence", "source": "marker_v2", "marker_block_id": "/page/15/Text/78"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0172", "section": "E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS", "page_start": 16, "page_end": 16, "type": "Equation", "text": "\\sum (\\epsilon_{X_i} - \\bar{\\epsilon_X})(\\epsilon_{Y_i} - \\bar{\\epsilon_Y}) = \\frac{1}{4}(2U_{XY} - 2V_{XY}) (29)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/79"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0173", "section": "E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS", "page_start": 16, "page_end": 16, "type": "Text", "text": "where U_{XY} \\sim \\chi^2(n-1) and V_{XY} \\sim \\chi^2(n-1) and these are independent.", "source": "marker_v2", "marker_block_id": "/page/15/Text/80"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0174", "section": "E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS", "page_start": 16, "page_end": 16, "type": "Text", "text": "Following the same pattern, it can be shown that", "source": "marker_v2", "marker_block_id": "/page/15/Text/81"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0175", "section": "E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS", "page_start": 16, "page_end": 16, "type": "Equation", "text": "S_9 = \\sum_{i} \\epsilon_{X_i} \\epsilon_{A_i} = \\frac{S_X S_A}{n} + \\frac{1}{2} (U_{XA} - V_{XA}) (30)", "source": "marker_v2", "marker_block_id": "/page/15/Equation/82"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0176", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Text", "text": "To generate one observation of ϕ from the agent's internal model, sample αXµ, αXσ, αY µ, αY σ, αAµ, αAσ, and NE. Then sample one observation for", "source": "marker_v2", "marker_block_id": "/page/16/Text/2"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0177", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_X \\sim \\mathcal{N}(0, n) \\tag{31}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/3"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0178", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_Y \\sim \\mathcal{N}(0, n) \\tag{32}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/4"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0179", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_A \\sim \\mathcal{N}(0, n) \\tag{33}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/5"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0180", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "Z_{XX} \\sim \\chi^2(n-1) \\tag{34}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/6"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0181", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "U_{XY} \\sim \\chi^2(n-1) \\tag{35}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/7"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0182", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "V_{XY} \\sim \\chi^2(n-1) \\tag{36}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/8"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0183", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "U_{XA} \\sim \\chi^2(n-1) \\tag{37}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/9"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0184", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "V_{XA} \\sim \\chi^2(n-1). \\tag{38}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/10"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0185", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Text", "text": "Insert the sampled values to calculate", "source": "marker_v2", "marker_block_id": "/page/16/Text/11"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0186", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_9 = \\frac{S_X S_A}{n} + \\frac{1}{2} (U_{XA} - V_{XA}) \\tag{39}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/12"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0187", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_8 = \\frac{S_X S_Y}{n} + \\frac{1}{2} (U_{XY} - V_{XY}) \\tag{40}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/13"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0188", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_7 = Z_{XX} - \\frac{S_X^2}{n} \\tag{41}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/14"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0189", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_6 = n\\alpha_{X\\mu}\\alpha_{Y\\mu} + \\alpha_{X\\mu}\\alpha_{Y\\sigma}S_Y + \\alpha_{Y\\mu}\\alpha_{X\\sigma}S_X + \\alpha_{X\\sigma}\\alpha_{Y\\sigma}S_8 \\tag{42}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/15"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0190", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_5 = N_E \\left( n\\alpha_{X\\mu}\\alpha_{A\\mu} + \\alpha_{X\\mu}\\alpha_{A\\sigma}S_A + \\alpha_{A\\mu}\\alpha_{X\\sigma}S_X + \\alpha_{X\\sigma}\\alpha_{A\\sigma}S_9 \\right) \\tag{43}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/16"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0191", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_4 = an\\alpha_{X\\mu} + a\\alpha_{X\\sigma}S_X \\tag{44}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/17"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0192", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_3 = \\alpha_{X\\sigma}^2 S_7 \\tag{45}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/18"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0193", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_2 = 2\\alpha_{X\\mu}\\alpha_{X\\sigma}S_X \\tag{46}", "source": "marker_v2", "marker_block_id": "/page/16/Equation/19"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0194", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "S_1 = n\\alpha_{X\\mu}^2. (47)", "source": "marker_v2", "marker_block_id": "/page/16/Equation/20"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0195", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Text", "text": "With these, we can finally generate one observation from the distribution of ϕ:", "source": "marker_v2", "marker_block_id": "/page/16/Text/21"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0196", "section": "E.3. Putting it all together", "page_start": 17, "page_end": 17, "type": "Equation", "text": "\\phi = b + dN_E + \\frac{S_4 + S_5 + S_6}{S_1 + S_2 + S_3}. (48)", "source": "marker_v2", "marker_block_id": "/page/16/Equation/22"}

icml26/a3c6aa1c-cfec-4174-aab8-a4cb5af0d892/appendix_text_v3.txt

@@ -0,0 +1,275 @@
+[p. 11 | section: A. Motivating example 2 | type: Text]
+In this section we report the posterior correlation plot obtained from an agent that learned from ML-DS as described in the second motivating example in Section 1. After observing a prediction that did not match expectation, the agent revises their beliefs, entertaining multiple hypotheses on how their beliefs should change to account for the situation. In this case, either the treatment effect is lower than anticipated and the dosage higher than assumed, or the other way around.
+
+[p. 11 | section: A. Motivating example 2 | type: FigureGroup]
+Figure 4. Plot showing the correlation of variables N E and µ A in the posterior beliefs of the agent after update with ML-DS information. The agent entertains multiple options
+
+[p. 11 | section: B. Sanity checks on the Bayesian model | type: Text]
+We have implemented a series of experiments to make sure that the framework is functioning as intended.
+
+[p. 11 | section: B. Sanity checks on the Bayesian model | type: ListGroup]
+1. We compare the posterior distributions with the explicit plate and with the sufficient statistics formulation given the same observations. We check if the posteriors of the two agents end up the same. This is repeated for different sizes of plate to control for potential effects of size. 2. We run an update with a single observation X o = 200 that is much higher than the prior mean of 80 (with variance 10). We verify that after the update the posterior of µ X has moved in the right direction. We also check if this creates correlations of µ X with other parameters. This is repeated for different plate sizes. 3. We run an update with many (1000) observations drawn from the real distribution of X and check that the (incorrect) prior of the agent about µ X converges to the real parameter in the generating process. 4. We set one observation with X equal to the mean of the agent's prior, with the priors on A and N E very peaked, but with P red o set much higher than the value Y = coeff ∗ X + N E ∗ A. We also setthe coeff of X in the structural equation of A to zero. We observe in the posterior that there is now a correlation between the parameters of N E and A.
+
+[p. 11 | section: C. Results obtained when varying prior µ Y | type: Text]
+In Figure 5 we report the CATE and Y plots obtained when varying the belief of the agent concerning the mean of the outcome in the historical population, µ Y .
+
+[p. 11 | section: D. Zoomed-in versions of the CATE plots | type: Text]
+The plots in Figure 3 encompass a wide sweep of options for the priors of the agent, which revealed interesting dynamics of CATE with ML-DS. In Figure 6 we 'zoom-in' on what happens in a more realistic range of priors within ± 3 std from the true SCM parameter.
+
+[p. 12 | section: D. Zoomed-in versions of the CATE plots | type: FigureGroup]
+Figure 5. Figures displaying the effects of different agent's priors when using ML-DS (in orange) and without it (in blue). In all plots, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameter for µ Y . The left plot displays the agent's prior and posterior (predictive) belief of CATE, while the right one shows the impact of the agent's decisions on the downstream outcome Y . In a rather wide band is the posterior within 1 std of the true mean (gray area).
+
+[p. 13 | section: D. Zoomed-in versions of the CATE plots | type: FigureGroup]
+Figure 6. Figures displaying the effects of different agent's priors on the posterior (predictive) belief of CATE when using ML-DS (in orange) and without it (in blue). In all plots except those for NE, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameters. The plots pertain to the belief on treatment effect, NE, the belief about past treatment policy (µA), the belief about past outcomes (µ Y ) and the belief about past covariate distribution (µX). The gray areas represents the portions of the belief in which the true CATE is within one std from the CATE learned from ML-DS (orange band.
+
+[p. 14 | section: E. Derivations for plate model | type: Text]
+We elaborate here on the strategy to reduce the plate with the help of sufficient statistics
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Text]
+In this section we will show how an observation of the model parameter ϕ from the agent's internal model can be generated. This is needed to estimate the posterior distributions of the agent's beliefs using a sampling approach. As running inference through a plate-model can be both computationally expensive and numerically unstable, we chose to reformulate the plate using low-dimensional auxiliary variables.
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Text]
+We start from the SCM (fig. 2) that defines the relation between the variables and we are interested in finding the lowdimensional auxiliary variables that define the regression coefficient ϕ of the model M. As we are using PyMC for inference – which does not have an implementation for the non-central χ 2 -distribution – all random variables (RVs) will be reformulated using zero-centered normally distributed RVs (standard normals where needed) and (central) χ 2 distributed RVs. Some of these RVs will be dependent and that dependence will be dealt with in section E.2.
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Text]
+We are specifically looking for a low-dimensional representation of the plate and treat a sample of the plate parameters {αXµ, αXσ, αAµ, αAσ, αY µ, αY σ, NE} as given. The relevant parts of the SCM, reproduced here for the reader's convenience, are defined as
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Equation]
+Y_{i} = a + bX_{i} + N_{E}A_{i} + N_{Y_{i}} A_{i} = dX_{i} + N_{A_{i}} X_{i} = N_{X_{i}} \forall i \in [1, ..., n] (1)
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Text]
+where n is the plate size, a, b, and d are constants and
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Equation]
+N_{X_i} \sim \mathcal{N}(\alpha_{X\mu}, \alpha_{X\sigma}^2) N_{A_i} \sim \mathcal{N}(\alpha_{A\mu}, \alpha_{A\sigma}^2) N_{Y_i} \sim \mathcal{N}(\alpha_{Y\mu}, \alpha_{Y\sigma}^2). (2)
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Text]
+We are modeling Y i without intercept as
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Equation]
+Y_i \approx \phi X_i \tag{3}
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Text]
+and looking for the ordinary least-squares estimate of ϕ, that is
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Equation]
+\min_{\phi} \left( \sum_{i} (Y_i - \phi X_i)^2 \right). \tag{4}
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Text]
+The least-squares estimate for this case has a well-known closed-form solution given by
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Equation]
+\phi = \frac{\sum_{i} X_i Y_i}{\sum_{i} X_i^2}. (5)
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Text]
+Our goal is to express this equation in terms of RVs that summarize the sums thus removing the need to sample the full plate n times. The denominator of this equation can be expressed as
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Equation]
+\sum_{i} X_i^2 = \sum_{i} N_{X_i}^2 \tag{6}
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Text]
+⇔
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Equation]
+\sum_{i} X_i^2 = \sum_{i} \alpha_{X\mu}^2 + 2\alpha_{X\mu}\alpha_{X\sigma}\epsilon_{X_i} + \alpha_{X\sigma}^2\epsilon_{X_i}^2 (7)
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Text]
+where
+
+[p. 14 | section: E.1. Plate model, agent's internal model | type: Equation]
+\epsilon_{X_i} \sim \mathcal{N}(0, 1). (8)
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Text]
+The denominator of (5) rewritten in (7) consists of three parts:
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_1 = \sum_{i} \alpha_{X\mu}^2 \tag{9}
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_2 = \sum_{i} 2\alpha_{X\mu}\alpha_{X\sigma}\epsilon_{X_i} \tag{10}
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_3 = \sum_i \alpha_{X\sigma}^2 \epsilon_{X_i}^2 \tag{11}
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Text]
+where
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_1 = n\alpha_{X\mu}^2 \tag{12}
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_2 = 2\alpha_{X\mu}\alpha_{X\sigma} \sum_{i} \epsilon_{X_i} \tag{13}
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_3 = \alpha_{X\sigma}^2 \sum_i \epsilon_{X_i}^2. \tag{14}
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Text]
+Given one set of parameters for the equations in (2), we see that S_1 is a constant, S_2 = 2\alpha_{X\mu}\alpha_{X\sigma}S_X where S_X \sim \mathcal{N}(0,n) , and S_3 \sim \alpha_{X\sigma}^2 Z_{S_3} where Z_{S_3} \sim \chi^2(n) . Here S_2 is the first low-dimensional auxiliary variable we need. Note that S_3 is dependent on S_2 and this has to be dealt with separately (see section E.2).
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Text]
+Next we decompose the numerator of equation (5) in a similar way. First we replace the RVs by the definitions from (1)
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+\sum_{i} X_{i} Y_{i} = \sum_{i} \left[ a N_{X_{i}} + b N_{X_{i}}^{2} + d N_{E} N_{X_{i}}^{2} + N_{E} N_{A_{i}} N_{X_{i}} + N_{Y_{i}} N_{X_{i}} \right]. (15)
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Text]
+As we are actually interested in equation (5), we can divide this expression by \sum_i N_{X_i}^2 (equation (6)) to cancel out some factors and get
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+\phi = b + dN_E + \frac{a\sum_{i} N_{X_i} + N_E \sum_{i} N_{X_i} N_{A_i} + \sum_{i} N_{X_i} N_{Y_i}}{\sum_{i} X_i^2} (16)
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Text]
+Remember that a, b, and d are constants. The denominator of the last term is equivalent to equation (7). We currently lack suitable expressions only for the numerator of the last term. The numerator consists of the three parts
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_4 = a \sum_i N_{X_i} \tag{17}
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_5 = N_E \sum_{i} N_{X_i} N_{A_i} \tag{18}
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_6 = \sum_i N_{X_i} N_{Y_i}. \tag{19}
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Text]
+S_4 can be expressed similarly to S_2 using S_X , i.e. S_4 = an\alpha_{X\mu} + a\alpha_{X\sigma}S_X . Both S_5 and S_6 are RVs following normal product distributions. We decompose S_6 as follows:
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_6 = \sum_{i} \left[ (\alpha_{X\mu} + \alpha_{X\sigma} \epsilon_{X_i})(\alpha_{Y\mu} + \alpha_{Y\sigma} \epsilon_{Y_i}) \right] (20)
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_6 = \sum_{i} \left[ \alpha_{X\mu} \alpha_{Y\mu} + \alpha_{X\mu} \alpha_{Y\sigma} \epsilon_{Y_i} + \alpha_{Y\mu} \alpha_{X\sigma} \epsilon_{X_i} + \alpha_{X\sigma} \alpha_{Y\sigma} \epsilon_{X_i} \epsilon_{Y_i} \right] (21)
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Text]
+\Leftrightarrow
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_6 = n\alpha_{X\mu}\alpha_{Y\mu} + \alpha_{X\mu}\alpha_{Y\sigma}S_Y + \alpha_{Y\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{Y\sigma}\sum_i \epsilon_{X_i}\epsilon_{Y_i} (22)
+
+[p. 15 | section: E.1. Plate model, agent's internal model | type: Text]
+where \epsilon_{Y_i} \sim \mathcal{N}(0,1) and S_Y \sim \mathcal{N}(0,n) . Note that S_X and S_Y are independent. We know that \sum_i \epsilon_{X_i} \epsilon_{Y_i} follows a normal product distribution that is dependent on both S_X and S_Y and deal with this dependence in section E.2 below.
+
+[p. 16 | section: E.1. Plate model, agent's internal model | type: Text]
+Following an almost identical derivation, we express
+
+[p. 16 | section: E.1. Plate model, agent's internal model | type: Equation]
+S_5 = N_E \left[ n\alpha_{X\mu}\alpha_{A\mu} + \alpha_{X\mu}\alpha_{A\sigma}S_A + \alpha_{A\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{A\sigma}\sum_i \epsilon_{X_i}\epsilon_{A_i} \right] (23)
+
+[p. 16 | section: E.1. Plate model, agent's internal model | type: Text]
+where
+
+[p. 16 | section: E.1. Plate model, agent's internal model | type: Text]
+E.2. Separating dependent and independent parts
+
+[p. 16 | section: E.1. Plate model, agent's internal model | type: Text]
+It can be shown that
+
+[p. 16 | section: E.1. Plate model, agent's internal model | type: Text]
+and sums of squares.
+
+[p. 16 | section: E.1. Plate model, agent's internal model | type: Text]
+It can be shown that
+
+[p. 16 | section: E.1. Plate model, agent's internal model | type: Text]
+where U_{XA} \sim \chi^2(n-1) and V_{XA} \sim \chi^2(n-1) .
+
+[p. 16 | section: E.1. Plate model, agent's internal model | type: Text]
+\epsilon_{A_i} \sim \mathcal{N}(0,1)
+
+[p. 16 | section: E.1. Plate model, agent's internal model | type: Equation]
+\epsilon_{A_i} \sim \mathcal{N}(0, 1) (24) S_A \sim \mathcal{N}(0, n) . (25)
+
+[p. 16 | section: S_7 = \sum_{i} \epsilon_{X_i}^2 = \sum_{i} (\epsilon_{X_i} - \bar{\epsilon_X})^2 - \frac{(\sum_{i} \epsilon_{X_i})^2}{n} (26) | type: Text]
+where \epsilon_{X_i} \sim \mathcal{N}(0,1) and \epsilon_X^- is the sample average, and that \sum_i (\epsilon_{X_i} - \epsilon_X^-)^2 is independent of \frac{(\sum_i \epsilon_{X_i})^2}{n} . The first term is a sum of squares of a normal and follows a \chi^2 -distribution with n-1 degrees of freedom, i.e. \sum_i (\epsilon_{X_i} - \epsilon_X^-)^2 = Z_{XX} where Z_{XX} \sim \chi^2(n-1) . The second term is the square of the sum, where the sum is over an RV that is normally distributed. This second term in equation (26) is \frac{\left(\sum_{i} \epsilon_{X_{i}}\right)^{2}}{n} = \frac{S_{X}^{2}}{n} .
+
+[p. 16 | section: E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS | type: Text]
+E.2.1. DEPENDENCE OF SUM OF SQUARES AND (SQUARE OF) SUM
+
+[p. 16 | section: E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS | type: Equation]
+S_8 = \sum_i \epsilon_{X_i} \epsilon_{Y_i} = n \bar{\epsilon_X} \bar{\epsilon_Y} + \sum_i (\epsilon_{X_i} - \bar{\epsilon_X}) (\epsilon_{Y_i} - \bar{\epsilon_Y}) (27)
+
+[p. 16 | section: E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS | type: Text]
+where bar denotes sample average so that \bar{\epsilon_X} = \frac{\sum_i \epsilon_{X_i}}{n} and \bar{\epsilon_Y} = \frac{\sum_i \epsilon_{Y_i}}{n} .
+
+[p. 16 | section: E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS | type: Text]
+The first term of equation (27) can be written with familiar quantities as \bar{\epsilon_X} = \frac{S_X}{n} and \bar{\epsilon_Y} = \frac{S_Y}{n} resulting in n\bar{\epsilon_X}\bar{\epsilon_Y} = \frac{S_XS_Y}{n} . The second part requires more deliberation. \sum_i (\epsilon_{X_i} - \bar{\epsilon_X})(\epsilon_{Y_i} - \bar{\epsilon_Y}) is clearly a sum of an RV that follows a normal product distribution. To facilitate sampling, we rewrite this as
+
+[p. 16 | section: E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS | type: Equation]
+\sum (\epsilon_{X_i} - \bar{\epsilon_X})(\epsilon_{Y_i} - \bar{\epsilon_Y}) = \frac{1}{4} \sum (\epsilon_{X_i} - \bar{\epsilon_X} + \epsilon_{Y_i} - \bar{\epsilon_Y})^2 + \frac{1}{4} \sum (\epsilon_{X_i} - \bar{\epsilon_X} - \epsilon_{Y_i} + \bar{\epsilon_Y})^2. (28)
+
+[p. 16 | section: E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS | type: Text]
+Both terms (sums of squares) in the right-hand side of this equation are sums of squared normals which are \chi^2 -distributed with n-1 degrees of freedom. Hence
+
+[p. 16 | section: E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS | type: Equation]
+\sum (\epsilon_{X_i} - \bar{\epsilon_X})(\epsilon_{Y_i} - \bar{\epsilon_Y}) = \frac{1}{4}(2U_{XY} - 2V_{XY}) (29)
+
+[p. 16 | section: E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS | type: Text]
+where U_{XY} \sim \chi^2(n-1) and V_{XY} \sim \chi^2(n-1) and these are independent.
+
+[p. 16 | section: E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS | type: Text]
+Following the same pattern, it can be shown that
+
+[p. 16 | section: E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS | type: Equation]
+S_9 = \sum_{i} \epsilon_{X_i} \epsilon_{A_i} = \frac{S_X S_A}{n} + \frac{1}{2} (U_{XA} - V_{XA}) (30)
+
+[p. 17 | section: E.3. Putting it all together | type: Text]
+To generate one observation of ϕ from the agent's internal model, sample αXµ, αXσ, αY µ, αY σ, αAµ, αAσ, and NE. Then sample one observation for
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_X \sim \mathcal{N}(0, n) \tag{31}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_Y \sim \mathcal{N}(0, n) \tag{32}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_A \sim \mathcal{N}(0, n) \tag{33}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+Z_{XX} \sim \chi^2(n-1) \tag{34}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+U_{XY} \sim \chi^2(n-1) \tag{35}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+V_{XY} \sim \chi^2(n-1) \tag{36}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+U_{XA} \sim \chi^2(n-1) \tag{37}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+V_{XA} \sim \chi^2(n-1). \tag{38}
+
+[p. 17 | section: E.3. Putting it all together | type: Text]
+Insert the sampled values to calculate
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_9 = \frac{S_X S_A}{n} + \frac{1}{2} (U_{XA} - V_{XA}) \tag{39}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_8 = \frac{S_X S_Y}{n} + \frac{1}{2} (U_{XY} - V_{XY}) \tag{40}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_7 = Z_{XX} - \frac{S_X^2}{n} \tag{41}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_6 = n\alpha_{X\mu}\alpha_{Y\mu} + \alpha_{X\mu}\alpha_{Y\sigma}S_Y + \alpha_{Y\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{Y\sigma}S_8 \tag{42}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_5 = N_E \left( n\alpha_{X\mu}\alpha_{A\mu} + \alpha_{X\mu}\alpha_{A\sigma}S_A + \alpha_{A\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{A\sigma}S_9 \right) \tag{43}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_4 = an\alpha_{X\mu} + a\alpha_{X\sigma}S_X \tag{44}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_3 = \alpha_{X\sigma}^2 S_7 \tag{45}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_2 = 2\alpha_{X\mu}\alpha_{X\sigma}S_X \tag{46}
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+S_1 = n\alpha_{X\mu}^2. (47)
+
+[p. 17 | section: E.3. Putting it all together | type: Text]
+With these, we can finally generate one observation from the distribution of ϕ:
+
+[p. 17 | section: E.3. Putting it all together | type: Equation]
+\phi = b + dN_E + \frac{S_4 + S_5 + S_6}{S_1 + S_2 + S_3}. (48)

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icml26/a3c6aa1c-cfec-4174-aab8-a4cb5af0d892/main_body_chunks.jsonl

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+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0001", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "Background. Prediction models are often used to support decision making: a model trained on past data produces a forecast for a future outcome of interest, which a agent can take into account when deciding how to act. For the rest of this article, we will refer to such decision support as ML-DS (Machine Learning Decision Support) to emphasize the fact that the model has been trained on data. This is sometimes also called 'predictive optimization' (Wang et al., 2024) . Contrary to full automation, in high-risk settings ML-DS is sometimes perceived as a way to leverage the power of machine learning while retaining human oversight (see for instance Article 14 of the EU AI Act (European Parliament & Council of the European Union, 2024) ).", "source": "marker_v2", "marker_block_id": "/page/0/Text/6"}
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+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0003", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "In this paper, we present a full-fledged computational models of ML-DS, the 2-Step Agent . While most research on effects of ML-DS focuses on the model, we unpack the inner working of the decision maker, studying how a rational agent incorporates information from ML-DS in order to make a decision. We demonstrate that the introduction of ML-DS can lead to harmful consequences due to a mismatch of the model with the users' expectations or training, even in an ideal scenario where the ML model is as good as it gets and the agent is a 'perfect' reasoner.", "source": "marker_v2", "marker_block_id": "/page/0/Text/10"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0004", "section": "Motivating example 1: Effect of the agent's prior beliefs.", "page_start": 1, "page_end": 1, "type": "Text", "text": "Consider the situation of an oncologist using ML-DS when deciding how to treat a cancer patient. The ML-model predicts that the patient has only two months left to live. We can then imagine two lines of thinking depending on the clinician's prior beliefs. If the model was trained on patients who received minimal treatment, a very heavy treatment may extend the survival time. On the other hand, if the model is trained on patients who received the heaviest treatment, it is best to put the patient on palliative care to avoid unnecessary suffering from side-effects.", "source": "marker_v2", "marker_block_id": "/page/0/Text/12"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0005", "section": "Motivating example 1: Effect of the agent's prior beliefs.", "page_start": 1, "page_end": 1, "type": "Text", "text": "This example showcases that, even in presence of the same information from ML-DS, different prior assumptions about the model and the data it was trained on can lead to opposite actions . This underscores the importance of an agent's priors in determining the effectiveness of ML-DS, 1 a point that has already been argued for in previous literature (see e.g., (van Geloven et al., 2025) ) but has not been studied", "source": "marker_v2", "marker_block_id": "/page/0/Text/13"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0006", "section": "Motivating example 1: Effect of the agent's prior beliefs.", "page_start": 1, "page_end": 1, "type": "Footnote", "text": "1 Setting the agent prior beliefs by proper training is indeed an important part of RCTs' protocols that is not always adhered to (Perepletchikova et al., 2007) . When it is not possible to create a common baseline of beliefs and behavior, a two-stage randomization might be required, both on the providers of the intervention and the subjects (Borm et al., 2005) .", "source": "marker_v2", "marker_block_id": "/page/0/Footnote/14"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0007", "section": "Motivating example 1: Effect of the agent's prior beliefs.", "page_start": 2, "page_end": 2, "type": "Text", "text": "quantitatively.", "source": "marker_v2", "marker_block_id": "/page/1/Text/1"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0008", "section": "Motivating example 2: Agent learning from ML-DS.", "page_start": 2, "page_end": 2, "type": "Text", "text": "Consider a variation of the previous scenario involving a clinician and a patient with a similar prognosis. Before seeing the model's prediction, the clinician believes that in the training data of the model (1) patients were always treated, and in general (2) patients ended up surviving between 6 and 12 months when treated. Seeing a prediction of two months of survival, and assuming the patient is not an outlier, this set of beliefs is challenged.", "source": "marker_v2", "marker_block_id": "/page/1/Text/3"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0009", "section": "Motivating example 2: Agent learning from ML-DS.", "page_start": 2, "page_end": 2, "type": "Text", "text": "The clinician can then revise their beliefs in at least two different ways. They can conclude the model's training data had in fact some untreated patients, and that this patient might be similar to one of the untreated patients. In this case, it might still be beneficial to treat. On the other hand, they might conclude they were wrong about the effect of treatment on patients. In this case, the treatment must be less beneficial than previously thought by the clinician. Therefore it might be best to abstain from treatment.", "source": "marker_v2", "marker_block_id": "/page/1/Text/4"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0010", "section": "Motivating example 2: Agent learning from ML-DS.", "page_start": 2, "page_end": 2, "type": "Text", "text": "Hence the way in which ML-DS changes the agent's be liefs will influence the action of the agent . Capturing this phenomenon means solving a challenging leaning problem where the ML prediction is used as proxy information about the causal process the agent wants to intervene on.", "source": "marker_v2", "marker_block_id": "/page/1/Text/5"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0011", "section": "Motivating example 2: Agent learning from ML-DS.", "page_start": 2, "page_end": 2, "type": "Text", "text": "Contributions of the paper These examples show the importance of an agent's prior knowledge for the impact of ML-DS. In this paper, we introduce a general computational framework to study the interaction of an agent with an ML-DS system.", "source": "marker_v2", "marker_block_id": "/page/1/Text/6"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0012", "section": "Motivating example 2: Agent learning from ML-DS.", "page_start": 2, "page_end": 2, "type": "Text", "text": "We make three main contributions. First, we provide a rigorous formalism to analyze how a user interacts with ML-DS system in order to act. Our framework captures said interaction in two steps (2-Step Agent): a Bayesian step in which the agent incorporates the information given by the decision support model into their model of the world, and a causal inference step in which the agent draws the conclusion on which action is best to take. This is schematically depicted in Fig 1 for the clinical example.", "source": "marker_v2", "marker_block_id": "/page/1/Text/7"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0013", "section": "Motivating example 2: Agent learning from ML-DS.", "page_start": 2, "page_end": 2, "type": "Text", "text": "Our second contribution is to run computational experiments to quantify the effect of using ML-DS on patient outcomes. We show that harmful effects can occur in realistic scenarios already when a single prior is misaligned. We also show that ML-DS can be beneficial in some settings.", "source": "marker_v2", "marker_block_id": "/page/1/Text/8"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0014", "section": "Motivating example 2: Agent learning from ML-DS.", "page_start": 2, "page_end": 2, "type": "Text", "text": "Finally, to the best of our knowledge we provide the first treatment of how a Bayesian agent can update their beliefs about a population after observing a prediction of an ML model trained on data from that population. This update involves a challenging inference through a set of latent interchangeable variables in the predictive model's training", "source": "marker_v2", "marker_block_id": "/page/1/Text/9"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0015", "section": "Motivating example 2: Agent learning from ML-DS.", "page_start": 2, "page_end": 2, "type": "FigureGroup", "text": "Figure 1. The diagram shows the overall flow of the model. A new patient with certain attributes (1) arrives for treatment, and the patient's data (2), potentially along with the output of a prediction model (2'), is observed by the agent, e.g., a doctor. The agent uses these observations to update their previous beliefs about the population and the predictive model (3), a step which we model as a Bayesian update. Based on their new beliefs, the agent estimates the effect of treatment (4) and makes a treatment decision (5). Finally, the treatment together with the patient's attributes determine a final outcome (6).", "source": "marker_v2", "marker_block_id": "/page/1/FigureGroup/453"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0016", "section": "Motivating example 2: Agent learning from ML-DS.", "page_start": 2, "page_end": 2, "type": "Text", "text": "data. We prove that for simple linear models it is possible to reduce this set to a few sufficient statistics, improving computational cost and numerical stability when applying our framework.", "source": "marker_v2", "marker_block_id": "/page/1/Text/12"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0017", "section": "Motivating example 2: Agent learning from ML-DS.", "page_start": 2, "page_end": 2, "type": "Text", "text": "While our running example is medical, the results are general and apply for every other field where prediction models are employed as decision support. Furthermore, our simulations are only concerned with simple (non-causal) prediction models; ML-DS with causal model is left for future work. The code and experiments are available open-source. 2", "source": "marker_v2", "marker_block_id": "/page/1/Text/13"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0018", "section": "2. The 2-Step Agent: Defining the framework", "page_start": 2, "page_end": 2, "type": "Text", "text": "In this section, we give a formal description of our 2-Step Agent for studying the effects of ML-DS. We first give a description of the process that generates the training data for the predictive model, as well as the possible types of predictive model (Sec. 2.1) . We formalize the first step, which consists of a Bayesian agent updating their beliefs about the world given a prediction from the predictive model for a new patient (Sec. 2.2) . We then formalize the second step, where the Bayesian agent uses the updated beliefs about the population to decide on a treatment for the new patient (Sec. 2.3) .", "source": "marker_v2", "marker_block_id": "/page/1/Text/15"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0019", "section": "2.1. The setup: Historical data and predictive model", "page_start": 2, "page_end": 2, "type": "Text", "text": "We start by formally defining a causal model of the domain. Let A be a variable denoting an intervention, such as the administration of a treatment. The treatment variable may", "source": "marker_v2", "marker_block_id": "/page/1/Text/17"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0020", "section": "2.1. The setup: Historical data and predictive model", "page_start": 2, "page_end": 2, "type": "Footnote", "text": "2 The code will be shared upon acceptance.", "source": "marker_v2", "marker_block_id": "/page/1/Footnote/18"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0021", "section": "2.1. The setup: Historical data and predictive model", "page_start": 3, "page_end": 3, "type": "Text", "text": "be continuous or categorical, but to lighten the notation we will present the formulas with a binary A. Let Y be a variable denoting an outcome of interest. In our example above, the treatment could be a fixed dose of chemotherapy and the outcome the months of survival, so in this case a higher outcome is better. We use X_i to refer to covariates involved in the causal mechanisms governing the treatment and outcome.", "source": "marker_v2", "marker_block_id": "/page/2/Text/1"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0022", "section": "2.1. The setup: Historical data and predictive model", "page_start": 3, "page_end": 3, "type": "Text", "text": "Definition 2.1 (Historical data generation). We assume Hist = (\\mathbf{S}, P_{\\mathbf{N}}) to be the minimal structural causal model (SCM) describing the causal mechanisms producing A and Y, where \\mathbf{S} is a set of assignments defining each variable in terms of its parents and P_{\\mathbf{N}} is a joint distribution over the noise variables. We use \\bar{X} = X_1, \\ldots, X_k to denote endogenous (non-noise) variables involved in the SCM that are different from A and Y.", "source": "marker_v2", "marker_block_id": "/page/2/Text/3"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0023", "section": "2.1. The setup: Historical data and predictive model", "page_start": 3, "page_end": 3, "type": "Text", "text": "An example of this SCM is depicted in Figure 2(b). The SCM Hist stands for the process in the world that generated the data before the introduction of decision support. Note that the assignment determining A might be due to a specific intervention protocol, e.g., the guidelines for administering the treatment that were in use in that particular time window. The SCM determines a unique joint distribution P^{Hist} over variables Y, A, \\bar{X} .", "source": "marker_v2", "marker_block_id": "/page/2/Text/5"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0024", "section": "2.1. The setup: Historical data and predictive model", "page_start": 3, "page_end": 3, "type": "Text", "text": "Next, we assume that a relevant dataset is generated by Hist and used to train a prediction model for decision support.", "source": "marker_v2", "marker_block_id": "/page/2/Text/7"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0025", "section": "2.1. The setup: Historical data and predictive model", "page_start": 3, "page_end": 3, "type": "Text", "text": "Definition 2.2 (Training data and prediction model). A dataset \\mathcal{D}_{Hist} = \\{(\\bar{x}_1, a_1, y_1), \\dots, (\\bar{x}_n, a_n, y_n)\\} of historical data is composed of tuples containing relevant characteristics, treatment, and outcome. The dataset contains n instances. This dataset is used to train a prediction model f: \\bar{X} \\to Y to estimate E^{Hist}(Y \\mid \\bar{X}) .", "source": "marker_v2", "marker_block_id": "/page/2/Text/9"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0026", "section": "2.1. The setup: Historical data and predictive model", "page_start": 3, "page_end": 3, "type": "Text", "text": "Other estimands we do not consider are the treatment dependent estimate E^{Hist}(Y\\mid A,\\bar{X}) and the interventional estimate E^{Hist}(Y\\mid do(A),\\bar{X}) ; these are left for future work.", "source": "marker_v2", "marker_block_id": "/page/2/Text/11"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0027", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 3, "page_end": 3, "type": "Text", "text": "We next introduce an agent \\mathcal{A} who decides on the intervention for new observations (e.g. new patients) with the help of ML-DS. Rather than mechanically applying the recommendation of the ML-DS system, this agent uses their own internal model of the world, capturing not only what they believe about the data generating process Hist, but also what they believe concerning the prediction model f and the training data \\mathcal{D}_{Hist} . We model an agent who is aware of the model documentation containing the model class (e.g., a linear model, a tree ensemble, etc.), the model's signature (i.e. that it is treatment naive and what covariates are", "source": "marker_v2", "marker_block_id": "/page/2/Text/15"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0028", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 3, "page_end": 3, "type": "Text", "text": "involved),<sup>3</sup> and the size of the training data, but who is uncertain about the model's parameters. The agent can update their world model, including beliefs concerning historical data and model training.", "source": "marker_v2", "marker_block_id": "/page/2/Text/16"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0029", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 3, "page_end": 3, "type": "Text", "text": "Definition 2.3 (Internal beliefs of the agent). Call A_{hist} the Bayesian network defining the agent's joint distribution over the following variables:", "source": "marker_v2", "marker_block_id": "/page/2/Text/17"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0030", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 3, "page_end": 3, "type": "ListGroup", "text": "1. N_{\\bar{X}}, N_Y, N_A : These are the variables encoding the population-level distributions over the various quantities of interest, respectively: the patient's characteristics (N_{\\bar{X}}) , the noise in the outcome (N_Y) , and the noise in the treatment (N_A) . Depending on how population-level uncertainty is parameterized, these can be vectors. These population-level variables are sampled to produce the historical data. Within this Bayesian network, the historical data is represented via a plate, a graphical notation device used to show repeated, independent, and identically-structured components of a probabilistic model. N<sub>E</sub>: This variable denotes the treatment effect. Contrary to the other noise variables, this is not sampled for each plate repetition but only once, so it remains the same across the plate. 3. A_i, Y_i, \\bar{X}_i : These represent the i-th (out of n) datapoint in the training data of the predictive model. The whole training set is represented as a plate of multiplicity n. The noise distributions corresponding to these variables are fixed and outside the plate, capturing the fact that they are population-level distributions. All variables Y_i share the noise term N_E . 4. M: The prediction model's parameters, obtained deterministically from the training data in the plate and possibly some additional known parameter N<sub>M</sub> which encodes, e.g., the hyperparameter settings or initialization values. M has as parents all the variables in the plate, representing the fact that it is trained on the whole data set. 5. \\bar{X}^o : An observed data instance for a new patient, for which the agent will make a treatment decision. This array of variables has no non-noise parents and it is sampled from the same noise distribution as \\bar{X} .", "source": "marker_v2", "marker_block_id": "/page/2/ListGroup/421"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0031", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 3, "page_end": 3, "type": "Footnote", "text": "& lt;sup>3</sup>Recent guidelines explicitly suggest to additionally inform model users about the historical treatment policy. Our framework can be extended to model such settings, but here we focus on agents who are uncertain about historical treatment policy, for two reasons. First, this is a common scenario in current practice. Second, this provides a baseline estimation for the effects of ML-DS, which can be compared to better informed users in the future. We come back to these points in the discussion.", "source": "marker_v2", "marker_block_id": "/page/2/Footnote/23"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0032", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "FigureGroup", "text": "the SCM generating the data in the real world, while models (a) and (c) are the first and second steps of the 2-Step Agent. Model (a) allows the agent to perform a belief update given new information from ML-DS, while model (c) allows the agent to estimate the", "source": "marker_v2", "marker_block_id": "/page/3/FigureGroup/324"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0033", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "Text", "text": "model for \\bar{X}^o . This is derived deterministically from \\bar{X}^o and M.", "source": "marker_v2", "marker_block_id": "/page/3/Text/3"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0034", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "Text", "text": "The Bayesian network encoding the conditional independencies of the model is shown in Figure 2 (a). To simplify presentation we depict the submodel of A_{hist} containing only the variables A, Y and X as coinciding with Hist, meaning that the agent has correct belief about the data generation process, although this does not have to be the case and the framework is general enough to allow these two structures to diverge.", "source": "marker_v2", "marker_block_id": "/page/3/Text/4"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0035", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "Text", "text": "N_E encodes the agent's beliefs concerning the effect of the intervention: if the agent were to fix all the coefficients in the structural equation of Y, the effect of treatment would be fixed and there could be no update of his/her beliefs on the interventional distribution E^{A_{hist}}(Y \\mid \\bar{X}, do(A = 1)) . By letting the mechanism be influenced by N_E , we can treat the posterior as encoding a distribution over SCMs, each with its own fixed assignment for Y (and thus each with its own treatment effect).", "source": "marker_v2", "marker_block_id": "/page/3/Text/5"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0036", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "Text", "text": "The agent's initial beliefs are represented via priors.", "source": "marker_v2", "marker_block_id": "/page/3/Text/6"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0037", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "Text", "text": "Definition 2.4 (Agent's priors). The agent is assumed to", "source": "marker_v2", "marker_block_id": "/page/3/Text/7"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0038", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "Text", "text": "have as prior a joint distribution over all variables in the model, which depends on the unconditional distribution P_{N_{\\bar{X}},N_Y,N_A,N_E}^{\\mathcal{A}_{hist}}.", "source": "marker_v2", "marker_block_id": "/page/3/Text/8"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0039", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "Text", "text": "For all i < n, the priors encode the agent's starting beliefs about various quantities. First, P_{N_{\\bar{X}}}^{\\mathcal{A}_{hist}} encodes the population-level distribution generating \\bar{X}_{i,} the agent's model of the historical population. Second, P_{N_A}^{\\mathcal{A}_{hist}} encodes the population-level distribution generating the noise of A_i , the agent's model of the historical protocol to administer the intervention. Third, P_{N_Y}^{\\mathcal{A}_{hist}} encodes the population-level distribution generating the noise of Y_i , the agent's representation of the outcome. Finally, P_{N_E}^{A_{hist}} encodes the effect of treatment, assumed to be fixed across the population. We assume the agent knows the values of the parameters of the distribution N_M , since he/she has access to the model documentation, therefore we omit this prior.", "source": "marker_v2", "marker_block_id": "/page/3/Text/9"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0040", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "Text", "text": "This prior is updated via a Bayesian update based on the observables X^o and Pred^o .", "source": "marker_v2", "marker_block_id": "/page/3/Text/10"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0041", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "Text", "text": "Definition 2.5 (Agent belief update). Given \\bar{X}^o and Predo the agent does a Bayesian update on the prior P^{\\mathcal{A}_{hist}}(N_{\\bar{X}}, N_Y, N_A, N_E) , obtaining a posterior distribu-", "source": "marker_v2", "marker_block_id": "/page/3/Text/11"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0042", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "Equation", "text": "P^{\\mathcal{A}_{hist}}(N_{\\bar{X}}, N_Y, N_A, N_E \\mid \\bar{X}^o, Pred^o)", "source": "marker_v2", "marker_block_id": "/page/3/Equation/12"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0043", "section": "2.2. The first step: Bayesian update of the agent's beliefs", "page_start": 4, "page_end": 4, "type": "Text", "text": "To simplify exposition, we marginalize over uncertainty in the plate variables (\\bar{X}_i, Y_i, A_i \\text{ with } i < n) and M.", "source": "marker_v2", "marker_block_id": "/page/3/Text/13"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0044", "section": "2.3. The second step: deciding on the best action after belief update", "page_start": 4, "page_end": 4, "type": "Text", "text": "We described the agent' world model and how the agent updates it upon observing predictions from an ML-DS system for a new patient, which we called the first step. In the second step, the agent estimates the effect of the treatment on the new patient using their updated world model. This is necessary for the agent to consider a 'fresh' iteration of the plate and calculate the correct interventional probability. Another conceptual reason to introduce a second model is that - when the agent does Bayesian inference - the values of A and Y for the new instance \\bar{X}^o are not realized yet.", "source": "marker_v2", "marker_block_id": "/page/3/Text/15"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0045", "section": "2.3. The second step: deciding on the best action after belief update", "page_start": 4, "page_end": 4, "type": "Text", "text": "Definition 2.6 (Agent model for decision making). This model contains variables N_E, N_{\\bar{X}}, N_A, N_Y, X, A, Y , arranged as the similarly-named variables in the sub-model of A_{hist} . This model is used for inference on new instances and we denote it as A_{inf} .", "source": "marker_v2", "marker_block_id": "/page/3/Text/16"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0046", "section": "2.3. The second step: deciding on the best action after belief update", "page_start": 4, "page_end": 4, "type": "Text", "text": "This second agent model is depicted in Figure 2 (c). The crucial difference between this model and a regular SCM is that the noise variables are not assumed to be independent but rather they are initialized with the joint posterior obtained in the previous Bayesian step. The agent is thus drawing an inference over a distribution of possible SCMs.", "source": "marker_v2", "marker_block_id": "/page/3/Text/17"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0047", "section": "2.3. The second step: deciding on the best action after belief update", "page_start": 4, "page_end": 4, "type": "Text", "text": "To decide what to do on the new instance, the agent then", "source": "marker_v2", "marker_block_id": "/page/3/Text/18"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0048", "section": "2.3. The second step: deciding on the best action after belief update", "page_start": 5, "page_end": 5, "type": "Text", "text": "calculates the effect of the intervention on the model A_{inf} where the joint distribution of the noise is set to the posterior calculated over A_{hist} .", "source": "marker_v2", "marker_block_id": "/page/4/Text/2"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0049", "section": "2.3. The second step: deciding on the best action after belief update", "page_start": 5, "page_end": 5, "type": "Text", "text": "Definition 2.7 (Decision rule). To make a decision, the agent first sets the joint distribution of the inference model according to the posterior on the historical model:", "source": "marker_v2", "marker_block_id": "/page/4/Text/3"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0050", "section": "2.3. The second step: deciding on the best action after belief update", "page_start": 5, "page_end": 5, "type": "Equation", "text": "P^{\\mathcal{A}_{inf}}(N_{\\bar{X}}, N_Y, N_A, N_E) := P^{\\mathcal{A}_{hist}}(N_{\\bar{X}}, N_Y, N_A, N_E \\mid \\bar{X}^o, Pred^o)", "source": "marker_v2", "marker_block_id": "/page/4/Equation/4"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0051", "section": "2.3. The second step: deciding on the best action after belief update", "page_start": 5, "page_end": 5, "type": "Text", "text": "Following this, treatment is given iff the Conditional Average Treatment Effect (CATE) for the new patient crosses a threshold value, meaning", "source": "marker_v2", "marker_block_id": "/page/4/Text/5"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0052", "section": "2.3. The second step: deciding on the best action after belief update", "page_start": 5, "page_end": 5, "type": "Equation", "text": "E_{P^{\\mathcal{A}_{inf}}}[\\mathrm{CATE}] > \\tau", "source": "marker_v2", "marker_block_id": "/page/4/Equation/6"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0053", "section": "2.3. The second step: deciding on the best action after belief update", "page_start": 5, "page_end": 5, "type": "Text", "text": "where \\tau is a threshold set by the agent beforehand and CATE = E^{\\mathcal{A}_{inf}}(Y \\mid \\bar{X} = \\bar{X}^o, do(A=0)) - E^{\\mathcal{A}_{inf}}(Y \\mid \\bar{X} = \\bar{X}^o, do(A=1)) . In other words, treatment is given if the expected effect of treatment is sufficiently large in the desired direction.", "source": "marker_v2", "marker_block_id": "/page/4/Text/7"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0054", "section": "2.3. The second step: deciding on the best action after belief update", "page_start": 5, "page_end": 5, "type": "Text", "text": "When the threshold is zero, the agent administers the treatment if the expected outcome is better in the scenario where they are giving the treatment, conditioned on the patient characteristics, versus the scenario where treatment is not given. Crucially, these interventional expectations are calculated on the internal model of the agent.", "source": "marker_v2", "marker_block_id": "/page/4/Text/8"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0055", "section": "2.4. Quantifying the effect of introducing decision support", "page_start": 5, "page_end": 5, "type": "Text", "text": "Given the machinery defined above, we are able to calculate the effect of introducing decision support. Given the prediction model and the specification of the agent - which is determined by spelling out \\mathcal{A}_{hist} and \\mathcal{A}_{inf} - the decision of the agent on a new data point (as described by covariates \\bar{X}^o ) is deterministic (i.e. we can implement the input-output process described in Figure 1). We can thus formalize it as a function Act^{\\mathcal{A},f}: \\bar{X} \\to A . The way the decision support changes the actions of the agent is then represented as an intervention on Hist changing the assignment of A:", "source": "marker_v2", "marker_block_id": "/page/4/Text/10"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0056", "section": "2.4. Quantifying the effect of introducing decision support", "page_start": 5, "page_end": 5, "type": "Equation", "text": "A := Act^{\\mathcal{A}, f}(\\bar{X})", "source": "marker_v2", "marker_block_id": "/page/4/Equation/11"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0057", "section": "2.4. Quantifying the effect of introducing decision support", "page_start": 5, "page_end": 5, "type": "Text", "text": "The post-interventional SCM is then indicated with the notation Hist^{A=Act^{\\mathcal{A},f}(\\bar{X})} , where f is the prediction model from Definition 2.2. We can now define the effect of introducing decision support as the risk difference in outcomes generated by Hist compared to Hist^{A=Act^{\\mathcal{A},f}(\\bar{X})} . This quantifies the effect of the intervention 'using MS-DL', compared to the historical policy.", "source": "marker_v2", "marker_block_id": "/page/4/Text/12"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0058", "section": "3. Experimental setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "We design experiments to answer two questions. First, how and how much does decision support influence (i) treatment", "source": "marker_v2", "marker_block_id": "/page/4/Text/14"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0059", "section": "3. Experimental setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "decisions, (ii) agent's beliefs, and (iii) downstream patient outcomes? Second, what role is played by the agent's priors? To address these questions, we vary the priors of the agent and compare the scenarios with and without ML-DS. For each prior setting, we plot the change in treatment decisions, in agent's beliefs, and in patient outcomes, with and without ML-DS, after observing a single patient.", "source": "marker_v2", "marker_block_id": "/page/4/Text/15"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0060", "section": "3. Experimental setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "Use case We consider a data-generating model with one continuous covariate X, continuous intervention A, and a continuous outcome Y. Higher values of the outcomes are better. One can think of the initial example and interpret X as body weight, A as dosage of chemotherapy, and Y as months of survival. We set n=1000 and we assume the prediction model to be a slope-only linear regression. Despite the continuous treatment, for simplicity we assume the agent compares two treatment options, A=10 or A=20, intuitively ten and twenty units of dosage respectively.", "source": "marker_v2", "marker_block_id": "/page/4/Text/16"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0061", "section": "3. Experimental setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "Historical SCM. Let the SCM Hist be defined as:", "source": "marker_v2", "marker_block_id": "/page/4/Text/17"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0062", "section": "3. Experimental setup", "page_start": 5, "page_end": 5, "type": "Equation", "text": "N_X \\sim \\mathcal{N}(80, 10) N_A \\sim \\mathcal{N}(2, 1) N_Y \\sim \\mathcal{N}(0, 0.1) X := N_X A := 0.125 * X + N_A Y := 12 - 0.1 * X + 1 * A + N_Y", "source": "marker_v2", "marker_block_id": "/page/4/Equation/18"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0063", "section": "3. Experimental setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "In this model, the treatment protocol is simply to give a dosage of treatment A which is an eight of the body weight X, and how well this dosing is followed is encoded by the noise term N_A . The mechanism of Y clarifies that the covariate X makes the outcome worse but the treatment improves the outcome. Of course, this equation is a great simplification of what happens in reality, but it is sufficient to demonstrate several important dynamics that emerge when introducing ML-DS. Average treatment effect is positive in this scenario, as witnessed by the positive coefficient of A in the equation for Y.", "source": "marker_v2", "marker_block_id": "/page/4/Text/19"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0064", "section": "3. Experimental setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "The internal model of the agent's beliefs. The internal model of the agent A_{hist} encodes the agent's uncertainty about Hist :", "source": "marker_v2", "marker_block_id": "/page/4/Text/20"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0065", "section": "3. Experimental setup", "page_start": 5, "page_end": 5, "type": "Equation", "text": "X_{i} \\sim \\mathcal{N}(\\boldsymbol{\\alpha}_{\\boldsymbol{X}\\boldsymbol{\\mu}}, \\boldsymbol{\\alpha}_{\\boldsymbol{X}\\boldsymbol{\\sigma}}) \\qquad \\text{with } i < n N_{A_{i}} \\sim \\mathcal{N}(\\boldsymbol{\\alpha}_{\\boldsymbol{A}\\boldsymbol{\\mu}}, \\boldsymbol{\\alpha}_{\\boldsymbol{A}\\boldsymbol{\\sigma}}) \\qquad \\text{with } i < n N_{Y_{i}} \\sim \\mathcal{N}(\\boldsymbol{\\alpha}_{\\boldsymbol{Y}\\boldsymbol{\\mu}}, \\boldsymbol{\\alpha}_{\\boldsymbol{Y}\\boldsymbol{\\sigma}}) \\qquad \\text{with } i < n A_{i} = 0.125 X_{i} + N_{A_{i}} \\qquad \\text{with } i < n Y_{i} = 12 - 0.1 X_{i} + N_{\\boldsymbol{E}} A_{i} + N_{Y_{i}} \\qquad \\text{with } i < n M = \\arg\\min_{\\boldsymbol{\\phi} \\in \\Phi} (MSE(\\boldsymbol{\\phi}\\bar{X}, \\bar{Y})) X^{o} \\sim \\mathcal{N}(\\boldsymbol{\\alpha}_{\\boldsymbol{X}\\boldsymbol{\\mu}}, \\boldsymbol{\\alpha}_{\\boldsymbol{X}\\boldsymbol{\\sigma}}) Pred^{o} = MX^{o}", "source": "marker_v2", "marker_block_id": "/page/4/Equation/21"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0066", "section": "3. Experimental setup", "page_start": 5, "page_end": 5, "type": "Text", "text": "where \\bar{Y} = (Y_1, \\dots, Y_n) , etc are the respective vectors of n variables, \\phi is the single parameter of a least squares regression without intercept, and the variables in bold font indicate", "source": "marker_v2", "marker_block_id": "/page/4/Text/22"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0067", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Text", "text": "population-level priors that are given different values in the different simulations below.<sup>4</sup>", "source": "marker_v2", "marker_block_id": "/page/5/Text/1"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0068", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Text", "text": "The agent knows some aspects of the data generating process, e.g., some of the parameters in the mechanisms of Y and A, but has uncertainty on several dimensions. \\alpha_{A\\mu} encodes the belief of the agent on how much historical dosages of treatment deviated from the standard protocol. \\alpha_{X\\mu}, \\alpha_{X\\sigma} represent the uncertainty of the agent on how the feature X of the training data was distributed. In the definition of Y there are two uncertainties at play. The prior on \\alpha_{Y\\sigma} describes the uncertainty of the agent concerning Y. The distribution of N_E , on the other hand, captures the belief of the agent concerning the effectiveness of treatment A.", "source": "marker_v2", "marker_block_id": "/page/5/Text/2"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0069", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Text", "text": "Defining different agents We consider two types of agents, one with 'correct' priors whose means are close to the historical SCM, and one with incorrect priors. This is an example of the former:", "source": "marker_v2", "marker_block_id": "/page/5/Text/3"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0070", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Equation", "text": "\\begin{aligned} &\\alpha_{X\\mu} \\sim \\mathcal{N}(80, 0.1) & \\alpha_{X\\sigma} \\sim \\mathcal{N}(10, 0.1) \\mid \\alpha_{X\\sigma} \\geq 0 \\\\ &\\alpha_{A\\mu} \\sim \\mathcal{N}(2, 1) & \\alpha_{A\\sigma} \\sim \\mathcal{N}(1, 0.1) \\mid \\alpha_{Y\\sigma} \\geq 0 \\\\ &\\alpha_{Y\\mu} \\sim \\mathcal{N}(0, 0.01) & \\alpha_{Y\\sigma} \\sim \\mathcal{N}(0.1, 0.01) \\mid \\alpha_{Y\\sigma} \\geq 0 \\\\ &N_E \\sim \\mathcal{N}(1, 0.1) \\end{aligned}", "source": "marker_v2", "marker_block_id": "/page/5/Equation/4"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0071", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Text", "text": "This agent essentially has the correct beliefs about the patient data distribution, historical treatment policy, outcome variance and treatment effect. They only have some uncertainty represented by some variance on these parameters. In our experiments, we vary each of the colored prior parameters, one at a time, to simulate various ways in which the agent's expectations may be incorrect.", "source": "marker_v2", "marker_block_id": "/page/5/Text/5"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0072", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Text", "text": "The internal model for the agent's effect estimation. Finally, the internal model of the agent used for inference, A_{inf} , is defined as:", "source": "marker_v2", "marker_block_id": "/page/5/Text/6"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0073", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Equation", "text": "\\begin{split} N_X &\\sim \\mathcal{N}(\\alpha_{X\\mu}, \\alpha_{X\\sigma}) \\quad N_A \\sim \\mathcal{N}(\\alpha_{A\\mu}, \\alpha_{A\\sigma}) \\\\ N_Y &\\sim \\mathcal{N}(\\alpha_{Y\\mu}, \\alpha_{Y\\sigma}) \\\\ X &:= N_X \\\\ A &:= 0.125 \\ X + N_A \\\\ Y &:= 12 - 0.1 \\ X + N_E \\ A + N_Y \\end{split}", "source": "marker_v2", "marker_block_id": "/page/5/Equation/7"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0074", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Text", "text": "The posterior defines a probability distribution over SCMs. Recall that the agent was interested in choosing between dosage A=10 or A=20 for the specific patient with", "source": "marker_v2", "marker_block_id": "/page/5/Text/8"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0075", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Text", "text": "weight equal to X^o . To compute the desired quantity, the agent aggregates the CATE produced by every SCM according to the posterior", "source": "marker_v2", "marker_block_id": "/page/5/Text/10"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0076", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Equation", "text": "C\\hat{ATE} = \\mathbb{E}_{P^{\\mathcal{A}_{inf}}}[E^{\\mathcal{A}_{inf}}(Y|X^o, do(A=20)) - E^{\\mathcal{A}_{inf}}(Y|X^o, do(A=10))]", "source": "marker_v2", "marker_block_id": "/page/5/Equation/11"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0077", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Text", "text": "where the expectation is over the joint posterior of the agent over the parameters, which was obtained updating A_{hist} .", "source": "marker_v2", "marker_block_id": "/page/5/Text/12"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0078", "section": "3. Experimental setup", "page_start": 6, "page_end": 6, "type": "Text", "text": "Reaching a decision. The agent will administer a dosage of 10 if the estimated CATE is below a threshold of 5 or a dosage of 20 otherwise. In other words, the higher dosage needs to improve the outcome by at least 5 months to be worth it. This completes the decision pipeline: given a patient X^o with a corresponding prediction and the type of agent, we can compute what the agent would do.", "source": "marker_v2", "marker_block_id": "/page/5/Text/13"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0079", "section": "4. Results", "page_start": 6, "page_end": 6, "type": "Text", "text": "We begin by showcasing how our framework can capture the two motivating examples presented in the Introduction. In the first example, the belief of the agent on the past policy determines how the prediction is interpreted and what action is taken.<sup>5</sup> Indeed, in the simulation, when X = 80, with a bad prognosis under the assumption of no treatment (\\alpha_{A\\mu} = 0) , the agent chooses a high dosage to improve the outcome (A = 20). With a bad prognosis under the assumption of heavy treatment ( \\alpha_{A\\mu}=20 ), the agent chooses a low dosage to reduce over-treatment (A = 10). In the second motivating example, new information from the prediction forces an agent to consider different options. In the model, this means that either belief in treatment effect (N_E) or in historical policy (\\mu_A) is revised, which appears as a correlation in the posterior belief in these two variables in Figure 4 in Appendix A.", "source": "marker_v2", "marker_block_id": "/page/5/Text/15"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0080", "section": "4. Results", "page_start": 6, "page_end": 6, "type": "Text", "text": "Next, for the use case described in Section 3, we report the effect of using ML-DS on two dimensions: 1) agent's belief of treatment effect and 2) downstream outcomes Y realized after the agent has taken an action, visualized in the top and bottom rows of Figure 3, respectively. In each of these plots, the outcomes obtained using ML-DS (orange boxplots) is contrasted with the outcomes obtained without it (blue boxplots), for different types of agents. The CATE plots sweep a wide range of options for the priors in order", "source": "marker_v2", "marker_block_id": "/page/5/Text/16"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0081", "section": "4. Results", "page_start": 6, "page_end": 6, "type": "Footnote", "text": "& lt;sup>4</sup>Note that for ease of exposition we introduced some variables (N_{A_i}, N_{Y_i}) which are not present in Figure 2 (a). These can in principle be removed by specifying distributions directly on the inverse transformations of the ones given above for A_i and Y_i , and applying the appropriate Jacobian adjustment. In the PyMC implementation, we also use a small (1e-3) observational noise around Pred^o to allow us to sample from an observed deterministic transformation.", "source": "marker_v2", "marker_block_id": "/page/5/Footnote/9"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0082", "section": "4. Results", "page_start": 6, "page_end": 6, "type": "Footnote", "text": "& lt;sup>5</sup>All simulations presented in this paper are ran with PyMC v5 (Salvatier et al., 2016). We used the NUTS sampler to get 1000 samples for each of 4 chains, with 1000 tuning samples and a 'target_accept' parameter of 0.85. Diagnostics and visual inspection of the traces showed good mixing and no convergence issues (Max \\hat{R}=1.01 , and only 16 out 4000 runs had > 5 divergences, due to fixed 'target_accept' and extreme parameter values.). Sanity checks on the Bayesian model are reported in Appendix B.", "source": "marker_v2", "marker_block_id": "/page/5/Footnote/17"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0083", "section": "4. Results", "page_start": 7, "page_end": 7, "type": "Text", "text": "to reveal interesting dynamics of the agent's beliefs; in Appendix D we also report the CATE results for the more realistic range of ± 3 stds from the SCM reference value.", "source": "marker_v2", "marker_block_id": "/page/6/Text/1"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0084", "section": "4. Results", "page_start": 7, "page_end": 7, "type": "Text", "text": "We begin with an agent whose priors beliefs coincide with the parameters of the historical SCM except for NE, i.e. treatment effect. Figure 3a and Figure 3d show what happens to CATE and Y when the agent's belief in N E varies. As expected, in the first figure we can see that the blue CATE has a slope that depends on the prior NE: without decision support, the belief on CATE is equal to N E multiplied by the difference in treatments. In the orange series however, we see an interesting behavior: after learning from the interaction with ML-DS, the agent corrects their belief in CATE re-aligning it to the correct value. In other words, when all other beliefs are correct, the agent can learn from ML-DS to estimate better CATE and take better actions. This is confirmed in Figure 3d where the orange series shows superior outcomes.", "source": "marker_v2", "marker_block_id": "/page/6/Text/2"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0085", "section": "4. Results", "page_start": 7, "page_end": 7, "type": "Text", "text": "Figure 3b and Figure 3e show what happens when the agent's prior belief in the treatment protocol, µA, varies. In this scenario the prior CATE estimates are flat, for the simple reason that the agent has a fixed belief in N E and they are not updating it. When interacting with ML-DS however, the belief update changes CATE estimates quite a lot; this is because the agent is attempting to explain the gap between P red o given by ML-DS and the expected outcome they had from their prior beliefs. This reflects on outcomes in 3e where the orange series is sometimes below the blue one: incorrect beliefs in µ A can result in CATE estimates that lead to wrong actions and worsen outcome Y.", "source": "marker_v2", "marker_block_id": "/page/6/Text/3"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0086", "section": "4. Results", "page_start": 7, "page_end": 7, "type": "Text", "text": "A similar pattern is observed in Figures 3c, 3f, 6c (last one in Appendix D) : while prior CATE and Y estimates are relatively stable without ML-DS, new dynamics emerge when when ML-DS is introduced. For CATE, incorrect beliefs can have large effects both on the magnitude and sign. For Y , in both cases incorrect beliefs can lead to bad actions and worse outcomes.", "source": "marker_v2", "marker_block_id": "/page/6/Text/4"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0087", "section": "5. Related work", "page_start": 7, "page_end": 7, "type": "Text", "text": "A large body of literature is concerned with the analysis of ML-DS from different angles. A strand of literature focuses on how ML-DS is entangled with (changes in) the data generating process. Works in this category study limitations of what can be learned from data generated from past policies, e.g. (Lakkaraju et al., 2017) , or the dynamics that occur with alternating cycles of ML-DS deployment and re-training, as in the performative prediction literature (Perdomo et al., 2020; Boeken et al., 2024) . Another batch of articles is concerned with evaluation of ML-DS, laying out identifiability conditions for the effects of ML-DS on decisions (Imai et al., 2023) or trying to optimize human-machine complementar-", "source": "marker_v2", "marker_block_id": "/page/6/Text/6"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0088", "section": "5. Related work", "page_start": 7, "page_end": 7, "type": "Text", "text": "ity by design (Donahue et al., 2022; McLaughlin & Spiess, 2024) . Finally, a different set of papers investigate what human input can add to artificial agents already reaching some level of optimality (Alur et al., 2024; Zhang & Barein boim, 2022; Stensrud et al., 2024) . The key difference of our work is that we offer a more fine-grained description of what happens inside the decision maker, namely we unpack the micro-mechanism leading a decision maker to take an action on a single observation paired with AI prediction.", "source": "marker_v2", "marker_block_id": "/page/6/Text/7"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0089", "section": "5. Related work", "page_start": 7, "page_end": 7, "type": "Text", "text": "Other papers have studied how information flows between agents influencing their actions. In (Kamenica & Gentzkow, 2011) a Sender of information influences a Receiver Bayesian agent in order to stir their action. What we add here is that in our context the signal is an ML prediction produced from training data drawn from the same environment the agent reasons about; this coupling yields the distinctive inference problem we study, i.e. updating through latent interchangeable training variables.", "source": "marker_v2", "marker_block_id": "/page/6/Text/8"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0090", "section": "5. Related work", "page_start": 7, "page_end": 7, "type": "Text", "text": "Another important strand of work focuses on problems related to inferring information about a population P given predictions from a model trained on samples from P in the context of confidentiality attacks. This literature studies whether it is possible to use models' predictions (and other behavior) to recover the presence of a specific data point, in so-called membership inference attacks (Hu et al., 2022; Huang, 2025) ), or to infer statistical properties of a training datasets, so-called property-inference attacks (Ate niese et al., 2015; Wang et al., 2019) . While these models aim to recover information about the training set, e.g., the proportion of data in each label, our challenge is to marginalize across datasets and directly infer information about the underlying population.", "source": "marker_v2", "marker_block_id": "/page/6/Text/9"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0091", "section": "6. Discussion", "page_start": 7, "page_end": 7, "type": "Text", "text": "We have presented the first explicit framework that includes an ML-DS system and a sophisticated user. Our 2-Step Agent framework captures the effect of an agent's prior beliefs, modeling learning effects due to the interaction and tracking agent's uncertainty. The framework allows for simulations comparing outcomes with and without ML-DS, in an RCT-like fashion, therefore quantifying the benefit/harm of ML-DS for different types of simulated agents.", "source": "marker_v2", "marker_block_id": "/page/6/Text/11"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0092", "section": "6. Discussion", "page_start": 7, "page_end": 7, "type": "Text", "text": "We provided an implementation of the framework for ML-DS with a linear model. Said implementation leverages a collapse of the plate via sufficient statistics (Appendix E.1) , greatly speeding up the simulations. Within this implementation, we have shown that incorrect beliefs about the distribution of the predictive model's training data may lead the agent to bad decisions, resulting in harmful outcomes.", "source": "marker_v2", "marker_block_id": "/page/6/Text/12"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0093", "section": "6. Discussion", "page_start": 7, "page_end": 7, "type": "Text", "text": "While previous literature had argued that incorrect beliefs about past treatment policy (µ A in our notation) might gener-", "source": "marker_v2", "marker_block_id": "/page/6/Text/13"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0094", "section": "6. Discussion", "page_start": 8, "page_end": 8, "type": "FigureGroup", "text": "Figure 3. Figures displaying the effects of different agent priors when using ML-DS (in orange) and without it (in blue). In all plots except from NE, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameters. The top row displays the agent's prior and posterior (predictive) belief of CATE, while the bottom row shows the impact of the agent's decisions on the downstream outcome Y . The left column pertains to the belief on treatment effect, NE, the central column the belief about past treatment policy (µA) and the right column about the belief on past covariate distribution (µX). The gray areas represents the portions of the belief in which the true CATE is within one std from the CATE learned from ML-DS (orange band).", "source": "marker_v2", "marker_block_id": "/page/7/FigureGroup/767"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0095", "section": "6. Discussion", "page_start": 8, "page_end": 8, "type": "Text", "text": "ate worse downstream outcomes, e.g. in (van Geloven et al., 2025) , this is the first quantitative estimation of harmful effects of ML-DS. In our experiments we show that ML-DS can be harmful if the agent has wrong beliefs not only about the past treatment policy, but also about the distribution of covariates and the distribution of the outcome.", "source": "marker_v2", "marker_block_id": "/page/7/Text/3"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0096", "section": "6. Discussion", "page_start": 8, "page_end": 8, "type": "Text", "text": "We also show that if the agent has otherwise correct beliefs, an incorrect belief about treatment effect may be corrected during updates that push the agent towards the true CATE. This suggests that in certain settings ML-DS building on a treatment-naive prediction model might still be beneficial.", "source": "marker_v2", "marker_block_id": "/page/7/Text/4"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0097", "section": "6. Discussion", "page_start": 8, "page_end": 8, "type": "Text", "text": "This finding highlights the extreme sensitivity of ML-DS to agent prior beliefs when using a treatment-naive prediction model, and underscore the importance of providing full documentation and proper training to users of ML-DS. Finally, we remark that these results apply to any field deploying such systems and to any agent interacting with ML-DS, be it human or artificial.", "source": "marker_v2", "marker_block_id": "/page/7/Text/5"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0098", "section": "6. Discussion", "page_start": 8, "page_end": 8, "type": "Text", "text": "Limitations and Future work The experiments in this paper explored only part of the potential of the framework. The experiments are limited to a use case in which the prediction model is linear and treatment-naive, and the data", "source": "marker_v2", "marker_block_id": "/page/7/Text/6"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0099", "section": "6. Discussion", "page_start": 8, "page_end": 8, "type": "Text", "text": "generating process is simple (e.g., every distribution is normal, there is only one confounder). Implementing the use case was not trivial and required pushing a Bayesian update through OLS and a large amount of variables (see Appendix E.1) , hence it remains to be seen whether similar results can be obtained when the complexity of the use case increases.", "source": "marker_v2", "marker_block_id": "/page/7/Text/7"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0100", "section": "6. Discussion", "page_start": 8, "page_end": 8, "type": "Text", "text": "We assume the agent reasons in a Bayesian fashion, which allows for a precise analysis of the agent's uncertainty. While this agent may not be realistic with respect to human behavior, it is useful to make this assumption to demonstrate what can go wrong in a best-case scenario, i.e., when the agent is a 'perfect' reasoner and the ML model has no flaws. Finally, this assumption is also interesting when considering how an artificial agent might learn from predictions of other ML models.", "source": "marker_v2", "marker_block_id": "/page/7/Text/8"}
+{"paper_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892", "chunk_id": "a3c6aa1c-cfec-4174-aab8-a4cb5af0d892:0101", "section": "6. Discussion", "page_start": 8, "page_end": 8, "type": "Text", "text": "In future work this line of research will be expanded across different dimensions, including i) different types of agents (e.g. agent with incorrect internal model of the world of different kinds of uncertainties) ii) more complex ML models for ML-DS, iii) cases with heterogeneity of treatment effect (i.e. the effect of the action is not the same for all instances), iv) repeated rounds of learning for the same agent and v) using causal models for ML-DS.", "source": "marker_v2", "marker_block_id": "/page/7/Text/9"}

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icml26/a3c6aa1c-cfec-4174-aab8-a4cb5af0d892/model_text_v3.txt

@@ -0,0 +1,305 @@
+[p. 1 | section: Abstract | type: Text]
+Across a growing number of fields, human decision making is supported by predictions from AI models. However, we still lack a deep understanding of the effects of adoption of these technologies. In this paper, we introduce a general computational framework, the 2-Step Agent, which models the effects of AI-assisted decision making. Our framework uses Bayesian methods for causal inference to model 1) how a prediction on a new observation affects the beliefs of a rational Bayesian agent, and 2) how this change in beliefs affects the downstream decision and subsequent outcome. Using this framework, we show by simulations how a single misaligned prior belief can be sufficient for decision support to result in worse downstream outcomes compared to no decision support. Our results reveal several potential pitfalls of AI-driven decision support and highlight the need for thorough model documentation and proper user training.
+
+[p. 1 | section: 1. Introduction | type: Text]
+Background. Prediction models are often used to support decision making: a model trained on past data produces a forecast for a future outcome of interest, which a agent can take into account when deciding how to act. For the rest of this article, we will refer to such decision support as ML-DS (Machine Learning Decision Support) to emphasize the fact that the model has been trained on data. This is sometimes also called 'predictive optimization' (Wang et al., 2024) . Contrary to full automation, in high-risk settings ML-DS is sometimes perceived as a way to leverage the power of machine learning while retaining human oversight (see for instance Article 14 of the EU AI Act (European Parliament & Council of the European Union, 2024) ).
+
+[p. 1 | section: 1. Introduction | type: Text]
+Understanding the practical impact of adopting ML-DS in a domain is essential. First, ML-DS is on the rise, and currently deployed in a variety of fields including healthcare, education (Smith et al., 2012) , policing and judicial system (Kleinberg et al., 2018) , as well as redistribution of welfare (Kube et al., 2019) . Second, allowing machines to influence decision making carries important risks, e.g. the incorporation of biased advice (Obermeyer et al., 2019) .
+
+[p. 1 | section: 1. Introduction | type: Text]
+In this paper, we present a full-fledged computational models of ML-DS, the 2-Step Agent . While most research on effects of ML-DS focuses on the model, we unpack the inner working of the decision maker, studying how a rational agent incorporates information from ML-DS in order to make a decision. We demonstrate that the introduction of ML-DS can lead to harmful consequences due to a mismatch of the model with the users' expectations or training, even in an ideal scenario where the ML model is as good as it gets and the agent is a 'perfect' reasoner.
+
+[p. 1 | section: Motivating example 1: Effect of the agent's prior beliefs. | type: Text]
+Consider the situation of an oncologist using ML-DS when deciding how to treat a cancer patient. The ML-model predicts that the patient has only two months left to live. We can then imagine two lines of thinking depending on the clinician's prior beliefs. If the model was trained on patients who received minimal treatment, a very heavy treatment may extend the survival time. On the other hand, if the model is trained on patients who received the heaviest treatment, it is best to put the patient on palliative care to avoid unnecessary suffering from side-effects.
+
+[p. 1 | section: Motivating example 1: Effect of the agent's prior beliefs. | type: Text]
+This example showcases that, even in presence of the same information from ML-DS, different prior assumptions about the model and the data it was trained on can lead to opposite actions . This underscores the importance of an agent's priors in determining the effectiveness of ML-DS, 1 a point that has already been argued for in previous literature (see e.g., (van Geloven et al., 2025) ) but has not been studied
+
+[p. 1 | section: Motivating example 1: Effect of the agent's prior beliefs. | type: Footnote]
+1 Setting the agent prior beliefs by proper training is indeed an important part of RCTs' protocols that is not always adhered to (Perepletchikova et al., 2007) . When it is not possible to create a common baseline of beliefs and behavior, a two-stage randomization might be required, both on the providers of the intervention and the subjects (Borm et al., 2005) .
+
+[p. 2 | section: Motivating example 1: Effect of the agent's prior beliefs. | type: Text]
+quantitatively.
+
+[p. 2 | section: Motivating example 2: Agent learning from ML-DS. | type: Text]
+Consider a variation of the previous scenario involving a clinician and a patient with a similar prognosis. Before seeing the model's prediction, the clinician believes that in the training data of the model (1) patients were always treated, and in general (2) patients ended up surviving between 6 and 12 months when treated. Seeing a prediction of two months of survival, and assuming the patient is not an outlier, this set of beliefs is challenged.
+
+[p. 2 | section: Motivating example 2: Agent learning from ML-DS. | type: Text]
+The clinician can then revise their beliefs in at least two different ways. They can conclude the model's training data had in fact some untreated patients, and that this patient might be similar to one of the untreated patients. In this case, it might still be beneficial to treat. On the other hand, they might conclude they were wrong about the effect of treatment on patients. In this case, the treatment must be less beneficial than previously thought by the clinician. Therefore it might be best to abstain from treatment.
+
+[p. 2 | section: Motivating example 2: Agent learning from ML-DS. | type: Text]
+Hence the way in which ML-DS changes the agent's be liefs will influence the action of the agent . Capturing this phenomenon means solving a challenging leaning problem where the ML prediction is used as proxy information about the causal process the agent wants to intervene on.
+
+[p. 2 | section: Motivating example 2: Agent learning from ML-DS. | type: Text]
+Contributions of the paper These examples show the importance of an agent's prior knowledge for the impact of ML-DS. In this paper, we introduce a general computational framework to study the interaction of an agent with an ML-DS system.
+
+[p. 2 | section: Motivating example 2: Agent learning from ML-DS. | type: Text]
+We make three main contributions. First, we provide a rigorous formalism to analyze how a user interacts with ML-DS system in order to act. Our framework captures said interaction in two steps (2-Step Agent): a Bayesian step in which the agent incorporates the information given by the decision support model into their model of the world, and a causal inference step in which the agent draws the conclusion on which action is best to take. This is schematically depicted in Fig 1 for the clinical example.
+
+[p. 2 | section: Motivating example 2: Agent learning from ML-DS. | type: Text]
+Our second contribution is to run computational experiments to quantify the effect of using ML-DS on patient outcomes. We show that harmful effects can occur in realistic scenarios already when a single prior is misaligned. We also show that ML-DS can be beneficial in some settings.
+
+[p. 2 | section: Motivating example 2: Agent learning from ML-DS. | type: Text]
+Finally, to the best of our knowledge we provide the first treatment of how a Bayesian agent can update their beliefs about a population after observing a prediction of an ML model trained on data from that population. This update involves a challenging inference through a set of latent interchangeable variables in the predictive model's training
+
+[p. 2 | section: Motivating example 2: Agent learning from ML-DS. | type: FigureGroup]
+Figure 1. The diagram shows the overall flow of the model. A new patient with certain attributes (1) arrives for treatment, and the patient's data (2), potentially along with the output of a prediction model (2'), is observed by the agent, e.g., a doctor. The agent uses these observations to update their previous beliefs about the population and the predictive model (3), a step which we model as a Bayesian update. Based on their new beliefs, the agent estimates the effect of treatment (4) and makes a treatment decision (5). Finally, the treatment together with the patient's attributes determine a final outcome (6).
+
+[p. 2 | section: Motivating example 2: Agent learning from ML-DS. | type: Text]
+data. We prove that for simple linear models it is possible to reduce this set to a few sufficient statistics, improving computational cost and numerical stability when applying our framework.
+
+[p. 2 | section: Motivating example 2: Agent learning from ML-DS. | type: Text]
+While our running example is medical, the results are general and apply for every other field where prediction models are employed as decision support. Furthermore, our simulations are only concerned with simple (non-causal) prediction models; ML-DS with causal model is left for future work. The code and experiments are available open-source. 2
+
+[p. 2 | section: 2. The 2-Step Agent: Defining the framework | type: Text]
+In this section, we give a formal description of our 2-Step Agent for studying the effects of ML-DS. We first give a description of the process that generates the training data for the predictive model, as well as the possible types of predictive model (Sec. 2.1) . We formalize the first step, which consists of a Bayesian agent updating their beliefs about the world given a prediction from the predictive model for a new patient (Sec. 2.2) . We then formalize the second step, where the Bayesian agent uses the updated beliefs about the population to decide on a treatment for the new patient (Sec. 2.3) .
+
+[p. 2 | section: 2.1. The setup: Historical data and predictive model | type: Text]
+We start by formally defining a causal model of the domain. Let A be a variable denoting an intervention, such as the administration of a treatment. The treatment variable may
+
+[p. 2 | section: 2.1. The setup: Historical data and predictive model | type: Footnote]
+2 The code will be shared upon acceptance.
+
+[p. 3 | section: 2.1. The setup: Historical data and predictive model | type: Text]
+be continuous or categorical, but to lighten the notation we will present the formulas with a binary A. Let Y be a variable denoting an outcome of interest. In our example above, the treatment could be a fixed dose of chemotherapy and the outcome the months of survival, so in this case a higher outcome is better. We use X_i to refer to covariates involved in the causal mechanisms governing the treatment and outcome.
+
+[p. 3 | section: 2.1. The setup: Historical data and predictive model | type: Text]
+Definition 2.1 (Historical data generation). We assume Hist = (\mathbf{S}, P_{\mathbf{N}}) to be the minimal structural causal model (SCM) describing the causal mechanisms producing A and Y, where \mathbf{S} is a set of assignments defining each variable in terms of its parents and P_{\mathbf{N}} is a joint distribution over the noise variables. We use \bar{X} = X_1, \ldots, X_k to denote endogenous (non-noise) variables involved in the SCM that are different from A and Y.
+
+[p. 3 | section: 2.1. The setup: Historical data and predictive model | type: Text]
+An example of this SCM is depicted in Figure 2(b). The SCM Hist stands for the process in the world that generated the data before the introduction of decision support. Note that the assignment determining A might be due to a specific intervention protocol, e.g., the guidelines for administering the treatment that were in use in that particular time window. The SCM determines a unique joint distribution P^{Hist} over variables Y, A, \bar{X} .
+
+[p. 3 | section: 2.1. The setup: Historical data and predictive model | type: Text]
+Next, we assume that a relevant dataset is generated by Hist and used to train a prediction model for decision support.
+
+[p. 3 | section: 2.1. The setup: Historical data and predictive model | type: Text]
+Definition 2.2 (Training data and prediction model). A dataset \mathcal{D}_{Hist} = \{(\bar{x}_1, a_1, y_1), \dots, (\bar{x}_n, a_n, y_n)\} of historical data is composed of tuples containing relevant characteristics, treatment, and outcome. The dataset contains n instances. This dataset is used to train a prediction model f: \bar{X} \to Y to estimate E^{Hist}(Y \mid \bar{X}) .
+
+[p. 3 | section: 2.1. The setup: Historical data and predictive model | type: Text]
+Other estimands we do not consider are the treatment dependent estimate E^{Hist}(Y\mid A,\bar{X}) and the interventional estimate E^{Hist}(Y\mid do(A),\bar{X}) ; these are left for future work.
+
+[p. 3 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+We next introduce an agent \mathcal{A} who decides on the intervention for new observations (e.g. new patients) with the help of ML-DS. Rather than mechanically applying the recommendation of the ML-DS system, this agent uses their own internal model of the world, capturing not only what they believe about the data generating process Hist, but also what they believe concerning the prediction model f and the training data \mathcal{D}_{Hist} . We model an agent who is aware of the model documentation containing the model class (e.g., a linear model, a tree ensemble, etc.), the model's signature (i.e. that it is treatment naive and what covariates are
+
+[p. 3 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+involved),<sup>3</sup> and the size of the training data, but who is uncertain about the model's parameters. The agent can update their world model, including beliefs concerning historical data and model training.
+
+[p. 3 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+Definition 2.3 (Internal beliefs of the agent). Call A_{hist} the Bayesian network defining the agent's joint distribution over the following variables:
+
+[p. 3 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: ListGroup]
+1. N_{\bar{X}}, N_Y, N_A : These are the variables encoding the population-level distributions over the various quantities of interest, respectively: the patient's characteristics (N_{\bar{X}}) , the noise in the outcome (N_Y) , and the noise in the treatment (N_A) . Depending on how population-level uncertainty is parameterized, these can be vectors. These population-level variables are sampled to produce the historical data. Within this Bayesian network, the historical data is represented via a plate, a graphical notation device used to show repeated, independent, and identically-structured components of a probabilistic model. N<sub>E</sub>: This variable denotes the treatment effect. Contrary to the other noise variables, this is not sampled for each plate repetition but only once, so it remains the same across the plate. 3. A_i, Y_i, \bar{X}_i : These represent the i-th (out of n) datapoint in the training data of the predictive model. The whole training set is represented as a plate of multiplicity n. The noise distributions corresponding to these variables are fixed and outside the plate, capturing the fact that they are population-level distributions. All variables Y_i share the noise term N_E . 4. M: The prediction model's parameters, obtained deterministically from the training data in the plate and possibly some additional known parameter N<sub>M</sub> which encodes, e.g., the hyperparameter settings or initialization values. M has as parents all the variables in the plate, representing the fact that it is trained on the whole data set. 5. \bar{X}^o : An observed data instance for a new patient, for which the agent will make a treatment decision. This array of variables has no non-noise parents and it is sampled from the same noise distribution as \bar{X} .
+
+[p. 3 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Footnote]
+& lt;sup>3</sup>Recent guidelines explicitly suggest to additionally inform model users about the historical treatment policy. Our framework can be extended to model such settings, but here we focus on agents who are uncertain about historical treatment policy, for two reasons. First, this is a common scenario in current practice. Second, this provides a baseline estimation for the effects of ML-DS, which can be compared to better informed users in the future. We come back to these points in the discussion.
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: FigureGroup]
+the SCM generating the data in the real world, while models (a) and (c) are the first and second steps of the 2-Step Agent. Model (a) allows the agent to perform a belief update given new information from ML-DS, while model (c) allows the agent to estimate the
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+model for \bar{X}^o . This is derived deterministically from \bar{X}^o and M.
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+The Bayesian network encoding the conditional independencies of the model is shown in Figure 2 (a). To simplify presentation we depict the submodel of A_{hist} containing only the variables A, Y and X as coinciding with Hist, meaning that the agent has correct belief about the data generation process, although this does not have to be the case and the framework is general enough to allow these two structures to diverge.
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+N_E encodes the agent's beliefs concerning the effect of the intervention: if the agent were to fix all the coefficients in the structural equation of Y, the effect of treatment would be fixed and there could be no update of his/her beliefs on the interventional distribution E^{A_{hist}}(Y \mid \bar{X}, do(A = 1)) . By letting the mechanism be influenced by N_E , we can treat the posterior as encoding a distribution over SCMs, each with its own fixed assignment for Y (and thus each with its own treatment effect).
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+The agent's initial beliefs are represented via priors.
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+Definition 2.4 (Agent's priors). The agent is assumed to
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+have as prior a joint distribution over all variables in the model, which depends on the unconditional distribution P_{N_{\bar{X}},N_Y,N_A,N_E}^{\mathcal{A}_{hist}}.
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+For all i < n, the priors encode the agent's starting beliefs about various quantities. First, P_{N_{\bar{X}}}^{\mathcal{A}_{hist}} encodes the population-level distribution generating \bar{X}_{i,} the agent's model of the historical population. Second, P_{N_A}^{\mathcal{A}_{hist}} encodes the population-level distribution generating the noise of A_i , the agent's model of the historical protocol to administer the intervention. Third, P_{N_Y}^{\mathcal{A}_{hist}} encodes the population-level distribution generating the noise of Y_i , the agent's representation of the outcome. Finally, P_{N_E}^{A_{hist}} encodes the effect of treatment, assumed to be fixed across the population. We assume the agent knows the values of the parameters of the distribution N_M , since he/she has access to the model documentation, therefore we omit this prior.
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+This prior is updated via a Bayesian update based on the observables X^o and Pred^o .
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+Definition 2.5 (Agent belief update). Given \bar{X}^o and Predo the agent does a Bayesian update on the prior P^{\mathcal{A}_{hist}}(N_{\bar{X}}, N_Y, N_A, N_E) , obtaining a posterior distribu-
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Equation]
+P^{\mathcal{A}_{hist}}(N_{\bar{X}}, N_Y, N_A, N_E \mid \bar{X}^o, Pred^o)
+
+[p. 4 | section: 2.2. The first step: Bayesian update of the agent's beliefs | type: Text]
+To simplify exposition, we marginalize over uncertainty in the plate variables (\bar{X}_i, Y_i, A_i \text{ with } i < n) and M.
+
+[p. 4 | section: 2.3. The second step: deciding on the best action after belief update | type: Text]
+We described the agent' world model and how the agent updates it upon observing predictions from an ML-DS system for a new patient, which we called the first step. In the second step, the agent estimates the effect of the treatment on the new patient using their updated world model. This is necessary for the agent to consider a 'fresh' iteration of the plate and calculate the correct interventional probability. Another conceptual reason to introduce a second model is that - when the agent does Bayesian inference - the values of A and Y for the new instance \bar{X}^o are not realized yet.
+
+[p. 4 | section: 2.3. The second step: deciding on the best action after belief update | type: Text]
+Definition 2.6 (Agent model for decision making). This model contains variables N_E, N_{\bar{X}}, N_A, N_Y, X, A, Y , arranged as the similarly-named variables in the sub-model of A_{hist} . This model is used for inference on new instances and we denote it as A_{inf} .
+
+[p. 4 | section: 2.3. The second step: deciding on the best action after belief update | type: Text]
+This second agent model is depicted in Figure 2 (c). The crucial difference between this model and a regular SCM is that the noise variables are not assumed to be independent but rather they are initialized with the joint posterior obtained in the previous Bayesian step. The agent is thus drawing an inference over a distribution of possible SCMs.
+
+[p. 4 | section: 2.3. The second step: deciding on the best action after belief update | type: Text]
+To decide what to do on the new instance, the agent then
+
+[p. 5 | section: 2.3. The second step: deciding on the best action after belief update | type: Text]
+calculates the effect of the intervention on the model A_{inf} where the joint distribution of the noise is set to the posterior calculated over A_{hist} .
+
+[p. 5 | section: 2.3. The second step: deciding on the best action after belief update | type: Text]
+Definition 2.7 (Decision rule). To make a decision, the agent first sets the joint distribution of the inference model according to the posterior on the historical model:
+
+[p. 5 | section: 2.3. The second step: deciding on the best action after belief update | type: Equation]
+P^{\mathcal{A}_{inf}}(N_{\bar{X}}, N_Y, N_A, N_E) := P^{\mathcal{A}_{hist}}(N_{\bar{X}}, N_Y, N_A, N_E \mid \bar{X}^o, Pred^o)
+
+[p. 5 | section: 2.3. The second step: deciding on the best action after belief update | type: Text]
+Following this, treatment is given iff the Conditional Average Treatment Effect (CATE) for the new patient crosses a threshold value, meaning
+
+[p. 5 | section: 2.3. The second step: deciding on the best action after belief update | type: Equation]
+E_{P^{\mathcal{A}_{inf}}}[\mathrm{CATE}] > \tau
+
+[p. 5 | section: 2.3. The second step: deciding on the best action after belief update | type: Text]
+where \tau is a threshold set by the agent beforehand and CATE = E^{\mathcal{A}_{inf}}(Y \mid \bar{X} = \bar{X}^o, do(A=0)) - E^{\mathcal{A}_{inf}}(Y \mid \bar{X} = \bar{X}^o, do(A=1)) . In other words, treatment is given if the expected effect of treatment is sufficiently large in the desired direction.
+
+[p. 5 | section: 2.3. The second step: deciding on the best action after belief update | type: Text]
+When the threshold is zero, the agent administers the treatment if the expected outcome is better in the scenario where they are giving the treatment, conditioned on the patient characteristics, versus the scenario where treatment is not given. Crucially, these interventional expectations are calculated on the internal model of the agent.
+
+[p. 5 | section: 2.4. Quantifying the effect of introducing decision support | type: Text]
+Given the machinery defined above, we are able to calculate the effect of introducing decision support. Given the prediction model and the specification of the agent - which is determined by spelling out \mathcal{A}_{hist} and \mathcal{A}_{inf} - the decision of the agent on a new data point (as described by covariates \bar{X}^o ) is deterministic (i.e. we can implement the input-output process described in Figure 1). We can thus formalize it as a function Act^{\mathcal{A},f}: \bar{X} \to A . The way the decision support changes the actions of the agent is then represented as an intervention on Hist changing the assignment of A:
+
+[p. 5 | section: 2.4. Quantifying the effect of introducing decision support | type: Equation]
+A := Act^{\mathcal{A}, f}(\bar{X})
+
+[p. 5 | section: 2.4. Quantifying the effect of introducing decision support | type: Text]
+The post-interventional SCM is then indicated with the notation Hist^{A=Act^{\mathcal{A},f}(\bar{X})} , where f is the prediction model from Definition 2.2. We can now define the effect of introducing decision support as the risk difference in outcomes generated by Hist compared to Hist^{A=Act^{\mathcal{A},f}(\bar{X})} . This quantifies the effect of the intervention 'using MS-DL', compared to the historical policy.
+
+[p. 5 | section: 3. Experimental setup | type: Text]
+We design experiments to answer two questions. First, how and how much does decision support influence (i) treatment
+
+[p. 5 | section: 3. Experimental setup | type: Text]
+decisions, (ii) agent's beliefs, and (iii) downstream patient outcomes? Second, what role is played by the agent's priors? To address these questions, we vary the priors of the agent and compare the scenarios with and without ML-DS. For each prior setting, we plot the change in treatment decisions, in agent's beliefs, and in patient outcomes, with and without ML-DS, after observing a single patient.
+
+[p. 5 | section: 3. Experimental setup | type: Text]
+Use case We consider a data-generating model with one continuous covariate X, continuous intervention A, and a continuous outcome Y. Higher values of the outcomes are better. One can think of the initial example and interpret X as body weight, A as dosage of chemotherapy, and Y as months of survival. We set n=1000 and we assume the prediction model to be a slope-only linear regression. Despite the continuous treatment, for simplicity we assume the agent compares two treatment options, A=10 or A=20, intuitively ten and twenty units of dosage respectively.
+
+[p. 5 | section: 3. Experimental setup | type: Text]
+Historical SCM. Let the SCM Hist be defined as:
+
+[p. 5 | section: 3. Experimental setup | type: Equation]
+N_X \sim \mathcal{N}(80, 10) N_A \sim \mathcal{N}(2, 1) N_Y \sim \mathcal{N}(0, 0.1) X := N_X A := 0.125 * X + N_A Y := 12 - 0.1 * X + 1 * A + N_Y
+
+[p. 5 | section: 3. Experimental setup | type: Text]
+In this model, the treatment protocol is simply to give a dosage of treatment A which is an eight of the body weight X, and how well this dosing is followed is encoded by the noise term N_A . The mechanism of Y clarifies that the covariate X makes the outcome worse but the treatment improves the outcome. Of course, this equation is a great simplification of what happens in reality, but it is sufficient to demonstrate several important dynamics that emerge when introducing ML-DS. Average treatment effect is positive in this scenario, as witnessed by the positive coefficient of A in the equation for Y.
+
+[p. 5 | section: 3. Experimental setup | type: Text]
+The internal model of the agent's beliefs. The internal model of the agent A_{hist} encodes the agent's uncertainty about Hist :
+
+[p. 5 | section: 3. Experimental setup | type: Equation]
+X_{i} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\sigma}}) \qquad \text{with } i < n N_{A_{i}} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{A}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{A}\boldsymbol{\sigma}}) \qquad \text{with } i < n N_{Y_{i}} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{Y}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{Y}\boldsymbol{\sigma}}) \qquad \text{with } i < n A_{i} = 0.125 X_{i} + N_{A_{i}} \qquad \text{with } i < n Y_{i} = 12 - 0.1 X_{i} + N_{\boldsymbol{E}} A_{i} + N_{Y_{i}} \qquad \text{with } i < n M = \arg\min_{\boldsymbol{\phi} \in \Phi} (MSE(\boldsymbol{\phi}\bar{X}, \bar{Y})) X^{o} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\sigma}}) Pred^{o} = MX^{o}
+
+[p. 5 | section: 3. Experimental setup | type: Text]
+where \bar{Y} = (Y_1, \dots, Y_n) , etc are the respective vectors of n variables, \phi is the single parameter of a least squares regression without intercept, and the variables in bold font indicate
+
+[p. 6 | section: 3. Experimental setup | type: Text]
+population-level priors that are given different values in the different simulations below.<sup>4</sup>
+
+[p. 6 | section: 3. Experimental setup | type: Text]
+The agent knows some aspects of the data generating process, e.g., some of the parameters in the mechanisms of Y and A, but has uncertainty on several dimensions. \alpha_{A\mu} encodes the belief of the agent on how much historical dosages of treatment deviated from the standard protocol. \alpha_{X\mu}, \alpha_{X\sigma} represent the uncertainty of the agent on how the feature X of the training data was distributed. In the definition of Y there are two uncertainties at play. The prior on \alpha_{Y\sigma} describes the uncertainty of the agent concerning Y. The distribution of N_E , on the other hand, captures the belief of the agent concerning the effectiveness of treatment A.
+
+[p. 6 | section: 3. Experimental setup | type: Text]
+Defining different agents We consider two types of agents, one with 'correct' priors whose means are close to the historical SCM, and one with incorrect priors. This is an example of the former:
+
+[p. 6 | section: 3. Experimental setup | type: Equation]
+\begin{aligned} &\alpha_{X\mu} \sim \mathcal{N}(80, 0.1) & \alpha_{X\sigma} \sim \mathcal{N}(10, 0.1) \mid \alpha_{X\sigma} \geq 0 \\ &\alpha_{A\mu} \sim \mathcal{N}(2, 1) & \alpha_{A\sigma} \sim \mathcal{N}(1, 0.1) \mid \alpha_{Y\sigma} \geq 0 \\ &\alpha_{Y\mu} \sim \mathcal{N}(0, 0.01) & \alpha_{Y\sigma} \sim \mathcal{N}(0.1, 0.01) \mid \alpha_{Y\sigma} \geq 0 \\ &N_E \sim \mathcal{N}(1, 0.1) \end{aligned}
+
+[p. 6 | section: 3. Experimental setup | type: Text]
+This agent essentially has the correct beliefs about the patient data distribution, historical treatment policy, outcome variance and treatment effect. They only have some uncertainty represented by some variance on these parameters. In our experiments, we vary each of the colored prior parameters, one at a time, to simulate various ways in which the agent's expectations may be incorrect.
+
+[p. 6 | section: 3. Experimental setup | type: Text]
+The internal model for the agent's effect estimation. Finally, the internal model of the agent used for inference, A_{inf} , is defined as:
+
+[p. 6 | section: 3. Experimental setup | type: Equation]
+\begin{split} N_X &\sim \mathcal{N}(\alpha_{X\mu}, \alpha_{X\sigma}) \quad N_A \sim \mathcal{N}(\alpha_{A\mu}, \alpha_{A\sigma}) \\ N_Y &\sim \mathcal{N}(\alpha_{Y\mu}, \alpha_{Y\sigma}) \\ X &:= N_X \\ A &:= 0.125 \ X + N_A \\ Y &:= 12 - 0.1 \ X + N_E \ A + N_Y \end{split}
+
+[p. 6 | section: 3. Experimental setup | type: Text]
+The posterior defines a probability distribution over SCMs. Recall that the agent was interested in choosing between dosage A=10 or A=20 for the specific patient with
+
+[p. 6 | section: 3. Experimental setup | type: Text]
+weight equal to X^o . To compute the desired quantity, the agent aggregates the CATE produced by every SCM according to the posterior
+
+[p. 6 | section: 3. Experimental setup | type: Equation]
+C\hat{ATE} = \mathbb{E}_{P^{\mathcal{A}_{inf}}}[E^{\mathcal{A}_{inf}}(Y|X^o, do(A=20)) - E^{\mathcal{A}_{inf}}(Y|X^o, do(A=10))]
+
+[p. 6 | section: 3. Experimental setup | type: Text]
+where the expectation is over the joint posterior of the agent over the parameters, which was obtained updating A_{hist} .
+
+[p. 6 | section: 3. Experimental setup | type: Text]
+Reaching a decision. The agent will administer a dosage of 10 if the estimated CATE is below a threshold of 5 or a dosage of 20 otherwise. In other words, the higher dosage needs to improve the outcome by at least 5 months to be worth it. This completes the decision pipeline: given a patient X^o with a corresponding prediction and the type of agent, we can compute what the agent would do.
+
+[p. 6 | section: 4. Results | type: Text]
+We begin by showcasing how our framework can capture the two motivating examples presented in the Introduction. In the first example, the belief of the agent on the past policy determines how the prediction is interpreted and what action is taken.<sup>5</sup> Indeed, in the simulation, when X = 80, with a bad prognosis under the assumption of no treatment (\alpha_{A\mu} = 0) , the agent chooses a high dosage to improve the outcome (A = 20). With a bad prognosis under the assumption of heavy treatment ( \alpha_{A\mu}=20 ), the agent chooses a low dosage to reduce over-treatment (A = 10). In the second motivating example, new information from the prediction forces an agent to consider different options. In the model, this means that either belief in treatment effect (N_E) or in historical policy (\mu_A) is revised, which appears as a correlation in the posterior belief in these two variables in Figure 4 in Appendix A.
+
+[p. 6 | section: 4. Results | type: Text]
+Next, for the use case described in Section 3, we report the effect of using ML-DS on two dimensions: 1) agent's belief of treatment effect and 2) downstream outcomes Y realized after the agent has taken an action, visualized in the top and bottom rows of Figure 3, respectively. In each of these plots, the outcomes obtained using ML-DS (orange boxplots) is contrasted with the outcomes obtained without it (blue boxplots), for different types of agents. The CATE plots sweep a wide range of options for the priors in order
+
+[p. 6 | section: 4. Results | type: Footnote]
+& lt;sup>4</sup>Note that for ease of exposition we introduced some variables (N_{A_i}, N_{Y_i}) which are not present in Figure 2 (a). These can in principle be removed by specifying distributions directly on the inverse transformations of the ones given above for A_i and Y_i , and applying the appropriate Jacobian adjustment. In the PyMC implementation, we also use a small (1e-3) observational noise around Pred^o to allow us to sample from an observed deterministic transformation.
+
+[p. 6 | section: 4. Results | type: Footnote]
+& lt;sup>5</sup>All simulations presented in this paper are ran with PyMC v5 (Salvatier et al., 2016). We used the NUTS sampler to get 1000 samples for each of 4 chains, with 1000 tuning samples and a 'target_accept' parameter of 0.85. Diagnostics and visual inspection of the traces showed good mixing and no convergence issues (Max \hat{R}=1.01 , and only 16 out 4000 runs had > 5 divergences, due to fixed 'target_accept' and extreme parameter values.). Sanity checks on the Bayesian model are reported in Appendix B.
+
+[p. 7 | section: 4. Results | type: Text]
+to reveal interesting dynamics of the agent's beliefs; in Appendix D we also report the CATE results for the more realistic range of ± 3 stds from the SCM reference value.
+
+[p. 7 | section: 4. Results | type: Text]
+We begin with an agent whose priors beliefs coincide with the parameters of the historical SCM except for NE, i.e. treatment effect. Figure 3a and Figure 3d show what happens to CATE and Y when the agent's belief in N E varies. As expected, in the first figure we can see that the blue CATE has a slope that depends on the prior NE: without decision support, the belief on CATE is equal to N E multiplied by the difference in treatments. In the orange series however, we see an interesting behavior: after learning from the interaction with ML-DS, the agent corrects their belief in CATE re-aligning it to the correct value. In other words, when all other beliefs are correct, the agent can learn from ML-DS to estimate better CATE and take better actions. This is confirmed in Figure 3d where the orange series shows superior outcomes.
+
+[p. 7 | section: 4. Results | type: Text]
+Figure 3b and Figure 3e show what happens when the agent's prior belief in the treatment protocol, µA, varies. In this scenario the prior CATE estimates are flat, for the simple reason that the agent has a fixed belief in N E and they are not updating it. When interacting with ML-DS however, the belief update changes CATE estimates quite a lot; this is because the agent is attempting to explain the gap between P red o given by ML-DS and the expected outcome they had from their prior beliefs. This reflects on outcomes in 3e where the orange series is sometimes below the blue one: incorrect beliefs in µ A can result in CATE estimates that lead to wrong actions and worsen outcome Y.
+
+[p. 7 | section: 4. Results | type: Text]
+A similar pattern is observed in Figures 3c, 3f, 6c (last one in Appendix D) : while prior CATE and Y estimates are relatively stable without ML-DS, new dynamics emerge when when ML-DS is introduced. For CATE, incorrect beliefs can have large effects both on the magnitude and sign. For Y , in both cases incorrect beliefs can lead to bad actions and worse outcomes.
+
+[p. 7 | section: 5. Related work | type: Text]
+A large body of literature is concerned with the analysis of ML-DS from different angles. A strand of literature focuses on how ML-DS is entangled with (changes in) the data generating process. Works in this category study limitations of what can be learned from data generated from past policies, e.g. (Lakkaraju et al., 2017) , or the dynamics that occur with alternating cycles of ML-DS deployment and re-training, as in the performative prediction literature (Perdomo et al., 2020; Boeken et al., 2024) . Another batch of articles is concerned with evaluation of ML-DS, laying out identifiability conditions for the effects of ML-DS on decisions (Imai et al., 2023) or trying to optimize human-machine complementar-
+
+[p. 7 | section: 5. Related work | type: Text]
+ity by design (Donahue et al., 2022; McLaughlin & Spiess, 2024) . Finally, a different set of papers investigate what human input can add to artificial agents already reaching some level of optimality (Alur et al., 2024; Zhang & Barein boim, 2022; Stensrud et al., 2024) . The key difference of our work is that we offer a more fine-grained description of what happens inside the decision maker, namely we unpack the micro-mechanism leading a decision maker to take an action on a single observation paired with AI prediction.
+
+[p. 7 | section: 5. Related work | type: Text]
+Other papers have studied how information flows between agents influencing their actions. In (Kamenica & Gentzkow, 2011) a Sender of information influences a Receiver Bayesian agent in order to stir their action. What we add here is that in our context the signal is an ML prediction produced from training data drawn from the same environment the agent reasons about; this coupling yields the distinctive inference problem we study, i.e. updating through latent interchangeable training variables.
+
+[p. 7 | section: 5. Related work | type: Text]
+Another important strand of work focuses on problems related to inferring information about a population P given predictions from a model trained on samples from P in the context of confidentiality attacks. This literature studies whether it is possible to use models' predictions (and other behavior) to recover the presence of a specific data point, in so-called membership inference attacks (Hu et al., 2022; Huang, 2025) ), or to infer statistical properties of a training datasets, so-called property-inference attacks (Ate niese et al., 2015; Wang et al., 2019) . While these models aim to recover information about the training set, e.g., the proportion of data in each label, our challenge is to marginalize across datasets and directly infer information about the underlying population.
+
+[p. 7 | section: 6. Discussion | type: Text]
+We have presented the first explicit framework that includes an ML-DS system and a sophisticated user. Our 2-Step Agent framework captures the effect of an agent's prior beliefs, modeling learning effects due to the interaction and tracking agent's uncertainty. The framework allows for simulations comparing outcomes with and without ML-DS, in an RCT-like fashion, therefore quantifying the benefit/harm of ML-DS for different types of simulated agents.
+
+[p. 7 | section: 6. Discussion | type: Text]
+We provided an implementation of the framework for ML-DS with a linear model. Said implementation leverages a collapse of the plate via sufficient statistics (Appendix E.1) , greatly speeding up the simulations. Within this implementation, we have shown that incorrect beliefs about the distribution of the predictive model's training data may lead the agent to bad decisions, resulting in harmful outcomes.
+
+[p. 7 | section: 6. Discussion | type: Text]
+While previous literature had argued that incorrect beliefs about past treatment policy (µ A in our notation) might gener-
+
+[p. 8 | section: 6. Discussion | type: FigureGroup]
+Figure 3. Figures displaying the effects of different agent priors when using ML-DS (in orange) and without it (in blue). In all plots except from NE, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameters. The top row displays the agent's prior and posterior (predictive) belief of CATE, while the bottom row shows the impact of the agent's decisions on the downstream outcome Y . The left column pertains to the belief on treatment effect, NE, the central column the belief about past treatment policy (µA) and the right column about the belief on past covariate distribution (µX). The gray areas represents the portions of the belief in which the true CATE is within one std from the CATE learned from ML-DS (orange band).
+
+[p. 8 | section: 6. Discussion | type: Text]
+ate worse downstream outcomes, e.g. in (van Geloven et al., 2025) , this is the first quantitative estimation of harmful effects of ML-DS. In our experiments we show that ML-DS can be harmful if the agent has wrong beliefs not only about the past treatment policy, but also about the distribution of covariates and the distribution of the outcome.
+
+[p. 8 | section: 6. Discussion | type: Text]
+We also show that if the agent has otherwise correct beliefs, an incorrect belief about treatment effect may be corrected during updates that push the agent towards the true CATE. This suggests that in certain settings ML-DS building on a treatment-naive prediction model might still be beneficial.
+
+[p. 8 | section: 6. Discussion | type: Text]
+This finding highlights the extreme sensitivity of ML-DS to agent prior beliefs when using a treatment-naive prediction model, and underscore the importance of providing full documentation and proper training to users of ML-DS. Finally, we remark that these results apply to any field deploying such systems and to any agent interacting with ML-DS, be it human or artificial.
+
+[p. 8 | section: 6. Discussion | type: Text]
+Limitations and Future work The experiments in this paper explored only part of the potential of the framework. The experiments are limited to a use case in which the prediction model is linear and treatment-naive, and the data
+
+[p. 8 | section: 6. Discussion | type: Text]
+generating process is simple (e.g., every distribution is normal, there is only one confounder). Implementing the use case was not trivial and required pushing a Bayesian update through OLS and a large amount of variables (see Appendix E.1) , hence it remains to be seen whether similar results can be obtained when the complexity of the use case increases.
+
+[p. 8 | section: 6. Discussion | type: Text]
+We assume the agent reasons in a Bayesian fashion, which allows for a precise analysis of the agent's uncertainty. While this agent may not be realistic with respect to human behavior, it is useful to make this assumption to demonstrate what can go wrong in a best-case scenario, i.e., when the agent is a 'perfect' reasoner and the ML model has no flaws. Finally, this assumption is also interesting when considering how an artificial agent might learn from predictions of other ML models.
+
+[p. 8 | section: 6. Discussion | type: Text]
+In future work this line of research will be expanded across different dimensions, including i) different types of agents (e.g. agent with incorrect internal model of the world of different kinds of uncertainties) ii) more complex ML models for ML-DS, iii) cases with heterogeneity of treatment effect (i.e. the effect of the action is not the same for all instances), iv) repeated rounds of learning for the same agent and v) using causal models for ML-DS.

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+# 2-Step Agent:
+
+## A Framework for the Interaction of a Decision Maker with AI Decision Support
+
+#### Anonymous Authors<sup>1</sup>
+
+### Abstract
+
+Across a growing number of fields, human decision making is supported by predictions from AI models. However, we still lack a deep understanding of the effects of adoption of these technologies. In this paper, we introduce a general computational framework, the 2-Step Agent, which models the effects of AI-assisted decision making. Our framework uses Bayesian methods for causal inference to model 1) how a prediction on a new observation affects the beliefs of a rational Bayesian agent, and 2) how this change in beliefs affects the downstream decision and subsequent outcome. Using this framework, we show by simulations how a single misaligned prior belief can be sufficient for decision support to result in worse downstream outcomes compared to no decision support. Our results reveal several potential pitfalls of AI-driven decision support and highlight the need for thorough model documentation and proper user training.
+
+#### <span id="page-0-1"></span>1. Introduction
+
+Background. Prediction models are often used to support decision making: a model trained on past data produces a forecast for a future outcome of interest, which a agent can take into account when deciding how to act. For the rest of this article, we will refer to such decision support as ML-DS (Machine Learning Decision Support) to emphasize the fact that the model has been trained on data. This is sometimes also called 'predictive optimization' [\(Wang et al.,](#page-8-0) [2024\)](#page-8-0). Contrary to full automation, in high-risk settings ML-DS is sometimes perceived as a way to leverage the power of machine learning while retaining human oversight (see for instance Article 14 of the EU AI Act [\(European Parliament](#page-8-1) [& Council of the European Union,](#page-8-1) [2024\)](#page-8-1)).
+
+Preliminary work. Under review by the International Conference on Machine Learning (ICML). Do not distribute.
+
+Understanding the practical impact of adopting ML-DS in a domain is essential. First, ML-DS is on the rise, and currently deployed in a variety of fields including healthcare, education [\(Smith et al.,](#page-8-2) [2012\)](#page-8-2), policing and judicial system [\(Kleinberg et al.,](#page-8-3) [2018\)](#page-8-3), as well as redistribution of welfare [\(Kube et al.,](#page-8-4) [2019\)](#page-8-4). Second, allowing machines to influence decision making carries important risks, e.g. the incorporation of biased advice [\(Obermeyer et al.,](#page-8-5) [2019\)](#page-8-5).
+
+In this paper, we present a full-fledged computational models of ML-DS, the *2-Step Agent*. While most research on effects of ML-DS focuses on the model, we unpack the inner working of the decision maker, studying how a rational agent incorporates information from ML-DS in order to make a decision. We demonstrate that the introduction of ML-DS can lead to harmful consequences due to a mismatch of the model with the users' expectations or training, even in an ideal scenario where the ML model is as good as it gets and the agent is a 'perfect' reasoner.
+
+#### Motivating example 1: Effect of the agent's prior beliefs.
+
+Consider the situation of an oncologist using ML-DS when deciding how to treat a cancer patient. The ML-model predicts that the patient has only two months left to live. We can then imagine two lines of thinking depending on the clinician's prior beliefs. If the model was trained on patients who received minimal treatment, a very heavy treatment may extend the survival time. On the other hand, if the model is trained on patients who received the heaviest treatment, it is best to put the patient on palliative care to avoid unnecessary suffering from side-effects.
+
+This example showcases that, even in presence of the same information from ML-DS, *different prior assumptions about the model and the data it was trained on can lead to opposite actions*. This underscores the importance of an agent's priors in determining the effectiveness of ML-DS,[1](#page-0-0) a point that has already been argued for in previous literature (see e.g., [\(van Geloven et al.,](#page-8-6) [2025\)](#page-8-6)) but has not been studied
+
+<sup>1</sup>Anonymous Institution, Anonymous City, Anonymous Region, Anonymous Country. Correspondence to: Anonymous Author <anon.email@domain.com>.
+
+<span id="page-0-0"></span><sup>1</sup> Setting the agent prior beliefs by proper training is indeed an important part of RCTs' protocols that is not always adhered to [\(Perepletchikova et al.,](#page-8-7) [2007\)](#page-8-7). When it is not possible to create a common baseline of beliefs and behavior, a two-stage randomization might be required, both on the providers of the intervention and the subjects [\(Borm et al.,](#page-8-8) [2005\)](#page-8-8).
+
+{1}------------------------------------------------
+
+quantitatively.
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+108 109
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+#### Motivating example 2: Agent learning from ML-DS.
+
+Consider a variation of the previous scenario involving a clinician and a patient with a similar prognosis. Before seeing the model's prediction, the clinician believes that in the training data of the model (1) patients were always treated, and in general (2) patients ended up surviving between 6 and 12 months when treated. Seeing a prediction of two months of survival, and assuming the patient is not an outlier, this set of beliefs is challenged.
+
+The clinician can then revise their beliefs in at least two different ways. They can conclude the model's training data had in fact some untreated patients, and that this patient might be similar to one of the untreated patients. In this case, it might still be beneficial to treat. On the other hand, they might conclude they were wrong about the effect of treatment on patients. In this case, the treatment must be less beneficial than previously thought by the clinician. Therefore it might be best to abstain from treatment.
+
+Hence *the way in which ML-DS changes the agent's beliefs will influence the action of the agent*. Capturing this phenomenon means solving a challenging leaning problem where the ML prediction is used as proxy information about the causal process the agent wants to intervene on.
+
+Contributions of the paper These examples show the importance of an agent's prior knowledge for the impact of ML-DS. In this paper, we introduce a general computational framework to study the interaction of an agent with an ML-DS system.
+
+We make three main contributions. First, we provide a rigorous formalism to analyze how a user interacts with ML-DS system in order to act. Our framework captures said interaction in two steps (2-Step Agent): a Bayesian step in which the agent incorporates the information given by the decision support model into their model of the world, and a causal inference step in which the agent draws the conclusion on which action is best to take. This is schematically depicted in Fig [1](#page-1-0) for the clinical example.
+
+Our second contribution is to run computational experiments to quantify the effect of using ML-DS on patient outcomes. We show that harmful effects can occur in realistic scenarios already when a single prior is misaligned. We also show that ML-DS can be beneficial in some settings.
+
+Finally, to the best of our knowledge we provide the first treatment of how a Bayesian agent can update their beliefs about a population after observing a prediction of an ML model trained on data from that population. This update involves a challenging inference through a set of latent interchangeable variables in the predictive model's training
+
+<span id="page-1-0"></span>![](_page_1_Figure_10.jpeg)
+
+*Figure 1.* The diagram shows the overall flow of the model. A new patient with certain attributes (1) arrives for treatment, and the patient's data (2), potentially along with the output of a prediction model (2'), is observed by the agent, e.g., a doctor. The agent uses these observations to update their previous beliefs about the population and the predictive model (3), a step which we model as a Bayesian update. Based on their new beliefs, the agent estimates the effect of treatment (4) and makes a treatment decision (5). Finally, the treatment together with the patient's attributes determine a final outcome (6).
+
+data. We prove that for simple linear models it is possible to reduce this set to a few sufficient statistics, improving computational cost and numerical stability when applying our framework.
+
+While our running example is medical, the results are general and apply for every other field where prediction models are employed as decision support. Furthermore, our simulations are only concerned with simple (non-causal) prediction models; ML-DS with causal model is left for future work. The code and experiments are available open-source.[2](#page-1-1)
+
+#### 2. The 2-Step Agent: Defining the framework
+
+In this section, we give a formal description of our 2-Step Agent for studying the effects of ML-DS. We first give a description of the process that generates the training data for the predictive model, as well as the possible types of predictive model (Sec. [2.1\)](#page-1-2). We formalize the first step, which consists of a Bayesian agent updating their beliefs about the world given a prediction from the predictive model for a new patient (Sec. [2.2\)](#page-2-0). We then formalize the second step, where the Bayesian agent uses the updated beliefs about the population to decide on a treatment for the new patient (Sec. [2.3\)](#page-3-0).
+
+#### <span id="page-1-2"></span>2.1. The setup: Historical data and predictive model
+
+We start by formally defining a causal model of the domain. Let A be a variable denoting an intervention, such as the administration of a treatment. The treatment variable may
+
+<span id="page-1-1"></span><sup>2</sup>The code will be shared upon acceptance.
+
+{2}------------------------------------------------
+
+be continuous or categorical, but to lighten the notation we will present the formulas with a binary A. Let Y be a variable denoting an outcome of interest. In our example above, the treatment could be a fixed dose of chemotherapy and the outcome the months of survival, so in this case a higher outcome is better. We use  $X_i$  to refer to covariates involved in the causal mechanisms governing the treatment and outcome.
+
+**Definition 2.1** (Historical data generation). We assume  $Hist = (\mathbf{S}, P_{\mathbf{N}})$  to be the minimal structural causal model (SCM) describing the causal mechanisms producing A and Y, where  $\mathbf{S}$  is a set of assignments defining each variable in terms of its parents and  $P_{\mathbf{N}}$  is a joint distribution over the noise variables. We use  $\bar{X} = X_1, \ldots, X_k$  to denote endogenous (non-noise) variables involved in the SCM that are different from A and Y.
+
+An example of this SCM is depicted in Figure 2(b). The SCM Hist stands for the process in the world that generated the data before the introduction of decision support. Note that the assignment determining A might be due to a specific intervention protocol, e.g., the guidelines for administering the treatment that were in use in that particular time window. The SCM determines a unique joint distribution  $P^{Hist}$  over variables  $Y, A, \bar{X}$ .
+
+Next, we assume that a relevant dataset is generated by Hist and used to train a prediction model for decision support.
+
+<span id="page-2-2"></span>**Definition 2.2** (Training data and prediction model). A dataset  $\mathcal{D}_{Hist} = \{(\bar{x}_1, a_1, y_1), \dots, (\bar{x}_n, a_n, y_n)\}$  of historical data is composed of tuples containing relevant characteristics, treatment, and outcome. The dataset contains n instances. This dataset is used to train a prediction model  $f: \bar{X} \to Y$  to estimate  $E^{Hist}(Y \mid \bar{X})$ .
+
+Other estimands we do not consider are the treatment dependent estimate  $E^{Hist}(Y\mid A,\bar{X})$  and the interventional estimate  $E^{Hist}(Y\mid do(A),\bar{X})$ ; these are left for future work.
+
+#### <span id="page-2-0"></span>2.2. The first step: Bayesian update of the agent's beliefs
+
+We next introduce an agent  $\mathcal{A}$  who decides on the intervention for new observations (e.g. new patients) with the help of ML-DS. Rather than mechanically applying the recommendation of the ML-DS system, this agent uses their own internal model of the world, capturing not only what they believe about the data generating process Hist, but also what they believe concerning the prediction model f and the training data  $\mathcal{D}_{Hist}$ . We model an agent who is aware of the model documentation containing the model class (e.g., a linear model, a tree ensemble, etc.), the model's signature (i.e. that it is treatment naive and what covariates are
+
+involved),<sup>3</sup> and the size of the training data, but who is uncertain about the model's parameters. The agent can update their world model, including beliefs concerning historical data and model training.
+
+**Definition 2.3** (Internal beliefs of the agent). Call  $A_{hist}$  the Bayesian network defining the agent's joint distribution over the following variables:
+
+- 1.  $N_{\bar{X}}, N_Y, N_A$ : These are the variables encoding the population-level distributions over the various quantities of interest, respectively: the patient's characteristics  $(N_{\bar{X}})$ , the noise in the outcome  $(N_Y)$ , and the noise in the treatment  $(N_A)$ . Depending on how population-level uncertainty is parameterized, these can be vectors. These population-level variables are sampled to produce the historical data. Within this Bayesian network, the historical data is represented via a plate, a graphical notation device used to show repeated, independent, and identically-structured components of a probabilistic model.
+- N<sub>E</sub>: This variable denotes the treatment effect. Contrary to the other noise variables, this is not sampled for each plate repetition but only once, so it remains the same across the plate.
+- 3.  $A_i, Y_i, \bar{X}_i$ : These represent the i-th (out of n) datapoint in the training data of the predictive model. The whole training set is represented as a plate of multiplicity n. The noise distributions corresponding to these variables are fixed and outside the plate, capturing the fact that they are population-level distributions. All variables  $Y_i$  share the noise term  $N_E$ .
+- 4. M: The prediction model's parameters, obtained deterministically from the training data in the plate and possibly some additional known parameter N<sub>M</sub> which encodes, e.g., the hyperparameter settings or initialization values. M has as parents all the variables in the plate, representing the fact that it is trained on the whole data set.
+- 5.  $\bar{X}^o$ : An observed data instance for a new patient, for which the agent will make a treatment decision. This array of variables has no non-noise parents and it is sampled from the same noise distribution as  $\bar{X}$ .
+
+<span id="page-2-1"></span><sup>&</sup>lt;sup>3</sup>Recent guidelines explicitly suggest to additionally inform model users about the historical treatment policy. Our framework can be extended to model such settings, but here we focus on agents who are uncertain about historical treatment policy, for two reasons. First, this is a common scenario in current practice. Second, this provides a baseline estimation for the effects of ML-DS, which can be compared to better informed users in the future. We come back to these points in the discussion.
+
+{3}------------------------------------------------
+
+<span id="page-3-1"></span>![](_page_3_Figure_1.jpeg)
+
+the SCM generating the data in the real world, while models (a) and (c) are the first and second steps of the 2-Step Agent. Model (a) allows the agent to perform a belief update given new information from ML-DS, while model (c) allows the agent to estimate the
+
+model for  $\bar{X}^o$ . This is derived deterministically from  $\bar{X}^o$  and M.
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+The Bayesian network encoding the conditional independencies of the model is shown in Figure 2 (a). To simplify presentation we depict the submodel of  $A_{hist}$  containing only the variables A, Y and X as coinciding with Hist, meaning that the agent has correct belief about the data generation process, although this does not have to be the case and the framework is general enough to allow these two structures to diverge.
+
+ $N_E$  encodes the agent's beliefs concerning the effect of the intervention: if the agent were to fix all the coefficients in the structural equation of Y, the effect of treatment would be fixed and there could be no update of his/her beliefs on the interventional distribution  $E^{A_{hist}}(Y \mid \bar{X}, do(A = 1))$ . By letting the mechanism be influenced by  $N_E$ , we can treat the posterior as encoding a distribution over SCMs, each with its own fixed assignment for Y (and thus each with its own treatment effect).
+
+The agent's initial beliefs are represented via priors.
+
+**Definition 2.4** (Agent's priors). The agent is assumed to
+
+have as prior a joint distribution over all variables in the model, which depends on the unconditional distribution  $P_{N_{\bar{X}},N_Y,N_A,N_E}^{\mathcal{A}_{hist}}.$ 
+
+For all i < n, the priors encode the agent's starting beliefs about various quantities. First,  $P_{N_{\bar{X}}}^{\mathcal{A}_{hist}}$  encodes the population-level distribution generating  $\bar{X}_{i,}$  the agent's model of the historical population. Second,  $P_{N_A}^{\mathcal{A}_{hist}}$  encodes the population-level distribution generating the noise of  $A_i$ , the agent's model of the historical protocol to administer the intervention. Third,  $P_{N_Y}^{\mathcal{A}_{hist}}$  encodes the population-level distribution generating the noise of  $Y_i$ , the agent's representation of the outcome. Finally,  $P_{N_E}^{A_{hist}}$  encodes the effect of treatment, assumed to be fixed across the population. We assume the agent knows the values of the parameters of the distribution  $N_M$ , since he/she has access to the model documentation, therefore we omit this prior.
+
+This prior is updated via a Bayesian update based on the observables  $X^o$  and  $Pred^o$ .
+
+**Definition 2.5** (Agent belief update). Given  $\bar{X}^o$  and Predo the agent does a Bayesian update on the prior  $P^{\mathcal{A}_{hist}}(N_{\bar{X}}, N_Y, N_A, N_E)$ , obtaining a posterior distribu-
+
+$$P^{\mathcal{A}_{hist}}(N_{\bar{X}}, N_Y, N_A, N_E \mid \bar{X}^o, Pred^o)$$
+
+To simplify exposition, we marginalize over uncertainty in the plate variables  $(\bar{X}_i, Y_i, A_i \text{ with } i < n)$  and M.
+
+#### <span id="page-3-0"></span>2.3. The second step: deciding on the best action after belief update
+
+We described the agent' world model and how the agent updates it upon observing predictions from an ML-DS system for a new patient, which we called the first step. In the second step, the agent estimates the effect of the treatment on the new patient using their updated world model. This is necessary for the agent to consider a 'fresh' iteration of the plate and calculate the correct interventional probability. Another conceptual reason to introduce a second model is that - when the agent does Bayesian inference - the values of A and Y for the new instance  $\bar{X}^o$  are not realized yet.
+
+**Definition 2.6** (Agent model for decision making). This model contains variables  $N_E, N_{\bar{X}}, N_A, N_Y, X, A, Y$ , arranged as the similarly-named variables in the sub-model of  $A_{hist}$ . This model is used for inference on new instances and we denote it as  $A_{inf}$ .
+
+This second agent model is depicted in Figure 2 (c). The crucial difference between this model and a regular SCM is that the noise variables are not assumed to be independent but rather they are initialized with the joint posterior obtained in the previous Bayesian step. The agent is thus drawing an inference over a distribution of possible SCMs.
+
+To decide what to do on the new instance, the agent then
+
+{4}------------------------------------------------
+
+calculates the effect of the intervention on the model  $A_{inf}$  where the joint distribution of the noise is set to the posterior calculated over  $A_{hist}$ .
+
+**Definition 2.7** (Decision rule). To make a decision, the agent first sets the joint distribution of the inference model according to the posterior on the historical model:
+
+$$P^{\mathcal{A}_{inf}}(N_{\bar{X}}, N_Y, N_A, N_E) :=$$
+
+$$P^{\mathcal{A}_{hist}}(N_{\bar{X}}, N_Y, N_A, N_E \mid \bar{X}^o, Pred^o)$$
+
+Following this, treatment is given iff the Conditional Average Treatment Effect (CATE) for the new patient crosses a threshold value, meaning
+
+$$E_{P^{\mathcal{A}_{inf}}}[\mathrm{CATE}] > \tau$$
+
+where  $\tau$  is a threshold set by the agent beforehand and CATE =  $E^{\mathcal{A}_{inf}}(Y \mid \bar{X} = \bar{X}^o, do(A=0)) - E^{\mathcal{A}_{inf}}(Y \mid \bar{X} = \bar{X}^o, do(A=1))$ . In other words, treatment is given if the expected effect of treatment is sufficiently large in the desired direction.
+
+When the threshold is zero, the agent administers the treatment if the expected outcome is better in the scenario where they are giving the treatment, conditioned on the patient characteristics, versus the scenario where treatment is not given. Crucially, these interventional expectations are calculated on the internal model of the agent.
+
+# 2.4. Quantifying the effect of introducing decision support
+
+Given the machinery defined above, we are able to calculate the effect of introducing decision support. Given the prediction model and the specification of the agent - which is determined by spelling out  $\mathcal{A}_{hist}$  and  $\mathcal{A}_{inf}$  - the decision of the agent on a new data point (as described by covariates  $\bar{X}^o$ ) is deterministic (i.e. we can implement the input-output process described in Figure 1). We can thus formalize it as a function  $Act^{\mathcal{A},f}: \bar{X} \to A$ . The way the decision support changes the actions of the agent is then represented as an intervention on Hist changing the assignment of A:
+
+$$A := Act^{\mathcal{A}, f}(\bar{X})$$
+
+The post-interventional SCM is then indicated with the notation  $Hist^{A=Act^{\mathcal{A},f}(\bar{X})}$ , where f is the prediction model from Definition 2.2. We can now define the effect of introducing decision support as the risk difference in outcomes generated by Hist compared to  $Hist^{A=Act^{\mathcal{A},f}(\bar{X})}$ . This quantifies the effect of the intervention 'using MS-DL', compared to the historical policy.
+
+#### <span id="page-4-0"></span>3. Experimental setup
+
+We design experiments to answer two questions. First, how and how much does decision support influence (i) treatment decisions, (ii) agent's beliefs, and (iii) downstream patient outcomes? Second, what role is played by the agent's priors? To address these questions, we vary the priors of the agent and compare the scenarios with and without ML-DS. For each prior setting, we plot the change in treatment decisions, in agent's beliefs, and in patient outcomes, with and without ML-DS, after observing a single patient.
+
+Use case We consider a data-generating model with one continuous covariate X, continuous intervention A, and a continuous outcome Y. Higher values of the outcomes are better. One can think of the initial example and interpret X as body weight, A as dosage of chemotherapy, and Y as months of survival. We set n=1000 and we assume the prediction model to be a slope-only linear regression. Despite the continuous treatment, for simplicity we assume the agent compares two treatment options, A=10 or A=20, intuitively ten and twenty units of dosage respectively.
+
+**Historical SCM.** Let the SCM *Hist* be defined as:
+
+$$N_X \sim \mathcal{N}(80, 10)$$
+  $N_A \sim \mathcal{N}(2, 1)$   $N_Y \sim \mathcal{N}(0, 0.1)$   $X := N_X$   $A := 0.125 * X + N_A$   $Y := 12 - 0.1 * X + 1 * A + N_Y$ 
+
+In this model, the treatment protocol is simply to give a dosage of treatment A which is an eight of the body weight X, and how well this dosing is followed is encoded by the noise term  $N_A$ . The mechanism of Y clarifies that the covariate X makes the outcome worse but the treatment improves the outcome. Of course, this equation is a great simplification of what happens in reality, but it is sufficient to demonstrate several important dynamics that emerge when introducing ML-DS. Average treatment effect is positive in this scenario, as witnessed by the positive coefficient of A in the equation for Y.
+
+The internal model of the agent's beliefs. The internal model of the agent  $A_{hist}$  encodes the agent's uncertainty about *Hist*:
+
+$$X_{i} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\sigma}}) \qquad \text{with } i < n$$
+
+$$N_{A_{i}} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{A}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{A}\boldsymbol{\sigma}}) \qquad \text{with } i < n$$
+
+$$N_{Y_{i}} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{Y}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{Y}\boldsymbol{\sigma}}) \qquad \text{with } i < n$$
+
+$$A_{i} = 0.125 X_{i} + N_{A_{i}} \qquad \text{with } i < n$$
+
+$$Y_{i} = 12 - 0.1 X_{i} + N_{\boldsymbol{E}} A_{i} + N_{Y_{i}} \qquad \text{with } i < n$$
+
+$$M = \arg\min_{\boldsymbol{\phi} \in \Phi} (MSE(\boldsymbol{\phi}\bar{X}, \bar{Y}))$$
+
+$$X^{o} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\sigma}})$$
+
+$$Pred^{o} = MX^{o}$$
+
+where  $\bar{Y} = (Y_1, \dots, Y_n)$ , etc are the respective vectors of n variables,  $\phi$  is the single parameter of a least squares regression without intercept, and the variables in bold font indicate
+
+{5}------------------------------------------------
+
+population-level priors that are given different values in the different simulations below.<sup>4</sup>
+
+The agent knows some aspects of the data generating process, e.g., some of the parameters in the mechanisms of Y and A, but has uncertainty on several dimensions.  $\alpha_{A\mu}$  encodes the belief of the agent on how much historical dosages of treatment deviated from the standard protocol.  $\alpha_{X\mu}, \alpha_{X\sigma}$  represent the uncertainty of the agent on how the feature X of the training data was distributed. In the definition of Y there are two uncertainties at play. The prior on  $\alpha_{Y\sigma}$  describes the uncertainty of the agent concerning Y. The distribution of  $N_E$ , on the other hand, captures the belief of the agent concerning the effectiveness of treatment A.
+
+**Defining different agents** We consider two types of agents, one with 'correct' priors whose means are close to the historical SCM, and one with incorrect priors. This is an example of the former:
+
+$$\begin{aligned} &\alpha_{X\mu} \sim \mathcal{N}(80, 0.1) & \alpha_{X\sigma} \sim \mathcal{N}(10, 0.1) \mid \alpha_{X\sigma} \geq 0 \\ &\alpha_{A\mu} \sim \mathcal{N}(2, 1) & \alpha_{A\sigma} \sim \mathcal{N}(1, 0.1) \mid \alpha_{Y\sigma} \geq 0 \\ &\alpha_{Y\mu} \sim \mathcal{N}(0, 0.01) & \alpha_{Y\sigma} \sim \mathcal{N}(0.1, 0.01) \mid \alpha_{Y\sigma} \geq 0 \\ &N_E \sim \mathcal{N}(1, 0.1) \end{aligned}$$
+
+This agent essentially has the correct beliefs about the patient data distribution, historical treatment policy, outcome variance and treatment effect. They only have some uncertainty represented by some variance on these parameters. In our experiments, we vary each of the colored prior parameters, one at a time, to simulate various ways in which the agent's expectations may be incorrect.
+
+The internal model for the agent's effect estimation. Finally, the internal model of the agent used for inference,  $A_{inf}$ , is defined as:
+
+$$\begin{split} N_X &\sim \mathcal{N}(\alpha_{X\mu}, \alpha_{X\sigma}) \quad N_A \sim \mathcal{N}(\alpha_{A\mu}, \alpha_{A\sigma}) \\ N_Y &\sim \mathcal{N}(\alpha_{Y\mu}, \alpha_{Y\sigma}) \\ X &:= N_X \\ A &:= 0.125 \ X + N_A \\ Y &:= 12 - 0.1 \ X + N_E \ A + N_Y \end{split}$$
+
+The posterior defines a probability distribution over SCMs. Recall that the agent was interested in choosing between dosage A=10 or A=20 for the specific patient with
+
+weight equal to  $X^o$ . To compute the desired quantity, the agent aggregates the CATE produced by every SCM according to the posterior
+
+$$C\hat{ATE} = \mathbb{E}_{P^{\mathcal{A}_{inf}}}[E^{\mathcal{A}_{inf}}(Y|X^o, do(A=20)) - E^{\mathcal{A}_{inf}}(Y|X^o, do(A=10))]$$
+
+where the expectation is over the joint posterior of the agent over the parameters, which was obtained updating  $A_{hist}$ .
+
+**Reaching a decision.** The agent will administer a dosage of 10 if the estimated CATE is below a threshold of 5 or a dosage of 20 otherwise. In other words, the higher dosage needs to improve the outcome by at least 5 months to be worth it. This completes the decision pipeline: given a patient  $X^o$  with a corresponding prediction and the type of agent, we can compute what the agent would do.
+
+#### 4. Results
+
+We begin by showcasing how our framework can capture the two motivating examples presented in the Introduction. In the first example, the belief of the agent on the past policy determines how the prediction is interpreted and what action is taken.<sup>5</sup> Indeed, in the simulation, when X = 80, with a bad prognosis under the assumption of no treatment  $(\alpha_{A\mu} = 0)$ , the agent chooses a high dosage to improve the outcome (A = 20). With a bad prognosis under the assumption of heavy treatment ( $\alpha_{A\mu}=20$ ), the agent chooses a low dosage to reduce over-treatment (A = 10). In the second motivating example, new information from the prediction forces an agent to consider different options. In the model, this means that either belief in treatment effect  $(N_E)$ or in historical policy  $(\mu_A)$  is revised, which appears as a correlation in the posterior belief in these two variables in Figure 4 in Appendix A.
+
+Next, for the use case described in Section 3, we report the effect of using ML-DS on two dimensions: 1) agent's belief of treatment effect and 2) downstream outcomes Y realized after the agent has taken an action, visualized in the top and bottom rows of Figure 3, respectively. In each of these plots, the outcomes obtained using ML-DS (orange boxplots) is contrasted with the outcomes obtained without it (blue boxplots), for different types of agents. The CATE plots sweep a wide range of options for the priors in order
+
+<span id="page-5-0"></span><sup>&</sup>lt;sup>4</sup>Note that for ease of exposition we introduced some variables  $(N_{A_i}, N_{Y_i})$  which are not present in Figure 2 (a). These can in principle be removed by specifying distributions directly on the inverse transformations of the ones given above for  $A_i$  and  $Y_i$ , and applying the appropriate Jacobian adjustment. In the PyMC implementation, we also use a small (1e-3) observational noise around  $Pred^o$  to allow us to sample from an observed deterministic transformation.
+
+<span id="page-5-1"></span><sup>&</sup>lt;sup>5</sup>All simulations presented in this paper are ran with PyMC v5 (Salvatier et al., 2016). We used the NUTS sampler to get 1000 samples for each of 4 chains, with 1000 tuning samples and a 'target\_accept' parameter of 0.85. Diagnostics and visual inspection of the traces showed good mixing and no convergence issues (Max  $\hat{R}=1.01$ , and only 16 out 4000 runs had > 5 divergences, due to fixed 'target\_accept' and extreme parameter values.). Sanity checks on the Bayesian model are reported in Appendix B.
+
+{6}------------------------------------------------
+
+to reveal interesting dynamics of the agent's beliefs; in Appendix [D](#page-10-3) we also report the CATE results for the more realistic range of ± 3 stds from the SCM reference value.
+
+330
+
+334
+
+336
+
+354
+
+356
+
+374
+
+376
+
+We begin with an agent whose priors beliefs coincide with the parameters of the historical SCM except for NE, i.e. treatment effect. Figure [3a](#page-7-0) and Figure [3d](#page-7-0) show what happens to CATE and Y when the agent's belief in N<sup>E</sup> varies. As expected, in the first figure we can see that the blue CATE has a slope that depends on the prior NE: without decision support, the belief on CATE is equal to N<sup>E</sup> multiplied by the difference in treatments. In the orange series however, we see an interesting behavior: after learning from the interaction with ML-DS, the agent corrects their belief in CATE re-aligning it to the correct value. In other words, *when all other beliefs are correct, the agent can learn from ML-DS to estimate better CATE* and take better actions. This is confirmed in Figure [3d](#page-7-0) where the orange series shows superior outcomes.
+
+Figure [3b](#page-7-0) and Figure [3e](#page-7-0) show what happens when the agent's prior belief in the treatment protocol, µA, varies. In this scenario the prior CATE estimates are flat, for the simple reason that the agent has a fixed belief in N<sup>E</sup> and they are not updating it. When interacting with ML-DS however, the belief update changes CATE estimates quite a lot; this is because the agent is attempting to explain the gap between P red<sup>o</sup> given by ML-DS and the expected outcome they had from their prior beliefs. This reflects on outcomes in [3e](#page-7-0) where the orange series is sometimes below the blue one: incorrect beliefs in µ<sup>A</sup> can result in CATE estimates that lead to wrong actions and worsen outcome Y.
+
+A similar pattern is observed in Figures [3c,](#page-7-0) [3f,](#page-7-0) [6c](#page-12-0) (last one in Appendix [D\)](#page-10-3): while prior CATE and Y estimates are relatively stable without ML-DS, new dynamics emerge when when ML-DS is introduced. For CATE, incorrect beliefs can have large effects both on the magnitude and sign. For Y , in both cases incorrect beliefs can lead to bad actions and worse outcomes.
+
+#### 5. Related work
+
+A large body of literature is concerned with the analysis of ML-DS from different angles. A strand of literature focuses on how ML-DS is entangled with (changes in) the data generating process. Works in this category study limitations of what can be learned from data generated from past policies, e.g. [\(Lakkaraju et al.,](#page-8-10) [2017\)](#page-8-10), or the dynamics that occur with alternating cycles of ML-DS deployment and re-training, as in the performative prediction literature [\(Perdomo et al.,](#page-8-11) [2020;](#page-8-11) [Boeken et al.,](#page-8-12) [2024\)](#page-8-12). Another batch of articles is concerned with evaluation of ML-DS, laying out identifiability conditions for the effects of ML-DS on decisions [\(Imai et al.,](#page-8-13) [2023\)](#page-8-13) or trying to optimize human-machine complementarity by design [\(Donahue et al.,](#page-8-14) [2022;](#page-8-14) [McLaughlin & Spiess,](#page-8-15) [2024\)](#page-8-15). Finally, a different set of papers investigate what human input can add to artificial agents already reaching some level of optimality [\(Alur et al.,](#page-8-16) [2024;](#page-8-16) [Zhang & Barein](#page-9-0)[boim,](#page-9-0) [2022;](#page-9-0) [Stensrud et al.,](#page-8-17) [2024\)](#page-8-17). The key difference of our work is that we offer a more fine-grained description of what happens inside the decision maker, namely we unpack the micro-mechanism leading a decision maker to take an action on a single observation paired with AI prediction.
+
+Other papers have studied how information flows between agents influencing their actions. In [\(Kamenica &](#page-8-18) [Gentzkow,](#page-8-18) [2011\)](#page-8-18) a Sender of information influences a Receiver Bayesian agent in order to stir their action. What we add here is that in our context the signal is an ML prediction produced from training data drawn from the same environment the agent reasons about; this coupling yields the distinctive inference problem we study, i.e. updating through latent interchangeable training variables.
+
+Another important strand of work focuses on problems related to inferring information about a population P given predictions from a model trained on samples from P in the context of confidentiality attacks. This literature studies whether it is possible to use models' predictions (and other behavior) to recover the presence of a specific data point, in so-called *membership inference attacks* [\(Hu et al.,](#page-8-19) [2022;](#page-8-19) [Huang,](#page-8-20) [2025\)](#page-8-20)), or to infer statistical properties of a training datasets, so-called *property-inference attacks* [\(Ate](#page-8-21)[niese et al.,](#page-8-21) [2015;](#page-8-21) [Wang et al.,](#page-9-1) [2019\)](#page-9-1). While these models aim to recover information about the training set, e.g., the proportion of data in each label, our challenge is to marginalize across datasets and directly infer information about the underlying population.
+
+#### 6. Discussion
+
+We have presented the first explicit framework that includes an ML-DS system and a sophisticated user. Our 2-Step Agent framework captures the effect of an agent's prior beliefs, modeling learning effects due to the interaction and tracking agent's uncertainty. The framework allows for simulations comparing outcomes with and without ML-DS, in an RCT-like fashion, therefore quantifying the benefit/harm of ML-DS for different types of simulated agents.
+
+We provided an implementation of the framework for ML-DS with a linear model. Said implementation leverages a collapse of the plate via sufficient statistics (Appendix [E.1\)](#page-13-0), greatly speeding up the simulations. Within this implementation, we have shown that incorrect beliefs about the distribution of the predictive model's training data may lead the agent to bad decisions, resulting in harmful outcomes.
+
+While previous literature had argued that incorrect beliefs about past treatment policy (µ<sup>A</sup> in our notation) might gener-
+
+{7}------------------------------------------------
+
+<span id="page-7-0"></span>![](_page_7_Figure_1.jpeg)
+
+*Figure 3.* Figures displaying the effects of different agent priors when using ML-DS (in orange) and without it (in blue). In all plots except from NE, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameters. The top row displays the agent's prior and posterior (predictive) belief of CATE, while the bottom row shows the impact of the agent's decisions on the downstream outcome Y . The left column pertains to the belief on treatment effect, NE, the central column the belief about past treatment policy (µA) and the right column about the belief on past covariate distribution (µX). The gray areas represents the portions of the belief in which the true CATE is within one std from the CATE learned from ML-DS (orange band).
+
+ate worse downstream outcomes, e.g. in [\(van Geloven et al.,](#page-8-6) [2025\)](#page-8-6), this is the first quantitative estimation of harmful effects of ML-DS. In our experiments we show that ML-DS can be harmful if the agent has wrong beliefs not only about the past treatment policy, but also about the distribution of covariates and the distribution of the outcome.
+
+394
+
+396
+
+We also show that if the agent has otherwise correct beliefs, an incorrect belief about treatment effect may be corrected during updates that push the agent towards the true CATE. This suggests that in certain settings ML-DS building on a treatment-naive prediction model might still be beneficial.
+
+This finding highlights the extreme sensitivity of ML-DS to agent prior beliefs when using a treatment-naive prediction model, and underscore the importance of providing full documentation and proper training to users of ML-DS. Finally, we remark that these results apply to *any* field deploying such systems and to any agent interacting with ML-DS, be it human or artificial.
+
+Limitations and Future work The experiments in this paper explored only part of the potential of the framework. The experiments are limited to a use case in which the prediction model is linear and treatment-naive, and the data
+
+generating process is simple (e.g., every distribution is normal, there is only one confounder). Implementing the use case was not trivial and required pushing a Bayesian update through OLS and a large amount of variables (see Appendix [E.1\)](#page-13-0), hence it remains to be seen whether similar results can be obtained when the complexity of the use case increases.
+
+We assume the agent reasons in a Bayesian fashion, which allows for a precise analysis of the agent's uncertainty. While this agent may not be realistic with respect to human behavior, it is useful to make this assumption to demonstrate what can go wrong in a best-case scenario, i.e., when the agent is a 'perfect' reasoner and the ML model has no flaws. Finally, this assumption is also interesting when considering how an artificial agent might learn from predictions of other ML models.
+
+In future work this line of research will be expanded across different dimensions, including i) different types of agents (e.g. agent with incorrect internal model of the world of different kinds of uncertainties) ii) more complex ML models for ML-DS, iii) cases with heterogeneity of treatment effect (i.e. the effect of the action is not the same for all instances), iv) repeated rounds of learning for the same agent and v) using causal models for ML-DS.
+
+{8}------------------------------------------------
+
+#### References
+
+- <span id="page-8-16"></span>Alur, R., Raghavan, M., and Shah, D. Human expertise in algorithmic prediction. *Advances in Neural Information Processing Systems*, 37:138088–138129, 2024.
+- <span id="page-8-21"></span>Ateniese, G., Mancini, L. V., Spognardi, A., Villani, A., Vitali, D., and Felici, G. Hacking smart machines with smarter ones: How to extract meaningful data from machine learning classifiers. *International Journal of Security and Networks*, 10(3):137–150, September 2015. ISSN 1747-8405. doi: 10.1504/IJSN.2015.071829.
+- <span id="page-8-12"></span>Boeken, P., Zoeter, O., and Mooij, J. Evaluating and correcting performative effects of decision support systems via causal domain shift. In *Causal Learning and Reasoning*, pp. 551–569. PMLR, 2024.
+- <span id="page-8-8"></span>Borm, G. F., Melis, R. J., Teerenstra, S., and Peer, P. G. Pseudo cluster randomization: a treatment allocation method to minimize contamination and selection bias. *Statistics in Medicine*, 24(23):3535–3547, 2005.
+- <span id="page-8-14"></span>Donahue, K., Chouldechova, A., and Kenthapadi, K. Human-algorithm collaboration: Achieving complementarity and avoiding unfairness. In *Proceedings of the 2022 ACM Conference on Fairness, Accountability, and Transparency*, pp. 1639–1656, 2022.
+- <span id="page-8-1"></span>European Parliament and Council of the European Union. Regulation (eu) 2024/1689 of the european parliament and of the council laying down harmonised rules on artificial intelligence (artificial intelligence act), 2024. URL [https://artificialintelligenceact.](https://artificialintelligenceact.eu/article/14/) [eu/article/14/](https://artificialintelligenceact.eu/article/14/). Published 12 July 2024; Article 14: Human oversight.
+- <span id="page-8-19"></span>Hu, H., Salcic, Z., Sun, L., Dobbie, G., Yu, P. S., and Zhang, X. Membership Inference Attacks on Machine Learning: A Survey. *Acm Computing Surveys*, 54(11s): 235:1–235:37, September 2022. ISSN 0360-0300. doi: 10.1145/3523273.
+- <span id="page-8-20"></span>Huang, Y. Bayesian Inference of Training Dataset Membership. May 2025.
+- <span id="page-8-13"></span>Imai, K., Jiang, Z., Greiner, D. J., Halen, R., and Shin, S. Experimental evaluation of algorithm-assisted human decision-making: Application to pretrial public safety assessment. *Journal of the Royal Statistical Society Series A: Statistics in Society*, 186(2):167–189, 2023.
+- <span id="page-8-18"></span>Kamenica, E. and Gentzkow, M. Bayesian persuasion. *American Economic Review*, 101(6):2590–2615, 2011.
+- <span id="page-8-3"></span>Kleinberg, J., Lakkaraju, H., Leskovec, J., Ludwig, J., and Mullainathan, S. Human decisions and machine predictions. *The quarterly journal of economics*, 133(1): 237–293, 2018.
+
+- <span id="page-8-4"></span>Kube, A., Das, S., and Fowler, P. J. Allocating interventions based on predicted outcomes: A case study on homelessness services. In *Proceedings of the AAAI Conference on Artificial Intelligence*, volume 33, pp. 622–629, 2019.
+- <span id="page-8-10"></span>Lakkaraju, H., Kleinberg, J., Leskovec, J., Ludwig, J., and Mullainathan, S. The selective labels problem: Evaluating algorithmic predictions in the presence of unobservables. In *Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining*, pp. 275–284, 2017.
+- <span id="page-8-15"></span>McLaughlin, B. and Spiess, J. Designing algorithmic recommendations to achieve human-ai complementarity. *arXiv preprint arXiv:2405.01484*, 2024.
+- <span id="page-8-5"></span>Obermeyer, Z., Powers, B., Vogeli, C., and Mullainathan, S. Dissecting racial bias in an algorithm used to manage the health of populations. *Science*, 366(6464):447–453, 2019.
+- <span id="page-8-11"></span>Perdomo, J., Zrnic, T., Mendler-Dunner, C., and Hardt, M. ¨ Performative prediction. In *International Conference on Machine Learning*, pp. 7599–7609. PMLR, 2020.
+- <span id="page-8-7"></span>Perepletchikova, F., Treat, T. A., and Kazdin, A. E. Treatment integrity in psychotherapy research: analysis of the studies and examination of the associated factors. *Journal of consulting and clinical psychology*, 75(6):829, 2007.
+- <span id="page-8-9"></span>Salvatier, J., Wiecki, T. V., and Fonnesbeck, C. Probabilistic programming in Python using PyMC3. *PeerJ Computer Science*, 2:e55, 2016. ISSN 2376-5992. doi: 10.7717/ peerj-cs.55.
+- <span id="page-8-2"></span>Smith, V. C., Lange, A., and Huston, D. R. Predictive modeling to forecast student outcomes and drive effective interventions in online community college courses. *Journal of asynchronous learning networks*, 16(3):51–61, 2012.
+- <span id="page-8-17"></span>Stensrud, M. J., Laurendeau, J., and Sarvet, A. L. Optimal regimes for algorithm-assisted human decision-making. *Biometrika*, 111(4):1089–1108, 2024.
+- <span id="page-8-6"></span>van Geloven, N., Keogh, R. H., van Amsterdam, W., Cina,` G., Krijthe, J. H., Peek, N., Luijken, K., Magliacane, S., Morzywołek, P., van Ommen, T., et al. The risks of risk assessment: causal blind spots when using prediction models for treatment decisions. *Annals of internal medicine*, 178(9):1326–1333, 2025.
+- <span id="page-8-0"></span>Wang, A., Kapoor, S., Barocas, S., and Narayanan, A. Against predictive optimization: On the legitimacy of decision-making algorithms that optimize predictive accuracy. *ACM J. Responsib. Comput.*, 1(1):9:1–9:45, March 2024. doi: 10.1145/3636509.
+
+{9}------------------------------------------------
+
+Wang, L., Xu, S., Wang, X., and Zhu, Q. Eavesdrop the Composition Proportion of Training Labels in Federated Learning. October 2019.
+
+<span id="page-9-1"></span>
+
+> <span id="page-9-0"></span>Zhang, J. and Bareinboim, E. Can humans be out of the loop? In *Conference on Causal Learning and Reasoning*, pp. 1010–1025. PMLR, 2022.
+
+{10}------------------------------------------------
+
+#### <span id="page-10-1"></span>A. Motivating example 2
+
+550
+
+554
+
+<span id="page-10-0"></span>556
+
+574
+
+576
+
+594
+
+596
+
+In this section we report the posterior correlation plot obtained from an agent that learned from ML-DS as described in the second motivating example in Section [1.](#page-0-1) After observing a prediction that did not match expectation, the agent revises their beliefs, entertaining multiple hypotheses on how their beliefs should change to account for the situation. In this case, either the treatment effect is lower than anticipated and the dosage higher than assumed, or the other way around.
+
+![](_page_10_Figure_3.jpeg)
+
+*Figure 4.* Plot showing the correlation of variables N<sup>E</sup> and µ<sup>A</sup> in the posterior beliefs of the agent after update with ML-DS information. The agent entertains multiple options
+
+### <span id="page-10-2"></span>B. Sanity checks on the Bayesian model
+
+We have implemented a series of experiments to make sure that the framework is functioning as intended.
+
+- 1. We compare the posterior distributions with the explicit plate and with the sufficient statistics formulation given the same observations. We check if the posteriors of the two agents end up the same. This is repeated for different sizes of plate to control for potential effects of size.
+- 2. We run an update with a single observation X<sup>o</sup> = 200 that is much higher than the prior mean of 80 (with variance 10). We verify that after the update the posterior of µ<sup>X</sup> has moved in the right direction. We also check if this creates correlations of µ<sup>X</sup> with other parameters. This is repeated for different plate sizes.
+- 3. We run an update with many (1000) observations drawn from the real distribution of X and check that the (incorrect) prior of the agent about µ<sup>X</sup> converges to the real parameter in the generating process.
+- 4. We set one observation with X equal to the mean of the agent's prior, with the priors on A and N<sup>E</sup> very peaked, but with P red<sup>o</sup> set much higher than the value Y = coeff ∗ X + N<sup>E</sup> ∗ A. We also setthe coeff of X in the structural equation of A to zero. We observe in the posterior that there is now a correlation between the parameters of N<sup>E</sup> and A.
+
+### C. Results obtained when varying prior µ<sup>Y</sup>
+
+In Figure [5](#page-11-0) we report the CATE and Y plots obtained when varying the belief of the agent concerning the mean of the outcome in the historical population, µ<sup>Y</sup> .
+
+#### <span id="page-10-3"></span>D. Zoomed-in versions of the CATE plots
+
+The plots in Figure [3](#page-7-0) encompass a wide sweep of options for the priors of the agent, which revealed interesting dynamics of CATE with ML-DS. In Figure [6](#page-12-0) we 'zoom-in' on what happens in a more realistic range of priors within ± 3 std from the true SCM parameter.
+
+{11}------------------------------------------------
+
+<span id="page-11-0"></span>![](_page_11_Figure_1.jpeg)
+
+![](_page_11_Figure_2.jpeg)
+
+*Figure 5.* Figures displaying the effects of different agent's priors when using ML-DS (in orange) and without it (in blue). In all plots, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameter for µ<sup>Y</sup> . The left plot displays the agent's prior and posterior (predictive) belief of CATE, while the right one shows the impact of the agent's decisions on the downstream outcome Y . In a rather wide band is the posterior within 1 std of the true mean (gray area).
+
+{12}------------------------------------------------
+
+<span id="page-12-0"></span>![](_page_12_Figure_1.jpeg)
+
+*Figure 6.* Figures displaying the effects of different agent's priors on the posterior (predictive) belief of CATE when using ML-DS (in orange) and without it (in blue). In all plots except those for NE, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameters. The plots pertain to the belief on treatment effect, NE, the belief about past treatment policy (µA), the belief about past outcomes (µ<sup>Y</sup> ) and the belief about past covariate distribution (µX). The gray areas represents the portions of the belief in which the true CATE is within one std from the CATE learned from ML-DS (orange band.
+
+{13}------------------------------------------------
+
+#### E. Derivations for plate model
+
+We elaborate here on the strategy to reduce the plate with the help of sufficient statistics
+
+716
+
+718 719 720
+
+724
+
+726
+
+734
+
+736
+
+754
+
+756
+
+#### <span id="page-13-0"></span>E.1. Plate model, agent's internal model
+
+In this section we will show how an observation of the model parameter ϕ from the agent's internal model can be generated. This is needed to estimate the posterior distributions of the agent's beliefs using a sampling approach. As running inference through a plate-model can be both computationally expensive and numerically unstable, we chose to reformulate the plate using low-dimensional auxiliary variables.
+
+We start from the SCM (fig. [2\)](#page-3-1) that defines the relation between the variables and we are interested in finding the lowdimensional auxiliary variables that define the regression coefficient ϕ of the model M. As we are using PyMC for inference – which does not have an implementation for the non-central χ 2 -distribution – all random variables (RVs) will be reformulated using zero-centered normally distributed RVs (standard normals where needed) and (central) χ <sup>2</sup> distributed RVs. Some of these RVs will be dependent and that dependence will be dealt with in section [E.2.](#page-15-0)
+
+We are specifically looking for a low-dimensional representation of the plate and treat a sample of the plate parameters {αXµ, αXσ, αAµ, αAσ, αY µ, αY σ, NE} as given. The relevant parts of the SCM, reproduced here for the reader's convenience, are defined as
+
+$$Y_{i} = a + bX_{i} + N_{E}A_{i} + N_{Y_{i}}$$
+
+$$A_{i} = dX_{i} + N_{A_{i}}$$
+
+$$X_{i} = N_{X_{i}}$$
+
+$$\forall i \in [1, ..., n]$$
+
+$$(1)$$
+
+where n is the plate size, a, b, and d are constants and
+
+<span id="page-13-4"></span>
+$$N_{X_i} \sim \mathcal{N}(\alpha_{X\mu}, \alpha_{X\sigma}^2)$$
+
+$$N_{A_i} \sim \mathcal{N}(\alpha_{A\mu}, \alpha_{A\sigma}^2)$$
+
+$$N_{Y_i} \sim \mathcal{N}(\alpha_{Y\mu}, \alpha_{Y\sigma}^2).$$
+(2)
+
+We are modeling Y<sup>i</sup> without intercept as
+
+<span id="page-13-3"></span>
+$$Y_i \approx \phi X_i \tag{3}$$
+
+and looking for the ordinary least-squares estimate of ϕ, that is
+
+$$\min_{\phi} \left( \sum_{i} (Y_i - \phi X_i)^2 \right). \tag{4}$$
+
+The least-squares estimate for this case has a well-known closed-form solution given by
+
+<span id="page-13-5"></span><span id="page-13-1"></span>
+$$\phi = \frac{\sum_{i} X_i Y_i}{\sum_{i} X_i^2}.$$
+ (5)
+
+Our goal is to express this equation in terms of RVs that summarize the sums thus removing the need to sample the full plate n times. The denominator of this equation can be expressed as
+
+$$\sum_{i} X_i^2 = \sum_{i} N_{X_i}^2 \tag{6}$$
+
+⇔
+
+$$\sum_{i} X_i^2 = \sum_{i} \alpha_{X\mu}^2 + 2\alpha_{X\mu}\alpha_{X\sigma}\epsilon_{X_i} + \alpha_{X\sigma}^2\epsilon_{X_i}^2$$
+ (7)
+
+where
+
+<span id="page-13-2"></span>
+$$\epsilon_{X_i} \sim \mathcal{N}(0, 1).$$
+ (8)
+
+{14}------------------------------------------------
+
+The denominator of (5) rewritten in (7) consists of three parts:
+
+$$S_1 = \sum_{i} \alpha_{X\mu}^2 \tag{9}$$
+
+$$S_2 = \sum_{i} 2\alpha_{X\mu}\alpha_{X\sigma}\epsilon_{X_i} \tag{10}$$
+
+$$S_3 = \sum_i \alpha_{X\sigma}^2 \epsilon_{X_i}^2 \tag{11}$$
+
+where
+
+$$S_1 = n\alpha_{X\mu}^2 \tag{12}$$
+
+$$S_2 = 2\alpha_{X\mu}\alpha_{X\sigma} \sum_{i} \epsilon_{X_i} \tag{13}$$
+
+$$S_3 = \alpha_{X\sigma}^2 \sum_i \epsilon_{X_i}^2. \tag{14}$$
+
+Given one set of parameters for the equations in (2), we see that  $S_1$  is a constant,  $S_2 = 2\alpha_{X\mu}\alpha_{X\sigma}S_X$  where  $S_X \sim \mathcal{N}(0,n)$ , and  $S_3 \sim \alpha_{X\sigma}^2 Z_{S_3}$  where  $Z_{S_3} \sim \chi^2(n)$ . Here  $S_2$  is the first low-dimensional auxiliary variable we need. Note that  $S_3$  is **dependent on**  $S_2$  and this has to be dealt with separately (see section E.2).
+
+Next we decompose the numerator of equation (5) in a similar way. First we replace the RVs by the definitions from (1)
+
+$$\sum_{i} X_{i} Y_{i} = \sum_{i} \left[ a N_{X_{i}} + b N_{X_{i}}^{2} + d N_{E} N_{X_{i}}^{2} + N_{E} N_{A_{i}} N_{X_{i}} + N_{Y_{i}} N_{X_{i}} \right].$$
+ (15)
+
+As we are actually interested in equation (5), we can divide this expression by  $\sum_i N_{X_i}^2$  (equation (6)) to cancel out some factors and get
+
+$$\phi = b + dN_E + \frac{a\sum_{i} N_{X_i} + N_E \sum_{i} N_{X_i} N_{A_i} + \sum_{i} N_{X_i} N_{Y_i}}{\sum_{i} X_i^2}$$
+(16)
+
+Remember that a, b, and d are constants. The denominator of the last term is equivalent to equation (7). We currently lack suitable expressions only for the numerator of the last term. The numerator consists of the three parts
+
+$$S_4 = a \sum_i N_{X_i} \tag{17}$$
+
+$$S_5 = N_E \sum_{i} N_{X_i} N_{A_i} \tag{18}$$
+
+$$S_6 = \sum_i N_{X_i} N_{Y_i}. \tag{19}$$
+
+ $S_4$  can be expressed similarly to  $S_2$  using  $S_X$ , i.e.  $S_4 = an\alpha_{X\mu} + a\alpha_{X\sigma}S_X$ . Both  $S_5$  and  $S_6$  are RVs following normal product distributions. We decompose  $S_6$  as follows:
+
+$$S_6 = \sum_{i} \left[ (\alpha_{X\mu} + \alpha_{X\sigma} \epsilon_{X_i})(\alpha_{Y\mu} + \alpha_{Y\sigma} \epsilon_{Y_i}) \right]$$
+ (20)
+
+$$S_6 = \sum_{i} \left[ \alpha_{X\mu} \alpha_{Y\mu} + \alpha_{X\mu} \alpha_{Y\sigma} \epsilon_{Y_i} + \alpha_{Y\mu} \alpha_{X\sigma} \epsilon_{X_i} + \alpha_{X\sigma} \alpha_{Y\sigma} \epsilon_{X_i} \epsilon_{Y_i} \right]$$
+ (21)
+
+ $\Leftrightarrow$ 
+
+$$S_6 = n\alpha_{X\mu}\alpha_{Y\mu} + \alpha_{X\mu}\alpha_{Y\sigma}S_Y + \alpha_{Y\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{Y\sigma}\sum_i \epsilon_{X_i}\epsilon_{Y_i}$$
+ (22)
+
+where  $\epsilon_{Y_i} \sim \mathcal{N}(0,1)$  and  $S_Y \sim \mathcal{N}(0,n)$ . Note that  $S_X$  and  $S_Y$  are independent. We know that  $\sum_i \epsilon_{X_i} \epsilon_{Y_i}$  follows a normal product distribution that is dependent on both  $S_X$  and  $S_Y$  and deal with this dependence in section E.2 below.
+
+{15}------------------------------------------------
+
+Following an almost identical derivation, we express
+
+$$S_5 = N_E \left[ n\alpha_{X\mu}\alpha_{A\mu} + \alpha_{X\mu}\alpha_{A\sigma}S_A + \alpha_{A\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{A\sigma}\sum_i \epsilon_{X_i}\epsilon_{A_i} \right]$$
+ (23)
+
+where 
+
+E.2. Separating dependent and independent parts 
+
+It can be shown that
+
+and sums of squares.
+
+It can be shown that
+
+where  $U_{XA} \sim \chi^2(n-1)$  and  $V_{XA} \sim \chi^2(n-1)$ .
+
+ $\epsilon_{A_i} \sim \mathcal{N}(0,1)$ 
+
+$$\epsilon_{A_i} \sim \mathcal{N}(0, 1)$$
+ (24)  
+ $S_A \sim \mathcal{N}(0, n)$ . (25)
+
+# (24)
+
+# <span id="page-15-0"></span>Cochran's theorem tells us that we can in certain situations separate the dependent and independent parts of sums of products
+
+# <span id="page-15-1"></span> $S_7 = \sum_{i} \epsilon_{X_i}^2 = \sum_{i} (\epsilon_{X_i} - \bar{\epsilon_X})^2 - \frac{(\sum_{i} \epsilon_{X_i})^2}{n}$ (26)
+
+where  $\epsilon_{X_i} \sim \mathcal{N}(0,1)$  and  $\epsilon_X^-$  is the sample average, and that  $\sum_i (\epsilon_{X_i} - \epsilon_X^-)^2$  is independent of  $\frac{(\sum_i \epsilon_{X_i})^2}{n}$ . The first term is a sum of squares of a normal and follows a  $\chi^2$ -distribution with n-1 degrees of freedom, i.e.  $\sum_i (\epsilon_{X_i} - \epsilon_X^-)^2 = Z_{XX}$  where  $Z_{XX} \sim \chi^2(n-1)$ . The second term is the square of the sum, where the sum is over an RV that is normally distributed. This second term in equation (26) is  $\frac{\left(\sum_{i} \epsilon_{X_{i}}\right)^{2}}{n} = \frac{S_{X}^{2}}{n}$ .
+
+## E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS
+
+E.2.1. DEPENDENCE OF SUM OF SQUARES AND (SQUARE OF) SUM
+
+<span id="page-15-2"></span>
+$$S_8 = \sum_i \epsilon_{X_i} \epsilon_{Y_i} = n \bar{\epsilon_X} \bar{\epsilon_Y} + \sum_i (\epsilon_{X_i} - \bar{\epsilon_X}) (\epsilon_{Y_i} - \bar{\epsilon_Y})$$
+ (27)
+
+where bar denotes sample average so that  $\bar{\epsilon_X} = \frac{\sum_i \epsilon_{X_i}}{n}$  and  $\bar{\epsilon_Y} = \frac{\sum_i \epsilon_{Y_i}}{n}$ .
+
+The first term of equation (27) can be written with familiar quantities as  $\bar{\epsilon_X} = \frac{S_X}{n}$  and  $\bar{\epsilon_Y} = \frac{S_Y}{n}$  resulting in  $n\bar{\epsilon_X}\bar{\epsilon_Y} = \frac{S_XS_Y}{n}$ . The second part requires more deliberation.  $\sum_i (\epsilon_{X_i} - \bar{\epsilon_X})(\epsilon_{Y_i} - \bar{\epsilon_Y})$  is clearly a sum of an RV that follows a normal product distribution. To facilitate sampling, we rewrite this as
+
+$$\sum (\epsilon_{X_i} - \bar{\epsilon_X})(\epsilon_{Y_i} - \bar{\epsilon_Y}) = \frac{1}{4} \sum (\epsilon_{X_i} - \bar{\epsilon_X} + \epsilon_{Y_i} - \bar{\epsilon_Y})^2 + \frac{1}{4} \sum (\epsilon_{X_i} - \bar{\epsilon_X} - \epsilon_{Y_i} + \bar{\epsilon_Y})^2.$$
+ (28)
+
+Both terms (sums of squares) in the right-hand side of this equation are sums of squared normals which are  $\chi^2$ -distributed with n-1 degrees of freedom. Hence
+
+$$\sum (\epsilon_{X_i} - \bar{\epsilon_X})(\epsilon_{Y_i} - \bar{\epsilon_Y}) = \frac{1}{4}(2U_{XY} - 2V_{XY})$$
+(29)
+
+where  $U_{XY} \sim \chi^2(n-1)$  and  $V_{XY} \sim \chi^2(n-1)$  and these are independent.
+
+Following the same pattern, it can be shown that
+
+$$S_9 = \sum_{i} \epsilon_{X_i} \epsilon_{A_i} = \frac{S_X S_A}{n} + \frac{1}{2} (U_{XA} - V_{XA})$$
+ (30)
+
+{16}------------------------------------------------
+
+#### E.3. Putting it all together
+
+To generate one observation of ϕ from the agent's internal model, sample αXµ, αXσ, αY µ, αY σ, αAµ, αAσ, and NE. Then sample one observation for
+
+$$S_X \sim \mathcal{N}(0, n) \tag{31}$$
+
+$$S_Y \sim \mathcal{N}(0, n) \tag{32}$$
+
+$$S_A \sim \mathcal{N}(0, n) \tag{33}$$
+
+$$Z_{XX} \sim \chi^2(n-1) \tag{34}$$
+
+$$U_{XY} \sim \chi^2(n-1) \tag{35}$$
+
+$$V_{XY} \sim \chi^2(n-1) \tag{36}$$
+
+$$U_{XA} \sim \chi^2(n-1) \tag{37}$$
+
+$$V_{XA} \sim \chi^2(n-1). \tag{38}$$
+
+Insert the sampled values to calculate
+
+$$S_9 = \frac{S_X S_A}{n} + \frac{1}{2} (U_{XA} - V_{XA}) \tag{39}$$
+
+$$S_8 = \frac{S_X S_Y}{n} + \frac{1}{2} (U_{XY} - V_{XY}) \tag{40}$$
+
+$$S_7 = Z_{XX} - \frac{S_X^2}{n} \tag{41}$$
+
+$$S_6 = n\alpha_{X\mu}\alpha_{Y\mu} + \alpha_{X\mu}\alpha_{Y\sigma}S_Y + \alpha_{Y\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{Y\sigma}S_8 \tag{42}$$
+
+$$S_5 = N_E \left( n\alpha_{X\mu}\alpha_{A\mu} + \alpha_{X\mu}\alpha_{A\sigma}S_A + \alpha_{A\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{A\sigma}S_9 \right) \tag{43}$$
+
+$$S_4 = an\alpha_{X\mu} + a\alpha_{X\sigma}S_X \tag{44}$$
+
+$$S_3 = \alpha_{X\sigma}^2 S_7 \tag{45}$$
+
+$$S_2 = 2\alpha_{X\mu}\alpha_{X\sigma}S_X \tag{46}$$
+
+$$S_1 = n\alpha_{X\mu}^2. (47)$$
+
+With these, we can finally generate one observation from the distribution of ϕ:
+
+$$\phi = b + dN_E + \frac{S_4 + S_5 + S_6}{S_1 + S_2 + S_3}. (48)$$

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+Kube, A., Das, S., and Fowler, P. J. Allocating interventions based on predicted outcomes: A case study on homelessness services. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 33, pp. 622–629, 2019. Lakkaraju, H., Kleinberg, J., Leskovec, J., Ludwig, J., and Mullainathan, S. The selective labels problem: Evaluating algorithmic predictions in the presence of unobservables. In Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining , pp. 275–284, 2017. McLaughlin, B. and Spiess, J. Designing algorithmic recommendations to achieve human-ai complementarity. arXiv preprint arXiv:2405.01484 , 2024. Obermeyer, Z., Powers, B., Vogeli, C., and Mullainathan, S. Dissecting racial bias in an algorithm used to manage the health of populations. Science , 366(6464):447–453, 2019. Perdomo, J., Zrnic, T., Mendler-Dunner, C., and Hardt, M. ¨ Performative prediction. In International Conference on Machine Learning , pp. 7599–7609. PMLR, 2020. Perepletchikova, F., Treat, T. A., and Kazdin, A. E. Treatment integrity in psychotherapy research: analysis of the studies and examination of the associated factors. Journal of consulting and clinical psychology , 75(6):829, 2007. Salvatier, J., Wiecki, T. V., and Fonnesbeck, C. Probabilistic programming in Python using PyMC3. PeerJ Computer Science , 2:e55, 2016. ISSN 2376-5992. doi: 10.7717/ peerj-cs.55. Smith, V. C., Lange, A., and Huston, D. R. Predictive modeling to forecast student outcomes and drive effective interventions in online community college courses. Journal of asynchronous learning networks , 16(3):51–61, 2012. Stensrud, M. J., Laurendeau, J., and Sarvet, A. L. Optimal regimes for algorithm-assisted human decision-making. Biometrika , 111(4):1089–1108, 2024. van Geloven, N., Keogh, R. H., van Amsterdam, W., Cina,` G., Krijthe, J. H., Peek, N., Luijken, K., Magliacane, S., Morzywołek, P., van Ommen, T., et al. The risks of risk assessment: causal blind spots when using prediction models for treatment decisions. Annals of internal medicine , 178(9):1326–1333, 2025. Wang, A., Kapoor, S., Barocas, S., and Narayanan, A. Against predictive optimization: On the legitimacy of decision-making algorithms that optimize predictive accuracy. ACM J. Responsib. Comput. , 1(1):9:1–9:45, March 2024. doi: 10.1145/3636509.
+
+[p. 10 | section: References | type: Text]
+Wang, L., Xu, S., Wang, X., and Zhu, Q. Eavesdrop the Composition Proportion of Training Labels in Federated Learning. October 2019.
+
+[p. 10 | section: References | type: Text]
+Zhang, J. and Bareinboim, E. Can humans be out of the loop? In Conference on Causal Learning and Reasoning , pp. 1010–1025. PMLR, 2022.

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+      "ACKNOWLEDGMENT": [],
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+      "\"accept_label\"": [],
+      "\"decision\"": [],
+      "\"decision_tier\"": [],
+      "\"source_status\"": [],
+      "Meta-review": [],
+      "Official Review": [],
+      "official_reviews": [],
+      "meta_reviews": [],
+      "suggested_verdict_score": []
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+  "default_model_input": "model_text_v3.txt",
+  "appendix_input": "appendix_text_v3.txt",
+  "reference_input": "reference_text_v3.txt"
+}

icml26/a3c6aa1c-cfec-4174-aab8-a4cb5af0d892/sanitized_v3.txt

@@ -0,0 +1,309 @@
+{0}
+### Abstract
+Across a growing number of fields, human decision making is supported by predictions from AI models. However, we still lack a deep understanding of the effects of adoption of these technologies. In this paper, we introduce a general computational framework, the 2-Step Agent, which models the effects of AI-assisted decision making. Our framework uses Bayesian methods for causal inference to model 1) how a prediction on a new observation affects the beliefs of a rational Bayesian agent, and 2) how this change in beliefs affects the downstream decision and subsequent outcome. Using this framework, we show by simulations how a single misaligned prior belief can be sufficient for decision support to result in worse downstream outcomes compared to no decision support. Our results reveal several potential pitfalls of AI-driven decision support and highlight the need for thorough model documentation and proper user training.
+#### <span id="page-0-1"></span>1. Introduction
+Background. Prediction models are often used to support decision making: a model trained on past data produces a forecast for a future outcome of interest, which a agent can take into account when deciding how to act. For the rest of this article, we will refer to such decision support as ML-DS (Machine Learning Decision Support) to emphasize the fact that the model has been trained on data. This is sometimes also called 'predictive optimization' [\(Wang et al.,](#page-8-0) [2024\)](#page-8-0). Contrary to full automation, in high-risk settings ML-DS is sometimes perceived as a way to leverage the power of machine learning while retaining human oversight (see for instance Article 14 of the EU AI Act [\(European Parliament](#page-8-1) [& Council of the European Union,](#page-8-1) [2024\)](#page-8-1)).
+Understanding the practical impact of adopting ML-DS in a domain is essential. First, ML-DS is on the rise, and currently deployed in a variety of fields including healthcare, education [\(Smith et al.,](#page-8-2) [2012\)](#page-8-2), policing and judicial system [\(Kleinberg et al.,](#page-8-3) [2018\)](#page-8-3), as well as redistribution of welfare [\(Kube et al.,](#page-8-4) [2019\)](#page-8-4). Second, allowing machines to influence decision making carries important risks, e.g. the incorporation of biased advice [\(Obermeyer et al.,](#page-8-5) [2019\)](#page-8-5).
+In this paper, we present a full-fledged computational models of ML-DS, the *2-Step Agent*. While most research on effects of ML-DS focuses on the model, we unpack the inner working of the decision maker, studying how a rational agent incorporates information from ML-DS in order to make a decision. We demonstrate that the introduction of ML-DS can lead to harmful consequences due to a mismatch of the model with the users' expectations or training, even in an ideal scenario where the ML model is as good as it gets and the agent is a 'perfect' reasoner.
+#### Motivating example 1: Effect of the agent's prior beliefs.
+Consider the situation of an oncologist using ML-DS when deciding how to treat a cancer patient. The ML-model predicts that the patient has only two months left to live. We can then imagine two lines of thinking depending on the clinician's prior beliefs. If the model was trained on patients who received minimal treatment, a very heavy treatment may extend the survival time. On the other hand, if the model is trained on patients who received the heaviest treatment, it is best to put the patient on palliative care to avoid unnecessary suffering from side-effects.
+This example showcases that, even in presence of the same information from ML-DS, *different prior assumptions about the model and the data it was trained on can lead to opposite actions*. This underscores the importance of an agent's priors in determining the effectiveness of ML-DS,[1](#page-0-0) a point that has already been argued for in previous literature (see e.g., [\(van Geloven et al.,](#page-8-6) [2025\)](#page-8-6)) but has not been studied
+<span id="page-0-0"></span><sup>1</sup> Setting the agent prior beliefs by proper training is indeed an important part of RCTs' protocols that is not always adhered to [\(Perepletchikova et al.,](#page-8-7) [2007\)](#page-8-7). When it is not possible to create a common baseline of beliefs and behavior, a two-stage randomization might be required, both on the providers of the intervention and the subjects [\(Borm et al.,](#page-8-8) [2005\)](#page-8-8).
+{1}------------------------------------------------
+quantitatively.
+#### Motivating example 2: Agent learning from ML-DS.
+Consider a variation of the previous scenario involving a clinician and a patient with a similar prognosis. Before seeing the model's prediction, the clinician believes that in the training data of the model (1) patients were always treated, and in general (2) patients ended up surviving between 6 and 12 months when treated. Seeing a prediction of two months of survival, and assuming the patient is not an outlier, this set of beliefs is challenged.
+The clinician can then revise their beliefs in at least two different ways. They can conclude the model's training data had in fact some untreated patients, and that this patient might be similar to one of the untreated patients. In this case, it might still be beneficial to treat. On the other hand, they might conclude they were wrong about the effect of treatment on patients. In this case, the treatment must be less beneficial than previously thought by the clinician. Therefore it might be best to abstain from treatment.
+Hence *the way in which ML-DS changes the agent's beliefs will influence the action of the agent*. Capturing this phenomenon means solving a challenging leaning problem where the ML prediction is used as proxy information about the causal process the agent wants to intervene on.
+Contributions of the paper These examples show the importance of an agent's prior knowledge for the impact of ML-DS. In this paper, we introduce a general computational framework to study the interaction of an agent with an ML-DS system.
+We make three main contributions. First, we provide a rigorous formalism to analyze how a user interacts with ML-DS system in order to act. Our framework captures said interaction in two steps (2-Step Agent): a Bayesian step in which the agent incorporates the information given by the decision support model into their model of the world, and a causal inference step in which the agent draws the conclusion on which action is best to take. This is schematically depicted in Fig [1](#page-1-0) for the clinical example.
+Our second contribution is to run computational experiments to quantify the effect of using ML-DS on patient outcomes. We show that harmful effects can occur in realistic scenarios already when a single prior is misaligned. We also show that ML-DS can be beneficial in some settings.
+Finally, to the best of our knowledge we provide the first treatment of how a Bayesian agent can update their beliefs about a population after observing a prediction of an ML model trained on data from that population. This update involves a challenging inference through a set of latent interchangeable variables in the predictive model's training
+<span id="page-1-0"></span>![](_page_1_Figure_10.jpeg)
+*Figure 1.* The diagram shows the overall flow of the model. A new patient with certain attributes (1) arrives for treatment, and the patient's data (2), potentially along with the output of a prediction model (2'), is observed by the agent, e.g., a doctor. The agent uses these observations to update their previous beliefs about the population and the predictive model (3), a step which we model as a Bayesian update. Based on their new beliefs, the agent estimates the effect of treatment (4) and makes a treatment decision (5). Finally, the treatment together with the patient's attributes determine a final outcome (6).
+data. We prove that for simple linear models it is possible to reduce this set to a few sufficient statistics, improving computational cost and numerical stability when applying our framework.
+While our running example is medical, the results are general and apply for every other field where prediction models are employed as decision support. Furthermore, our simulations are only concerned with simple (non-causal) prediction models; ML-DS with causal model is left for future work. The code and experiments are available open-source.[2](#page-1-1)
+#### 2. The 2-Step Agent: Defining the framework
+In this section, we give a formal description of our 2-Step Agent for studying the effects of ML-DS. We first give a description of the process that generates the training data for the predictive model, as well as the possible types of predictive model (Sec. [2.1\)](#page-1-2). We formalize the first step, which consists of a Bayesian agent updating their beliefs about the world given a prediction from the predictive model for a new patient (Sec. [2.2\)](#page-2-0). We then formalize the second step, where the Bayesian agent uses the updated beliefs about the population to decide on a treatment for the new patient (Sec. [2.3\)](#page-3-0).
+#### <span id="page-1-2"></span>2.1. The setup: Historical data and predictive model
+We start by formally defining a causal model of the domain. Let A be a variable denoting an intervention, such as the administration of a treatment. The treatment variable may
+<span id="page-1-1"></span><sup>2</sup>The code will be shared upon acceptance.
+{2}------------------------------------------------
+be continuous or categorical, but to lighten the notation we will present the formulas with a binary A. Let Y be a variable denoting an outcome of interest. In our example above, the treatment could be a fixed dose of chemotherapy and the outcome the months of survival, so in this case a higher outcome is better. We use $X_i$ to refer to covariates involved in the causal mechanisms governing the treatment and outcome.
+**Definition 2.1** (Historical data generation). We assume $Hist = (\mathbf{S}, P_{\mathbf{N}})$ to be the minimal structural causal model (SCM) describing the causal mechanisms producing A and Y, where $\mathbf{S}$ is a set of assignments defining each variable in terms of its parents and $P_{\mathbf{N}}$ is a joint distribution over the noise variables. We use $\bar{X} = X_1, \ldots, X_k$ to denote endogenous (non-noise) variables involved in the SCM that are different from A and Y.
+An example of this SCM is depicted in Figure 2(b). The SCM Hist stands for the process in the world that generated the data before the introduction of decision support. Note that the assignment determining A might be due to a specific intervention protocol, e.g., the guidelines for administering the treatment that were in use in that particular time window. The SCM determines a unique joint distribution $P^{Hist}$ over variables $Y, A, \bar{X}$ .
+Next, we assume that a relevant dataset is generated by Hist and used to train a prediction model for decision support.
+<span id="page-2-2"></span>**Definition 2.2** (Training data and prediction model). A dataset $\mathcal{D}_{Hist} = \{(\bar{x}_1, a_1, y_1), \dots, (\bar{x}_n, a_n, y_n)\}$ of historical data is composed of tuples containing relevant characteristics, treatment, and outcome. The dataset contains n instances. This dataset is used to train a prediction model $f: \bar{X} \to Y$ to estimate $E^{Hist}(Y \mid \bar{X})$ .
+Other estimands we do not consider are the treatment dependent estimate $E^{Hist}(Y\mid A,\bar{X})$ and the interventional estimate $E^{Hist}(Y\mid do(A),\bar{X})$ ; these are left for future work.
+#### <span id="page-2-0"></span>2.2. The first step: Bayesian update of the agent's beliefs
+We next introduce an agent $\mathcal{A}$ who decides on the intervention for new observations (e.g. new patients) with the help of ML-DS. Rather than mechanically applying the recommendation of the ML-DS system, this agent uses their own internal model of the world, capturing not only what they believe about the data generating process Hist, but also what they believe concerning the prediction model f and the training data $\mathcal{D}_{Hist}$ . We model an agent who is aware of the model documentation containing the model class (e.g., a linear model, a tree ensemble, etc.), the model's signature (i.e. that it is treatment naive and what covariates are
+involved),<sup>3</sup> and the size of the training data, but who is uncertain about the model's parameters. The agent can update their world model, including beliefs concerning historical data and model training.
+**Definition 2.3** (Internal beliefs of the agent). Call $A_{hist}$ the Bayesian network defining the agent's joint distribution over the following variables:
+- 1. $N_{\bar{X}}, N_Y, N_A$ : These are the variables encoding the population-level distributions over the various quantities of interest, respectively: the patient's characteristics $(N_{\bar{X}})$ , the noise in the outcome $(N_Y)$ , and the noise in the treatment $(N_A)$ . Depending on how population-level uncertainty is parameterized, these can be vectors. These population-level variables are sampled to produce the historical data. Within this Bayesian network, the historical data is represented via a plate, a graphical notation device used to show repeated, independent, and identically-structured components of a probabilistic model.
+- N<sub>E</sub>: This variable denotes the treatment effect. Contrary to the other noise variables, this is not sampled for each plate repetition but only once, so it remains the same across the plate.
+- 3. $A_i, Y_i, \bar{X}_i$ : These represent the i-th (out of n) datapoint in the training data of the predictive model. The whole training set is represented as a plate of multiplicity n. The noise distributions corresponding to these variables are fixed and outside the plate, capturing the fact that they are population-level distributions. All variables $Y_i$ share the noise term $N_E$ .
+- 4. M: The prediction model's parameters, obtained deterministically from the training data in the plate and possibly some additional known parameter N<sub>M</sub> which encodes, e.g., the hyperparameter settings or initialization values. M has as parents all the variables in the plate, representing the fact that it is trained on the whole data set.
+- 5. $\bar{X}^o$ : An observed data instance for a new patient, for which the agent will make a treatment decision. This array of variables has no non-noise parents and it is sampled from the same noise distribution as $\bar{X}$ .
+<span id="page-2-1"></span><sup>&</sup>lt;sup>3</sup>Recent guidelines explicitly suggest to additionally inform model users about the historical treatment policy. Our framework can be extended to model such settings, but here we focus on agents who are uncertain about historical treatment policy, for two reasons. First, this is a common scenario in current practice. Second, this provides a baseline estimation for the effects of ML-DS, which can be compared to better informed users in the future. We come back to these points in the discussion.
+{3}------------------------------------------------
+<span id="page-3-1"></span>![](_page_3_Figure_1.jpeg)
+the SCM generating the data in the real world, while models (a) and (c) are the first and second steps of the 2-Step Agent. Model (a) allows the agent to perform a belief update given new information from ML-DS, while model (c) allows the agent to estimate the
+model for $\bar{X}^o$ . This is derived deterministically from $\bar{X}^o$ and M.
+The Bayesian network encoding the conditional independencies of the model is shown in Figure 2 (a). To simplify presentation we depict the submodel of $A_{hist}$ containing only the variables A, Y and X as coinciding with Hist, meaning that the agent has correct belief about the data generation process, although this does not have to be the case and the framework is general enough to allow these two structures to diverge.
+$N_E$ encodes the agent's beliefs concerning the effect of the intervention: if the agent were to fix all the coefficients in the structural equation of Y, the effect of treatment would be fixed and there could be no update of his/her beliefs on the interventional distribution $E^{A_{hist}}(Y \mid \bar{X}, do(A = 1))$ . By letting the mechanism be influenced by $N_E$ , we can treat the posterior as encoding a distribution over SCMs, each with its own fixed assignment for Y (and thus each with its own treatment effect).
+The agent's initial beliefs are represented via priors.
+**Definition 2.4** (Agent's priors). The agent is assumed to
+have as prior a joint distribution over all variables in the model, which depends on the unconditional distribution $P_{N_{\bar{X}},N_Y,N_A,N_E}^{\mathcal{A}_{hist}}.$
+For all i < n, the priors encode the agent's starting beliefs about various quantities. First, $P_{N_{\bar{X}}}^{\mathcal{A}_{hist}}$ encodes the population-level distribution generating $\bar{X}_{i,}$ the agent's model of the historical population. Second, $P_{N_A}^{\mathcal{A}_{hist}}$ encodes the population-level distribution generating the noise of $A_i$ , the agent's model of the historical protocol to administer the intervention. Third, $P_{N_Y}^{\mathcal{A}_{hist}}$ encodes the population-level distribution generating the noise of $Y_i$ , the agent's representation of the outcome. Finally, $P_{N_E}^{A_{hist}}$ encodes the effect of treatment, assumed to be fixed across the population. We assume the agent knows the values of the parameters of the distribution $N_M$ , since he/she has access to the model documentation, therefore we omit this prior.
+This prior is updated via a Bayesian update based on the observables $X^o$ and $Pred^o$ .
+**Definition 2.5** (Agent belief update). Given $\bar{X}^o$ and Predo the agent does a Bayesian update on the prior $P^{\mathcal{A}_{hist}}(N_{\bar{X}}, N_Y, N_A, N_E)$ , obtaining a posterior distribu-
+$$P^{\mathcal{A}_{hist}}(N_{\bar{X}}, N_Y, N_A, N_E \mid \bar{X}^o, Pred^o)$$
+To simplify exposition, we marginalize over uncertainty in the plate variables $(\bar{X}_i, Y_i, A_i \text{ with } i < n)$ and M.
+#### <span id="page-3-0"></span>2.3. The second step: deciding on the best action after belief update
+We described the agent' world model and how the agent updates it upon observing predictions from an ML-DS system for a new patient, which we called the first step. In the second step, the agent estimates the effect of the treatment on the new patient using their updated world model. This is necessary for the agent to consider a 'fresh' iteration of the plate and calculate the correct interventional probability. Another conceptual reason to introduce a second model is that - when the agent does Bayesian inference - the values of A and Y for the new instance $\bar{X}^o$ are not realized yet.
+**Definition 2.6** (Agent model for decision making). This model contains variables $N_E, N_{\bar{X}}, N_A, N_Y, X, A, Y$ , arranged as the similarly-named variables in the sub-model of $A_{hist}$ . This model is used for inference on new instances and we denote it as $A_{inf}$ .
+This second agent model is depicted in Figure 2 (c). The crucial difference between this model and a regular SCM is that the noise variables are not assumed to be independent but rather they are initialized with the joint posterior obtained in the previous Bayesian step. The agent is thus drawing an inference over a distribution of possible SCMs.
+To decide what to do on the new instance, the agent then
+{4}------------------------------------------------
+calculates the effect of the intervention on the model $A_{inf}$ where the joint distribution of the noise is set to the posterior calculated over $A_{hist}$ .
+**Definition 2.7** (Decision rule). To make a decision, the agent first sets the joint distribution of the inference model according to the posterior on the historical model:
+$$P^{\mathcal{A}_{inf}}(N_{\bar{X}}, N_Y, N_A, N_E) :=$$
+$$P^{\mathcal{A}_{hist}}(N_{\bar{X}}, N_Y, N_A, N_E \mid \bar{X}^o, Pred^o)$$
+Following this, treatment is given iff the Conditional Average Treatment Effect (CATE) for the new patient crosses a threshold value, meaning
+$$E_{P^{\mathcal{A}_{inf}}}[\mathrm{CATE}] > \tau$$
+where $\tau$ is a threshold set by the agent beforehand and CATE = $E^{\mathcal{A}_{inf}}(Y \mid \bar{X} = \bar{X}^o, do(A=0)) - E^{\mathcal{A}_{inf}}(Y \mid \bar{X} = \bar{X}^o, do(A=1))$ . In other words, treatment is given if the expected effect of treatment is sufficiently large in the desired direction.
+When the threshold is zero, the agent administers the treatment if the expected outcome is better in the scenario where they are giving the treatment, conditioned on the patient characteristics, versus the scenario where treatment is not given. Crucially, these interventional expectations are calculated on the internal model of the agent.
+# 2.4. Quantifying the effect of introducing decision support
+Given the machinery defined above, we are able to calculate the effect of introducing decision support. Given the prediction model and the specification of the agent - which is determined by spelling out $\mathcal{A}_{hist}$ and $\mathcal{A}_{inf}$ - the decision of the agent on a new data point (as described by covariates $\bar{X}^o$ ) is deterministic (i.e. we can implement the input-output process described in Figure 1). We can thus formalize it as a function $Act^{\mathcal{A},f}: \bar{X} \to A$ . The way the decision support changes the actions of the agent is then represented as an intervention on Hist changing the assignment of A:
+$$A := Act^{\mathcal{A}, f}(\bar{X})$$
+The post-interventional SCM is then indicated with the notation $Hist^{A=Act^{\mathcal{A},f}(\bar{X})}$ , where f is the prediction model from Definition 2.2. We can now define the effect of introducing decision support as the risk difference in outcomes generated by Hist compared to $Hist^{A=Act^{\mathcal{A},f}(\bar{X})}$ . This quantifies the effect of the intervention 'using MS-DL', compared to the historical policy.
+#### <span id="page-4-0"></span>3. Experimental setup
+We design experiments to answer two questions. First, how and how much does decision support influence (i) treatment decisions, (ii) agent's beliefs, and (iii) downstream patient outcomes? Second, what role is played by the agent's priors? To address these questions, we vary the priors of the agent and compare the scenarios with and without ML-DS. For each prior setting, we plot the change in treatment decisions, in agent's beliefs, and in patient outcomes, with and without ML-DS, after observing a single patient.
+Use case We consider a data-generating model with one continuous covariate X, continuous intervention A, and a continuous outcome Y. Higher values of the outcomes are better. One can think of the initial example and interpret X as body weight, A as dosage of chemotherapy, and Y as months of survival. We set n=1000 and we assume the prediction model to be a slope-only linear regression. Despite the continuous treatment, for simplicity we assume the agent compares two treatment options, A=10 or A=20, intuitively ten and twenty units of dosage respectively.
+**Historical SCM.** Let the SCM *Hist* be defined as:
+$$N_X \sim \mathcal{N}(80, 10)$$
+$N_A \sim \mathcal{N}(2, 1)$ $N_Y \sim \mathcal{N}(0, 0.1)$ $X := N_X$ $A := 0.125 * X + N_A$ $Y := 12 - 0.1 * X + 1 * A + N_Y$
+In this model, the treatment protocol is simply to give a dosage of treatment A which is an eight of the body weight X, and how well this dosing is followed is encoded by the noise term $N_A$ . The mechanism of Y clarifies that the covariate X makes the outcome worse but the treatment improves the outcome. Of course, this equation is a great simplification of what happens in reality, but it is sufficient to demonstrate several important dynamics that emerge when introducing ML-DS. Average treatment effect is positive in this scenario, as witnessed by the positive coefficient of A in the equation for Y.
+The internal model of the agent's beliefs. The internal model of the agent $A_{hist}$ encodes the agent's uncertainty about *Hist*:
+$$X_{i} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\sigma}}) \qquad \text{with } i < n$$
+$$N_{A_{i}} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{A}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{A}\boldsymbol{\sigma}}) \qquad \text{with } i < n$$
+$$N_{Y_{i}} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{Y}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{Y}\boldsymbol{\sigma}}) \qquad \text{with } i < n$$
+$$A_{i} = 0.125 X_{i} + N_{A_{i}} \qquad \text{with } i < n$$
+$$Y_{i} = 12 - 0.1 X_{i} + N_{\boldsymbol{E}} A_{i} + N_{Y_{i}} \qquad \text{with } i < n$$
+$$M = \arg\min_{\boldsymbol{\phi} \in \Phi} (MSE(\boldsymbol{\phi}\bar{X}, \bar{Y}))$$
+$$X^{o} \sim \mathcal{N}(\boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\mu}}, \boldsymbol{\alpha}_{\boldsymbol{X}\boldsymbol{\sigma}})$$
+$$Pred^{o} = MX^{o}$$
+where $\bar{Y} = (Y_1, \dots, Y_n)$ , etc are the respective vectors of n variables, $\phi$ is the single parameter of a least squares regression without intercept, and the variables in bold font indicate
+{5}------------------------------------------------
+population-level priors that are given different values in the different simulations below.<sup>4</sup>
+The agent knows some aspects of the data generating process, e.g., some of the parameters in the mechanisms of Y and A, but has uncertainty on several dimensions. $\alpha_{A\mu}$ encodes the belief of the agent on how much historical dosages of treatment deviated from the standard protocol. $\alpha_{X\mu}, \alpha_{X\sigma}$ represent the uncertainty of the agent on how the feature X of the training data was distributed. In the definition of Y there are two uncertainties at play. The prior on $\alpha_{Y\sigma}$ describes the uncertainty of the agent concerning Y. The distribution of $N_E$ , on the other hand, captures the belief of the agent concerning the effectiveness of treatment A.
+**Defining different agents** We consider two types of agents, one with 'correct' priors whose means are close to the historical SCM, and one with incorrect priors. This is an example of the former:
+$$\begin{aligned} &\alpha_{X\mu} \sim \mathcal{N}(80, 0.1) & \alpha_{X\sigma} \sim \mathcal{N}(10, 0.1) \mid \alpha_{X\sigma} \geq 0 \\ &\alpha_{A\mu} \sim \mathcal{N}(2, 1) & \alpha_{A\sigma} \sim \mathcal{N}(1, 0.1) \mid \alpha_{Y\sigma} \geq 0 \\ &\alpha_{Y\mu} \sim \mathcal{N}(0, 0.01) & \alpha_{Y\sigma} \sim \mathcal{N}(0.1, 0.01) \mid \alpha_{Y\sigma} \geq 0 \\ &N_E \sim \mathcal{N}(1, 0.1) \end{aligned}$$
+This agent essentially has the correct beliefs about the patient data distribution, historical treatment policy, outcome variance and treatment effect. They only have some uncertainty represented by some variance on these parameters. In our experiments, we vary each of the colored prior parameters, one at a time, to simulate various ways in which the agent's expectations may be incorrect.
+The internal model for the agent's effect estimation. Finally, the internal model of the agent used for inference, $A_{inf}$ , is defined as:
+$$\begin{split} N_X &\sim \mathcal{N}(\alpha_{X\mu}, \alpha_{X\sigma}) \quad N_A \sim \mathcal{N}(\alpha_{A\mu}, \alpha_{A\sigma}) \\ N_Y &\sim \mathcal{N}(\alpha_{Y\mu}, \alpha_{Y\sigma}) \\ X &:= N_X \\ A &:= 0.125 \ X + N_A \\ Y &:= 12 - 0.1 \ X + N_E \ A + N_Y \end{split}$$
+The posterior defines a probability distribution over SCMs. Recall that the agent was interested in choosing between dosage A=10 or A=20 for the specific patient with
+weight equal to $X^o$ . To compute the desired quantity, the agent aggregates the CATE produced by every SCM according to the posterior
+$$C\hat{ATE} = \mathbb{E}_{P^{\mathcal{A}_{inf}}}[E^{\mathcal{A}_{inf}}(Y|X^o, do(A=20)) - E^{\mathcal{A}_{inf}}(Y|X^o, do(A=10))]$$
+where the expectation is over the joint posterior of the agent over the parameters, which was obtained updating $A_{hist}$ .
+**Reaching a decision.** The agent will administer a dosage of 10 if the estimated CATE is below a threshold of 5 or a dosage of 20 otherwise. In other words, the higher dosage needs to improve the outcome by at least 5 months to be worth it. This completes the decision pipeline: given a patient $X^o$ with a corresponding prediction and the type of agent, we can compute what the agent would do.
+#### 4. Results
+We begin by showcasing how our framework can capture the two motivating examples presented in the Introduction. In the first example, the belief of the agent on the past policy determines how the prediction is interpreted and what action is taken.<sup>5</sup> Indeed, in the simulation, when X = 80, with a bad prognosis under the assumption of no treatment $(\alpha_{A\mu} = 0)$ , the agent chooses a high dosage to improve the outcome (A = 20). With a bad prognosis under the assumption of heavy treatment ( $\alpha_{A\mu}=20$ ), the agent chooses a low dosage to reduce over-treatment (A = 10). In the second motivating example, new information from the prediction forces an agent to consider different options. In the model, this means that either belief in treatment effect $(N_E)$ or in historical policy $(\mu_A)$ is revised, which appears as a correlation in the posterior belief in these two variables in Figure 4 in Appendix A.
+Next, for the use case described in Section 3, we report the effect of using ML-DS on two dimensions: 1) agent's belief of treatment effect and 2) downstream outcomes Y realized after the agent has taken an action, visualized in the top and bottom rows of Figure 3, respectively. In each of these plots, the outcomes obtained using ML-DS (orange boxplots) is contrasted with the outcomes obtained without it (blue boxplots), for different types of agents. The CATE plots sweep a wide range of options for the priors in order
+<span id="page-5-0"></span><sup>&</sup>lt;sup>4</sup>Note that for ease of exposition we introduced some variables $(N_{A_i}, N_{Y_i})$ which are not present in Figure 2 (a). These can in principle be removed by specifying distributions directly on the inverse transformations of the ones given above for $A_i$ and $Y_i$ , and applying the appropriate Jacobian adjustment. In the PyMC implementation, we also use a small (1e-3) observational noise around $Pred^o$ to allow us to sample from an observed deterministic transformation.
+<span id="page-5-1"></span><sup>&</sup>lt;sup>5</sup>All simulations presented in this paper are ran with PyMC v5 (Salvatier et al., 2016). We used the NUTS sampler to get 1000 samples for each of 4 chains, with 1000 tuning samples and a 'target\_accept' parameter of 0.85. Diagnostics and visual inspection of the traces showed good mixing and no convergence issues (Max $\hat{R}=1.01$ , and only 16 out 4000 runs had > 5 divergences, due to fixed 'target\_accept' and extreme parameter values.). Sanity checks on the Bayesian model are reported in Appendix B.
+{6}------------------------------------------------
+to reveal interesting dynamics of the agent's beliefs; in Appendix [D](#page-10-3) we also report the CATE results for the more realistic range of ± 3 stds from the SCM reference value.
+We begin with an agent whose priors beliefs coincide with the parameters of the historical SCM except for NE, i.e. treatment effect. Figure [3a](#page-7-0) and Figure [3d](#page-7-0) show what happens to CATE and Y when the agent's belief in N<sup>E</sup> varies. As expected, in the first figure we can see that the blue CATE has a slope that depends on the prior NE: without decision support, the belief on CATE is equal to N<sup>E</sup> multiplied by the difference in treatments. In the orange series however, we see an interesting behavior: after learning from the interaction with ML-DS, the agent corrects their belief in CATE re-aligning it to the correct value. In other words, *when all other beliefs are correct, the agent can learn from ML-DS to estimate better CATE* and take better actions. This is confirmed in Figure [3d](#page-7-0) where the orange series shows superior outcomes.
+Figure [3b](#page-7-0) and Figure [3e](#page-7-0) show what happens when the agent's prior belief in the treatment protocol, µA, varies. In this scenario the prior CATE estimates are flat, for the simple reason that the agent has a fixed belief in N<sup>E</sup> and they are not updating it. When interacting with ML-DS however, the belief update changes CATE estimates quite a lot; this is because the agent is attempting to explain the gap between P red<sup>o</sup> given by ML-DS and the expected outcome they had from their prior beliefs. This reflects on outcomes in [3e](#page-7-0) where the orange series is sometimes below the blue one: incorrect beliefs in µ<sup>A</sup> can result in CATE estimates that lead to wrong actions and worsen outcome Y.
+A similar pattern is observed in Figures [3c,](#page-7-0) [3f,](#page-7-0) [6c](#page-12-0) (last one in Appendix [D\)](#page-10-3): while prior CATE and Y estimates are relatively stable without ML-DS, new dynamics emerge when when ML-DS is introduced. For CATE, incorrect beliefs can have large effects both on the magnitude and sign. For Y , in both cases incorrect beliefs can lead to bad actions and worse outcomes.
+#### 5. Related work
+A large body of literature is concerned with the analysis of ML-DS from different angles. A strand of literature focuses on how ML-DS is entangled with (changes in) the data generating process. Works in this category study limitations of what can be learned from data generated from past policies, e.g. [\(Lakkaraju et al.,](#page-8-10) [2017\)](#page-8-10), or the dynamics that occur with alternating cycles of ML-DS deployment and re-training, as in the performative prediction literature [\(Perdomo et al.,](#page-8-11) [2020;](#page-8-11) [Boeken et al.,](#page-8-12) [2024\)](#page-8-12). Another batch of articles is concerned with evaluation of ML-DS, laying out identifiability conditions for the effects of ML-DS on decisions [\(Imai et al.,](#page-8-13) [2023\)](#page-8-13) or trying to optimize human-machine complementarity by design [\(Donahue et al.,](#page-8-14) [2022;](#page-8-14) [McLaughlin & Spiess,](#page-8-15) [2024\)](#page-8-15). Finally, a different set of papers investigate what human input can add to artificial agents already reaching some level of optimality [\(Alur et al.,](#page-8-16) [2024;](#page-8-16) [Zhang & Barein](#page-9-0)[boim,](#page-9-0) [2022;](#page-9-0) [Stensrud et al.,](#page-8-17) [2024\)](#page-8-17). The key difference of our work is that we offer a more fine-grained description of what happens inside the decision maker, namely we unpack the micro-mechanism leading a decision maker to take an action on a single observation paired with AI prediction.
+Other papers have studied how information flows between agents influencing their actions. In [\(Kamenica &](#page-8-18) [Gentzkow,](#page-8-18) [2011\)](#page-8-18) a Sender of information influences a Receiver Bayesian agent in order to stir their action. What we add here is that in our context the signal is an ML prediction produced from training data drawn from the same environment the agent reasons about; this coupling yields the distinctive inference problem we study, i.e. updating through latent interchangeable training variables.
+Another important strand of work focuses on problems related to inferring information about a population P given predictions from a model trained on samples from P in the context of confidentiality attacks. This literature studies whether it is possible to use models' predictions (and other behavior) to recover the presence of a specific data point, in so-called *membership inference attacks* [\(Hu et al.,](#page-8-19) [2022;](#page-8-19) [Huang,](#page-8-20) [2025\)](#page-8-20)), or to infer statistical properties of a training datasets, so-called *property-inference attacks* [\(Ate](#page-8-21)[niese et al.,](#page-8-21) [2015;](#page-8-21) [Wang et al.,](#page-9-1) [2019\)](#page-9-1). While these models aim to recover information about the training set, e.g., the proportion of data in each label, our challenge is to marginalize across datasets and directly infer information about the underlying population.
+#### 6. Discussion
+We have presented the first explicit framework that includes an ML-DS system and a sophisticated user. Our 2-Step Agent framework captures the effect of an agent's prior beliefs, modeling learning effects due to the interaction and tracking agent's uncertainty. The framework allows for simulations comparing outcomes with and without ML-DS, in an RCT-like fashion, therefore quantifying the benefit/harm of ML-DS for different types of simulated agents.
+We provided an implementation of the framework for ML-DS with a linear model. Said implementation leverages a collapse of the plate via sufficient statistics (Appendix [E.1\)](#page-13-0), greatly speeding up the simulations. Within this implementation, we have shown that incorrect beliefs about the distribution of the predictive model's training data may lead the agent to bad decisions, resulting in harmful outcomes.
+While previous literature had argued that incorrect beliefs about past treatment policy (µ<sup>A</sup> in our notation) might gener-
+{7}------------------------------------------------
+<span id="page-7-0"></span>![](_page_7_Figure_1.jpeg)
+*Figure 3.* Figures displaying the effects of different agent priors when using ML-DS (in orange) and without it (in blue). In all plots except from NE, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameters. The top row displays the agent's prior and posterior (predictive) belief of CATE, while the bottom row shows the impact of the agent's decisions on the downstream outcome Y . The left column pertains to the belief on treatment effect, NE, the central column the belief about past treatment policy (µA) and the right column about the belief on past covariate distribution (µX). The gray areas represents the portions of the belief in which the true CATE is within one std from the CATE learned from ML-DS (orange band).
+ate worse downstream outcomes, e.g. in [\(van Geloven et al.,](#page-8-6) [2025\)](#page-8-6), this is the first quantitative estimation of harmful effects of ML-DS. In our experiments we show that ML-DS can be harmful if the agent has wrong beliefs not only about the past treatment policy, but also about the distribution of covariates and the distribution of the outcome.
+We also show that if the agent has otherwise correct beliefs, an incorrect belief about treatment effect may be corrected during updates that push the agent towards the true CATE. This suggests that in certain settings ML-DS building on a treatment-naive prediction model might still be beneficial.
+This finding highlights the extreme sensitivity of ML-DS to agent prior beliefs when using a treatment-naive prediction model, and underscore the importance of providing full documentation and proper training to users of ML-DS. Finally, we remark that these results apply to *any* field deploying such systems and to any agent interacting with ML-DS, be it human or artificial.
+Limitations and Future work The experiments in this paper explored only part of the potential of the framework. The experiments are limited to a use case in which the prediction model is linear and treatment-naive, and the data
+generating process is simple (e.g., every distribution is normal, there is only one confounder). Implementing the use case was not trivial and required pushing a Bayesian update through OLS and a large amount of variables (see Appendix [E.1\)](#page-13-0), hence it remains to be seen whether similar results can be obtained when the complexity of the use case increases.
+We assume the agent reasons in a Bayesian fashion, which allows for a precise analysis of the agent's uncertainty. While this agent may not be realistic with respect to human behavior, it is useful to make this assumption to demonstrate what can go wrong in a best-case scenario, i.e., when the agent is a 'perfect' reasoner and the ML model has no flaws. Finally, this assumption is also interesting when considering how an artificial agent might learn from predictions of other ML models.
+In future work this line of research will be expanded across different dimensions, including i) different types of agents (e.g. agent with incorrect internal model of the world of different kinds of uncertainties) ii) more complex ML models for ML-DS, iii) cases with heterogeneity of treatment effect (i.e. the effect of the action is not the same for all instances), iv) repeated rounds of learning for the same agent and v) using causal models for ML-DS.
+{8}------------------------------------------------
+#### References
+- <span id="page-8-16"></span>Alur, R., Raghavan, M., and Shah, D. Human expertise in algorithmic prediction. *Advances in Neural Information Processing Systems*, 37:138088–138129, 2024.
+- <span id="page-8-21"></span>Ateniese, G., Mancini, L. V., Spognardi, A., Villani, A., Vitali, D., and Felici, G. Hacking smart machines with smarter ones: How to extract meaningful data from machine learning classifiers. *International Journal of Security and Networks*, 10(3):137–150, September 2015. ISSN 1747-8405. doi: 10.1504/IJSN.2015.071829.
+- <span id="page-8-12"></span>Boeken, P., Zoeter, O., and Mooij, J. Evaluating and correcting performative effects of decision support systems via causal domain shift. In *Causal Learning and Reasoning*, pp. 551–569. PMLR, 2024.
+- <span id="page-8-8"></span>Borm, G. F., Melis, R. J., Teerenstra, S., and Peer, P. G. Pseudo cluster randomization: a treatment allocation method to minimize contamination and selection bias. *Statistics in Medicine*, 24(23):3535–3547, 2005.
+- <span id="page-8-14"></span>Donahue, K., Chouldechova, A., and Kenthapadi, K. Human-algorithm collaboration: Achieving complementarity and avoiding unfairness. In *Proceedings of the 2022 ACM Conference on Fairness, Accountability, and Transparency*, pp. 1639–1656, 2022.
+- <span id="page-8-19"></span>Hu, H., Salcic, Z., Sun, L., Dobbie, G., Yu, P. S., and Zhang, X. Membership Inference Attacks on Machine Learning: A Survey. *Acm Computing Surveys*, 54(11s): 235:1–235:37, September 2022. ISSN 0360-0300. doi: 10.1145/3523273.
+- <span id="page-8-20"></span>Huang, Y. Bayesian Inference of Training Dataset Membership. May 2025.
+- <span id="page-8-13"></span>Imai, K., Jiang, Z., Greiner, D. J., Halen, R., and Shin, S. Experimental evaluation of algorithm-assisted human decision-making: Application to pretrial public safety assessment. *Journal of the Royal Statistical Society Series A: Statistics in Society*, 186(2):167–189, 2023.
+- <span id="page-8-18"></span>Kamenica, E. and Gentzkow, M. Bayesian persuasion. *American Economic Review*, 101(6):2590–2615, 2011.
+- <span id="page-8-3"></span>Kleinberg, J., Lakkaraju, H., Leskovec, J., Ludwig, J., and Mullainathan, S. Human decisions and machine predictions. *The quarterly journal of economics*, 133(1): 237–293, 2018.
+- <span id="page-8-4"></span>Kube, A., Das, S., and Fowler, P. J. Allocating interventions based on predicted outcomes: A case study on homelessness services. In *Proceedings of the AAAI Conference on Artificial Intelligence*, volume 33, pp. 622–629, 2019.
+- <span id="page-8-10"></span>Lakkaraju, H., Kleinberg, J., Leskovec, J., Ludwig, J., and Mullainathan, S. The selective labels problem: Evaluating algorithmic predictions in the presence of unobservables. In *Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining*, pp. 275–284, 2017.
+- <span id="page-8-15"></span>McLaughlin, B. and Spiess, J. Designing algorithmic recommendations to achieve human-ai complementarity. *arXiv preprint arXiv:2405.01484*, 2024.
+- <span id="page-8-5"></span>Obermeyer, Z., Powers, B., Vogeli, C., and Mullainathan, S. Dissecting racial bias in an algorithm used to manage the health of populations. *Science*, 366(6464):447–453, 2019.
+- <span id="page-8-11"></span>Perdomo, J., Zrnic, T., Mendler-Dunner, C., and Hardt, M. ¨ Performative prediction. In *International Conference on Machine Learning*, pp. 7599–7609. PMLR, 2020.
+- <span id="page-8-7"></span>Perepletchikova, F., Treat, T. A., and Kazdin, A. E. Treatment integrity in psychotherapy research: analysis of the studies and examination of the associated factors. *Journal of consulting and clinical psychology*, 75(6):829, 2007.
+- <span id="page-8-9"></span>Salvatier, J., Wiecki, T. V., and Fonnesbeck, C. Probabilistic programming in Python using PyMC3. *PeerJ Computer Science*, 2:e55, 2016. ISSN 2376-5992. doi: 10.7717/ peerj-cs.55.
+- <span id="page-8-2"></span>Smith, V. C., Lange, A., and Huston, D. R. Predictive modeling to forecast student outcomes and drive effective interventions in online community college courses. *Journal of asynchronous learning networks*, 16(3):51–61, 2012.
+- <span id="page-8-17"></span>Stensrud, M. J., Laurendeau, J., and Sarvet, A. L. Optimal regimes for algorithm-assisted human decision-making. *Biometrika*, 111(4):1089–1108, 2024.
+- <span id="page-8-6"></span>van Geloven, N., Keogh, R. H., van Amsterdam, W., Cina,` G., Krijthe, J. H., Peek, N., Luijken, K., Magliacane, S., Morzywołek, P., van Ommen, T., et al. The risks of risk assessment: causal blind spots when using prediction models for treatment decisions. *Annals of internal medicine*, 178(9):1326–1333, 2025.
+- <span id="page-8-0"></span>Wang, A., Kapoor, S., Barocas, S., and Narayanan, A. Against predictive optimization: On the legitimacy of decision-making algorithms that optimize predictive accuracy. *ACM J. Responsib. Comput.*, 1(1):9:1–9:45, March 2024. doi: 10.1145/3636509.
+{9}------------------------------------------------
+Wang, L., Xu, S., Wang, X., and Zhu, Q. Eavesdrop the Composition Proportion of Training Labels in Federated Learning. October 2019.
+<span id="page-9-1"></span>
+> <span id="page-9-0"></span>Zhang, J. and Bareinboim, E. Can humans be out of the loop? In *Conference on Causal Learning and Reasoning*, pp. 1010–1025. PMLR, 2022.
+{10}------------------------------------------------
+#### <span id="page-10-1"></span>A. Motivating example 2
+<span id="page-10-0"></span>556
+In this section we report the posterior correlation plot obtained from an agent that learned from ML-DS as described in the second motivating example in Section [1.](#page-0-1) After observing a prediction that did not match expectation, the agent revises their beliefs, entertaining multiple hypotheses on how their beliefs should change to account for the situation. In this case, either the treatment effect is lower than anticipated and the dosage higher than assumed, or the other way around.
+![](_page_10_Figure_3.jpeg)
+*Figure 4.* Plot showing the correlation of variables N<sup>E</sup> and µ<sup>A</sup> in the posterior beliefs of the agent after update with ML-DS information. The agent entertains multiple options
+### <span id="page-10-2"></span>B. Sanity checks on the Bayesian model
+We have implemented a series of experiments to make sure that the framework is functioning as intended.
+- 1. We compare the posterior distributions with the explicit plate and with the sufficient statistics formulation given the same observations. We check if the posteriors of the two agents end up the same. This is repeated for different sizes of plate to control for potential effects of size.
+- 2. We run an update with a single observation X<sup>o</sup> = 200 that is much higher than the prior mean of 80 (with variance 10). We verify that after the update the posterior of µ<sup>X</sup> has moved in the right direction. We also check if this creates correlations of µ<sup>X</sup> with other parameters. This is repeated for different plate sizes.
+- 3. We run an update with many (1000) observations drawn from the real distribution of X and check that the (incorrect) prior of the agent about µ<sup>X</sup> converges to the real parameter in the generating process.
+- 4. We set one observation with X equal to the mean of the agent's prior, with the priors on A and N<sup>E</sup> very peaked, but with P red<sup>o</sup> set much higher than the value Y = coeff ∗ X + N<sup>E</sup> ∗ A. We also setthe coeff of X in the structural equation of A to zero. We observe in the posterior that there is now a correlation between the parameters of N<sup>E</sup> and A.
+### C. Results obtained when varying prior µ<sup>Y</sup>
+In Figure [5](#page-11-0) we report the CATE and Y plots obtained when varying the belief of the agent concerning the mean of the outcome in the historical population, µ<sup>Y</sup> .
+#### <span id="page-10-3"></span>D. Zoomed-in versions of the CATE plots
+The plots in Figure [3](#page-7-0) encompass a wide sweep of options for the priors of the agent, which revealed interesting dynamics of CATE with ML-DS. In Figure [6](#page-12-0) we 'zoom-in' on what happens in a more realistic range of priors within ± 3 std from the true SCM parameter.
+{11}------------------------------------------------
+<span id="page-11-0"></span>![](_page_11_Figure_1.jpeg)
+![](_page_11_Figure_2.jpeg)
+*Figure 5.* Figures displaying the effects of different agent's priors when using ML-DS (in orange) and without it (in blue). In all plots, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameter for µ<sup>Y</sup> . The left plot displays the agent's prior and posterior (predictive) belief of CATE, while the right one shows the impact of the agent's decisions on the downstream outcome Y . In a rather wide band is the posterior within 1 std of the true mean (gray area).
+{12}------------------------------------------------
+<span id="page-12-0"></span>![](_page_12_Figure_1.jpeg)
+*Figure 6.* Figures displaying the effects of different agent's priors on the posterior (predictive) belief of CATE when using ML-DS (in orange) and without it (in blue). In all plots except those for NE, the x-axis represents the deviation of the agent's prior belief from the historical SCM's parameters. The plots pertain to the belief on treatment effect, NE, the belief about past treatment policy (µA), the belief about past outcomes (µ<sup>Y</sup> ) and the belief about past covariate distribution (µX). The gray areas represents the portions of the belief in which the true CATE is within one std from the CATE learned from ML-DS (orange band.
+{13}------------------------------------------------
+#### E. Derivations for plate model
+We elaborate here on the strategy to reduce the plate with the help of sufficient statistics
+#### <span id="page-13-0"></span>E.1. Plate model, agent's internal model
+In this section we will show how an observation of the model parameter ϕ from the agent's internal model can be generated. This is needed to estimate the posterior distributions of the agent's beliefs using a sampling approach. As running inference through a plate-model can be both computationally expensive and numerically unstable, we chose to reformulate the plate using low-dimensional auxiliary variables.
+We start from the SCM (fig. [2\)](#page-3-1) that defines the relation between the variables and we are interested in finding the lowdimensional auxiliary variables that define the regression coefficient ϕ of the model M. As we are using PyMC for inference – which does not have an implementation for the non-central χ 2 -distribution – all random variables (RVs) will be reformulated using zero-centered normally distributed RVs (standard normals where needed) and (central) χ <sup>2</sup> distributed RVs. Some of these RVs will be dependent and that dependence will be dealt with in section [E.2.](#page-15-0)
+We are specifically looking for a low-dimensional representation of the plate and treat a sample of the plate parameters {αXµ, αXσ, αAµ, αAσ, αY µ, αY σ, NE} as given. The relevant parts of the SCM, reproduced here for the reader's convenience, are defined as
+$$Y_{i} = a + bX_{i} + N_{E}A_{i} + N_{Y_{i}}$$
+$$A_{i} = dX_{i} + N_{A_{i}}$$
+$$X_{i} = N_{X_{i}}$$
+$$\forall i \in [1, ..., n]$$
+$$(1)$$
+where n is the plate size, a, b, and d are constants and
+<span id="page-13-4"></span>
+$$N_{X_i} \sim \mathcal{N}(\alpha_{X\mu}, \alpha_{X\sigma}^2)$$
+$$N_{A_i} \sim \mathcal{N}(\alpha_{A\mu}, \alpha_{A\sigma}^2)$$
+$$N_{Y_i} \sim \mathcal{N}(\alpha_{Y\mu}, \alpha_{Y\sigma}^2).$$
+(2)
+We are modeling Y<sup>i</sup> without intercept as
+<span id="page-13-3"></span>
+$$Y_i \approx \phi X_i \tag{3}$$
+and looking for the ordinary least-squares estimate of ϕ, that is
+$$\min_{\phi} \left( \sum_{i} (Y_i - \phi X_i)^2 \right). \tag{4}$$
+The least-squares estimate for this case has a well-known closed-form solution given by
+<span id="page-13-5"></span><span id="page-13-1"></span>
+$$\phi = \frac{\sum_{i} X_i Y_i}{\sum_{i} X_i^2}.$$
+(5)
+Our goal is to express this equation in terms of RVs that summarize the sums thus removing the need to sample the full plate n times. The denominator of this equation can be expressed as
+$$\sum_{i} X_i^2 = \sum_{i} N_{X_i}^2 \tag{6}$$
+⇔
+$$\sum_{i} X_i^2 = \sum_{i} \alpha_{X\mu}^2 + 2\alpha_{X\mu}\alpha_{X\sigma}\epsilon_{X_i} + \alpha_{X\sigma}^2\epsilon_{X_i}^2$$
+(7)
+where
+<span id="page-13-2"></span>
+$$\epsilon_{X_i} \sim \mathcal{N}(0, 1).$$
+(8)
+{14}------------------------------------------------
+The denominator of (5) rewritten in (7) consists of three parts:
+$$S_1 = \sum_{i} \alpha_{X\mu}^2 \tag{9}$$
+$$S_2 = \sum_{i} 2\alpha_{X\mu}\alpha_{X\sigma}\epsilon_{X_i} \tag{10}$$
+$$S_3 = \sum_i \alpha_{X\sigma}^2 \epsilon_{X_i}^2 \tag{11}$$
+where
+$$S_1 = n\alpha_{X\mu}^2 \tag{12}$$
+$$S_2 = 2\alpha_{X\mu}\alpha_{X\sigma} \sum_{i} \epsilon_{X_i} \tag{13}$$
+$$S_3 = \alpha_{X\sigma}^2 \sum_i \epsilon_{X_i}^2. \tag{14}$$
+Given one set of parameters for the equations in (2), we see that $S_1$ is a constant, $S_2 = 2\alpha_{X\mu}\alpha_{X\sigma}S_X$ where $S_X \sim \mathcal{N}(0,n)$ , and $S_3 \sim \alpha_{X\sigma}^2 Z_{S_3}$ where $Z_{S_3} \sim \chi^2(n)$ . Here $S_2$ is the first low-dimensional auxiliary variable we need. Note that $S_3$ is **dependent on** $S_2$ and this has to be dealt with separately (see section E.2).
+Next we decompose the numerator of equation (5) in a similar way. First we replace the RVs by the definitions from (1)
+$$\sum_{i} X_{i} Y_{i} = \sum_{i} \left[ a N_{X_{i}} + b N_{X_{i}}^{2} + d N_{E} N_{X_{i}}^{2} + N_{E} N_{A_{i}} N_{X_{i}} + N_{Y_{i}} N_{X_{i}} \right].$$
+(15)
+As we are actually interested in equation (5), we can divide this expression by $\sum_i N_{X_i}^2$ (equation (6)) to cancel out some factors and get
+$$\phi = b + dN_E + \frac{a\sum_{i} N_{X_i} + N_E \sum_{i} N_{X_i} N_{A_i} + \sum_{i} N_{X_i} N_{Y_i}}{\sum_{i} X_i^2}$$
+(16)
+Remember that a, b, and d are constants. The denominator of the last term is equivalent to equation (7). We currently lack suitable expressions only for the numerator of the last term. The numerator consists of the three parts
+$$S_4 = a \sum_i N_{X_i} \tag{17}$$
+$$S_5 = N_E \sum_{i} N_{X_i} N_{A_i} \tag{18}$$
+$$S_6 = \sum_i N_{X_i} N_{Y_i}. \tag{19}$$
+$S_4$ can be expressed similarly to $S_2$ using $S_X$ , i.e. $S_4 = an\alpha_{X\mu} + a\alpha_{X\sigma}S_X$ . Both $S_5$ and $S_6$ are RVs following normal product distributions. We decompose $S_6$ as follows:
+$$S_6 = \sum_{i} \left[ (\alpha_{X\mu} + \alpha_{X\sigma} \epsilon_{X_i})(\alpha_{Y\mu} + \alpha_{Y\sigma} \epsilon_{Y_i}) \right]$$
+(20)
+$$S_6 = \sum_{i} \left[ \alpha_{X\mu} \alpha_{Y\mu} + \alpha_{X\mu} \alpha_{Y\sigma} \epsilon_{Y_i} + \alpha_{Y\mu} \alpha_{X\sigma} \epsilon_{X_i} + \alpha_{X\sigma} \alpha_{Y\sigma} \epsilon_{X_i} \epsilon_{Y_i} \right]$$
+(21)
+$\Leftrightarrow$
+$$S_6 = n\alpha_{X\mu}\alpha_{Y\mu} + \alpha_{X\mu}\alpha_{Y\sigma}S_Y + \alpha_{Y\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{Y\sigma}\sum_i \epsilon_{X_i}\epsilon_{Y_i}$$
+(22)
+where $\epsilon_{Y_i} \sim \mathcal{N}(0,1)$ and $S_Y \sim \mathcal{N}(0,n)$ . Note that $S_X$ and $S_Y$ are independent. We know that $\sum_i \epsilon_{X_i} \epsilon_{Y_i}$ follows a normal product distribution that is dependent on both $S_X$ and $S_Y$ and deal with this dependence in section E.2 below.
+{15}------------------------------------------------
+Following an almost identical derivation, we express
+$$S_5 = N_E \left[ n\alpha_{X\mu}\alpha_{A\mu} + \alpha_{X\mu}\alpha_{A\sigma}S_A + \alpha_{A\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{A\sigma}\sum_i \epsilon_{X_i}\epsilon_{A_i} \right]$$
+(23)
+where
+E.2. Separating dependent and independent parts
+It can be shown that
+and sums of squares.
+It can be shown that
+where $U_{XA} \sim \chi^2(n-1)$ and $V_{XA} \sim \chi^2(n-1)$ .
+$\epsilon_{A_i} \sim \mathcal{N}(0,1)$
+$$\epsilon_{A_i} \sim \mathcal{N}(0, 1)$$
+(24)
+$S_A \sim \mathcal{N}(0, n)$ . (25)
+# (24)
+# <span id="page-15-0"></span>Cochran's theorem tells us that we can in certain situations separate the dependent and independent parts of sums of products
+# <span id="page-15-1"></span> $S_7 = \sum_{i} \epsilon_{X_i}^2 = \sum_{i} (\epsilon_{X_i} - \bar{\epsilon_X})^2 - \frac{(\sum_{i} \epsilon_{X_i})^2}{n}$ (26)
+where $\epsilon_{X_i} \sim \mathcal{N}(0,1)$ and $\epsilon_X^-$ is the sample average, and that $\sum_i (\epsilon_{X_i} - \epsilon_X^-)^2$ is independent of $\frac{(\sum_i \epsilon_{X_i})^2}{n}$ . The first term is a sum of squares of a normal and follows a $\chi^2$ -distribution with n-1 degrees of freedom, i.e. $\sum_i (\epsilon_{X_i} - \epsilon_X^-)^2 = Z_{XX}$ where $Z_{XX} \sim \chi^2(n-1)$ . The second term is the square of the sum, where the sum is over an RV that is normally distributed. This second term in equation (26) is $\frac{\left(\sum_{i} \epsilon_{X_{i}}\right)^{2}}{n} = \frac{S_{X}^{2}}{n}$ .
+## E.2.2. DEPENDENCE OF SUM OF NORMAL PRODUCT AND SUM OF NORMALS
+E.2.1. DEPENDENCE OF SUM OF SQUARES AND (SQUARE OF) SUM
+<span id="page-15-2"></span>
+$$S_8 = \sum_i \epsilon_{X_i} \epsilon_{Y_i} = n \bar{\epsilon_X} \bar{\epsilon_Y} + \sum_i (\epsilon_{X_i} - \bar{\epsilon_X}) (\epsilon_{Y_i} - \bar{\epsilon_Y})$$
+(27)
+where bar denotes sample average so that $\bar{\epsilon_X} = \frac{\sum_i \epsilon_{X_i}}{n}$ and $\bar{\epsilon_Y} = \frac{\sum_i \epsilon_{Y_i}}{n}$ .
+The first term of equation (27) can be written with familiar quantities as $\bar{\epsilon_X} = \frac{S_X}{n}$ and $\bar{\epsilon_Y} = \frac{S_Y}{n}$ resulting in $n\bar{\epsilon_X}\bar{\epsilon_Y} = \frac{S_XS_Y}{n}$ . The second part requires more deliberation. $\sum_i (\epsilon_{X_i} - \bar{\epsilon_X})(\epsilon_{Y_i} - \bar{\epsilon_Y})$ is clearly a sum of an RV that follows a normal product distribution. To facilitate sampling, we rewrite this as
+$$\sum (\epsilon_{X_i} - \bar{\epsilon_X})(\epsilon_{Y_i} - \bar{\epsilon_Y}) = \frac{1}{4} \sum (\epsilon_{X_i} - \bar{\epsilon_X} + \epsilon_{Y_i} - \bar{\epsilon_Y})^2 + \frac{1}{4} \sum (\epsilon_{X_i} - \bar{\epsilon_X} - \epsilon_{Y_i} + \bar{\epsilon_Y})^2.$$
+(28)
+Both terms (sums of squares) in the right-hand side of this equation are sums of squared normals which are $\chi^2$ -distributed with n-1 degrees of freedom. Hence
+$$\sum (\epsilon_{X_i} - \bar{\epsilon_X})(\epsilon_{Y_i} - \bar{\epsilon_Y}) = \frac{1}{4}(2U_{XY} - 2V_{XY})$$
+(29)
+where $U_{XY} \sim \chi^2(n-1)$ and $V_{XY} \sim \chi^2(n-1)$ and these are independent.
+Following the same pattern, it can be shown that
+$$S_9 = \sum_{i} \epsilon_{X_i} \epsilon_{A_i} = \frac{S_X S_A}{n} + \frac{1}{2} (U_{XA} - V_{XA})$$
+(30)
+{16}------------------------------------------------
+#### E.3. Putting it all together
+To generate one observation of ϕ from the agent's internal model, sample αXµ, αXσ, αY µ, αY σ, αAµ, αAσ, and NE. Then sample one observation for
+$$S_X \sim \mathcal{N}(0, n) \tag{31}$$
+$$S_Y \sim \mathcal{N}(0, n) \tag{32}$$
+$$S_A \sim \mathcal{N}(0, n) \tag{33}$$
+$$Z_{XX} \sim \chi^2(n-1) \tag{34}$$
+$$U_{XY} \sim \chi^2(n-1) \tag{35}$$
+$$V_{XY} \sim \chi^2(n-1) \tag{36}$$
+$$U_{XA} \sim \chi^2(n-1) \tag{37}$$
+$$V_{XA} \sim \chi^2(n-1). \tag{38}$$
+Insert the sampled values to calculate
+$$S_9 = \frac{S_X S_A}{n} + \frac{1}{2} (U_{XA} - V_{XA}) \tag{39}$$
+$$S_8 = \frac{S_X S_Y}{n} + \frac{1}{2} (U_{XY} - V_{XY}) \tag{40}$$
+$$S_7 = Z_{XX} - \frac{S_X^2}{n} \tag{41}$$
+$$S_6 = n\alpha_{X\mu}\alpha_{Y\mu} + \alpha_{X\mu}\alpha_{Y\sigma}S_Y + \alpha_{Y\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{Y\sigma}S_8 \tag{42}$$
+$$S_5 = N_E \left( n\alpha_{X\mu}\alpha_{A\mu} + \alpha_{X\mu}\alpha_{A\sigma}S_A + \alpha_{A\mu}\alpha_{X\sigma}S_X + \alpha_{X\sigma}\alpha_{A\sigma}S_9 \right) \tag{43}$$
+$$S_4 = an\alpha_{X\mu} + a\alpha_{X\sigma}S_X \tag{44}$$
+$$S_3 = \alpha_{X\sigma}^2 S_7 \tag{45}$$
+$$S_2 = 2\alpha_{X\mu}\alpha_{X\sigma}S_X \tag{46}$$
+$$S_1 = n\alpha_{X\mu}^2. (47)$$
+With these, we can finally generate one observation from the distribution of ϕ:
+$$\phi = b + dN_E + \frac{S_4 + S_5 + S_6}{S_1 + S_2 + S_3}. (48)$$