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x/bracerightbigg =P/braceleftbiggXn(k) σn(k)σn(k) ˆσn(k)> x/bracerightbigg ≥P/braceleftbigg/braceleftigXn(k) σn(k)>x 1+ǫ/bracerightig ∩/braceleftigσn(k) ˆσn(k)≥1+ǫ/bracerightig/bracerightbigg =P/braceleftbiggXn(k) σn(k)>x 1+ǫ/bracerightbigg −P/braceleftbigg/braceleftigXn(k) σn(k)>x 1+ǫ/bracerightig ∩/braceleftig... | https://arxiv.org/abs/2503.10851v3 |
H´ajek, J. (1960). Limiting distributions in simple random sampling from a finite pop ulation. Publications of the Mathematical Institute of the Hungarian Academy of Sc iences,5361–374. Imai, K. (2008). Variance identification and efficiency analysis in randomized e xperiments under the matched-pair design. Statistics in m... | https://arxiv.org/abs/2503.10851v3 |
Statistical Impossibility and Possibility of Aligning LLMs with Human Preferences: From Condorcet Paradox to Nash Equilibrium Kaizhao Liu1, Qi Long2, Zhekun Shi1, Weijie J. Su2, and Jiancong Xiao∗2 1Peking University 2University of Pennsylvania March 17, 2025 Abstract Aligning large language models (LLMs) with diverse ... | https://arxiv.org/abs/2503.10990v1 |
. . . . . . . . . . . . . 7 2.2 Reward Model and Condorcet Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Probability of Condorcet Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Possibility of Preserving Diverse Preferences 11 3.1 Preliminaries of NLHF . . . . . . . . . . . . ... | https://arxiv.org/abs/2503.10990v1 |
. . . . . . . . . . . . . . . . . . . . . . . 35 A.2 Preference Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 A.3 Proof of Proposition A.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 B Auxiliary Lemmas 38 B.1 Properties of Tournament Graphs . . . . . . . . .... | https://arxiv.org/abs/2503.10990v1 |
52 D.6 Proof of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 D.7 Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 D.8 Proof of Lemma 3.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2 E Proofs of Main... | https://arxiv.org/abs/2503.10990v1 |
relatively preferred by human labelers, thereby presumably representing the broader human population. The LLM is then fine-tuned with respect to this reward model to generate responses that reflect high preferences in distribution. From a statistical perspective, a critical question is whether reward models can suffici... | https://arxiv.org/abs/2503.10990v1 |
two LLMs are trained to generate responses, each aiming to maximize the probability that its own response is preferred over the other’s. As such, any optimal solution of NLHF must be a Nash equilibrium of this two-player game. Although the pairwise preference modeling in NLHF can, by design, represent allhuman preferen... | https://arxiv.org/abs/2503.10990v1 |
1), which illustrates the statistical limits of aligning LLMs with human preferences. Whether human preferences can be captured by reward models depends on the presence of Condorcet cycles. In scenarios where Condorcet cycles exist (with high probability), an NLHF-aligned LLM would output a mixed strategy if no Condorc... | https://arxiv.org/abs/2503.10990v1 |
type, a phenomenon known as preference collapse. Additionally, a recent study (Li et al., 2024a) analyzed diversity in the supervised fine-tuning phase, further highlighting the challenges in maintaining preference diversity. Several preference fine-tuning methods have been proposed for RLHF. The work of Li et al. (202... | https://arxiv.org/abs/2503.10990v1 |
of candidates (responses) nis relatively small, while the number of voters (labelers) mis large and is often assumed to tend to infinity in analysis. As noted by Gehrlein (2006), analysis in the case of a small or finite m, which is our focus, is significantly more challenging. The remainder of the paper is structured ... | https://arxiv.org/abs/2503.10990v1 |
responses to a wide range of prompts. For certain prompts, preferences can be highly diverse. For instance, in subjective questions where preferences are influenced by factors such as age, gender, or cultural background, individual choices may vary significantly, even approaching a uniform distribution over all possibl... | https://arxiv.org/abs/2503.10990v1 |
welfare theory, where the reward model r(y)can be interpreted as a social welfare function. Arrow’s impossibility theorem demonstrates that when there are at least three choices, no social welfare function can simultaneously satisfy three conditions: non-imposition, non-dictatorship, and independence of irrelevant alte... | https://arxiv.org/abs/2503.10990v1 |
which is the typical regime for LLMs. Blin (1973) noted that this case is more challenging, as the events y1≻y2andy1≻y3are no longer independent. Below, we present our solution. Theorem 2.2. Suppose the population satisfies Assumption 2.2. Let the number of responses n⩾3 and the number of labelers m⩾3. Then, Pm,n(Condo... | https://arxiv.org/abs/2503.10990v1 |
methods, NLHF ensures that the learned policy is resistant to adversarial preference comparisons, leveraging the concept of Nash equilibrium from game theory. For a given prompt x, the LLM’s policy πcompetes against an opposing policy π′in a pairwise preference contest, where the objective is to find a policy that maxi... | https://arxiv.org/abs/2503.10990v1 |
there exists a Condorcet winning response, the Nash equilibrium is unique and corresponds to this Condorcet winning response. A similar result by Duersch et al. (2012) showed that Nash equilibrium in pure strategies exists in a symmetric two-player zero-sum game if and only if it is not a generalized rock-paper-scissor... | https://arxiv.org/abs/2503.10990v1 |
fixed and n→∞, our main results simplify to the following corollary. Corollary 3.8. Suppose the population satisfies Assumption 2.2. Let the number of responses be n+ 1and the number of labelers m⩾3. Then, there exists a prompt with its nresponses, Pm,n(Condorcet winning response ) =˜Θ/parenleftig n1−m l/parenrightig... | https://arxiv.org/abs/2503.10990v1 |
the highest average ranking, corresponding to the highest approximated reward. As a result, a fully optimized RLHF-trained model will collapse to always selecting y1. Thus, Example 3.10 directly leads to the following proposition: Proposition 3.11. The optimal solution of RLHF, when considered as a function of human pr... | https://arxiv.org/abs/2503.10990v1 |
contains a Condorcet cycle including all responses approaches one as n→∞(Erdős and Rényi, 1959). Therefore, the probability of |S1|=napproaches one, and all responses could possibly be included in the Nash equilibrium strategies. This demonstrates NLHF’s potential ability to preserve preferences for every single respon... | https://arxiv.org/abs/2503.10990v1 |
relaxed event {Noi∈{2,···,n+ 1}s.t.{Uj i> Uj 1}1⩽j⩽l}can be rewritten into {N1∩···∩Nl=∅}. We also define the cardinality of the random sets Xj=|Nj|, which counts the number of responses preferred over y1by individual j. We can impose the condition X1⩾···⩾Xm without loss of generality. Now, conditioning on a set of prof... | https://arxiv.org/abs/2503.10990v1 |
1 m/summationtextm j=11{Uj i>Uj 1}<1 2for anyi∈{2,...,n + 1}. •Suppose that1 m/summationtextm j=11{Uj i>Uj 1}<1 2for anyi∈ {2,...,n + 1}, but there exists t⋆:= {j⋆ 1,...,j⋆ l}∈Tsuch thatTt⋆̸=∅. Taking any i⋆∈Tt⋆, we haveUj⋆ k i⋆> Uj⋆ k 1,∀k∈[l]. Thus, we have 1 mm/summationdisplay j=11{Uj i⋆>Uj 1}⩾l m⩾1 2, which causes... | https://arxiv.org/abs/2503.10990v1 |
can be reformulated as a maximization problem. In practice, we also add an additional KL regularization term, yielding maxπmin π′Pτ/parenleftbigπ≻π′/parenrightbig, (4.1) wherePτis the regularized preference defined by Pτ(π≻π′) :=P(π≻π′)−τEx∼ρKL(π(·|x)∥πref(·|x)) +τEx∼ρKL(π′(·|x)∥πref(·|x)), 22 andτ >0is a a parameter c... | https://arxiv.org/abs/2503.10990v1 |
the rejection sampling technique to sample from a probability distribution f(·). Suppose that there exists a constant c>0and another probability distribution g(·), which is called the proposal distribution, such that f⩽c·g, then the following rejection sampling procedure (Forsythe, 1972) produces unbiased samples from ... | https://arxiv.org/abs/2503.10990v1 |
original NLHF problem (Calandriello et al., 2024). The comparison is provided in Table 4. Table 4: Comparison between different preference fine-tuning methods. Method Loop Structure On-policy Training Convergence to Nash Solution IPO-MD Double % % Online-IPO Double ! ! Nash-MD ( β̸= 0)Double ! % Nash-EMA Double ! % Nas... | https://arxiv.org/abs/2503.10990v1 |
as our main evaluation measure. Given a prompt in the test set, let y1andy2be two responses generated by two models π1andπ2. The win rate of π1overπ2is the percentage of cases where the reward for y1is greater than the reward for y2. Evaluation and Results To evaluate the fine-tuned model, we calculate the mean reward ... | https://arxiv.org/abs/2503.10990v1 |
Another important direction involves extending NLHF so that, when human preferences follow the BTL model, the aligned policy precisely matches the BTL distribution—perhaps by regularizing the payoff in the NLHF two-player game. In this context, it is noteworthy that RLHF with appropriate regularization can match the BT... | https://arxiv.org/abs/2503.10990v1 |
Shmitchell. On the dangers of stochastic parrots: Can language models be too big? In Proceedings of the 2021 ACM conference on fairness, accountability, and transparency , pages 610–623, 2021. J.-M. Blin. Intransitive social orderings and the probability of the condorcet effect. Kyklos, 26(1):25–35, 1973. R. A. Bradley... | https://arxiv.org/abs/2503.10990v1 |
J. Wei, K. Meier-Hellstern, D. Eck, J. Dean, S. Petrov, and N. Fiedel. Palm: Scaling language modeling with pathways. Journal of Machine Learning Research , 24(240):1–113, 2023. H. Dong, W. Xiong, D. Goyal, Y. Zhang, W. Chow, R. Pan, S. Diao, J. Zhang, S. KaShun, and T. Zhang. Raft: Reward ranked finetuning for generat... | https://arxiv.org/abs/2503.10990v1 |
A. Karpathy, A. Braunstein, A. Cann, A. Codispoti, A. Galu, A. Kondrich, A. Tulloch, A. Mishchenko, A. Baek, A. Jiang, A. Pelisse, A. Woodford, A. Gosalia, A. Dhar, A. Pantuliano, A. Nayak, A. Oliver, B. Zoph, B. Ghorbani, B. Leimberger, B. Rossen, B. Sokolowsky, B. Wang, B. Zweig, B. Hoover, B. Samic, B. McGrew, B. Sp... | https://arxiv.org/abs/2503.10990v1 |
Bavarian, M. Lin, M. Yesildal, N. Soto, N. Gimelshein, N. Cone, N. Staudacher, N. Summers, N. LaFontaine, N. Chowdhury, N. Ryder, N. Stathas, N. Turley, N. Tezak, N. Felix, N. Kudige, N. Keskar, N. Deutsch, N. Bundick, N. Puckett, O. Nachum, O. Okelola, O. Boiko, O. Murk, O. Jaffe, O. Watkins, O. Godement, O. Campbell-... | https://arxiv.org/abs/2503.10990v1 |
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approach to preference rankings. In ICML 2023 Workshop The Many Facets of Preference-Based Learning , 2023. G. Swamy, C. Dann, R. Kidambi, S. Wu, and A. Agarwal. A minimaximalist approach to reinforcement learning from human feedback. In International Conference on Machine Learning , pages 47345–47377. PMLR, 2024. S. T... | https://arxiv.org/abs/2503.10990v1 |
modeling with preference representations for aligning language models. arXiv preprint arXiv:2410.02197 , 2024. Y. Zhang, D. Yu, B. Peng, L. Song, Y. Tian, M. Huo, N. Jiang, H. Mi, and D. Yu. Iterative nash policy optimization: Aligning llms with general preferences via no-regret learning. In The Thirteenth Internationa... | https://arxiv.org/abs/2503.10990v1 |
to maximize a nonlinear transformation of the preference relative to a fixed reference policy πref: maxπEx∼ρ[Ey∼π(·|x)Ey′∼πref(·|x)Ψ/parenleftbigP(y≻y′|x)/parenrightbig−τKL(π(·|x)∥πref(·|x))] (A.1) whereτ >0is a parameter controlling the deviation from the base reference policy πref. Above, the second term is the regul... | https://arxiv.org/abs/2503.10990v1 |
Lemma B.1. Consider a tournament graph Gwithn(n⩾3)pointsy1,...,yn. If for any y∈G, there exists y′̸=ysuch thaty′→y, then there exists a cycle in this graph. In other words, if there is no cycle in this graph, there exists y⋆∈Gsuch thaty⋆→y′for anyy′̸=y⋆. Proof of Lemma B.1. Starting from an arbitrary vertex yi1∈G, we c... | https://arxiv.org/abs/2503.10990v1 |
Case 2: there exists a cycle in GWe choose S⋆to be a maximum cycle, i.e. we cannot add any pointy∈G\S⋆intoS⋆such thatS⋆∪{y}also forms a cycle7. Then we introduce the following crucial conclusion. Conclusion 1 GivenS⋆, for anyy∈G\S⋆, it must hold either y→y⋆for anyy⋆∈S⋆ory⋆→y for anyy⋆∈S⋆. Proof of the Conclusion 1 We p... | https://arxiv.org/abs/2503.10990v1 |
Proposition 2.7 Proof.We reformulate this problem in terms of a directed graph G. Specifically, we represent the nresponsesy1,...,ynas vertices 1,...,n, and we introduce a directed edge i→jif and only ifP(yi≻yj)>1/2. Under this reformulation, a Condorcet cycle in the preference structure corresponds to a cycle in G. No... | https://arxiv.org/abs/2503.10990v1 |
12−1 12 −1 121 4−1 12 −1 12−1 121 4 is the covariance matrix. Consider (Z1,Z2,Z3)∼N(0,Σ), then we obtain that lim m→∞P/parenleftig P(y1≻y2)>1 2,P(y2≻y3)>1 2,P(y3≻y1)>1 2/parenrightig = lim m→∞P/parenleftigg 1√mm/summationdisplay j=1/parenleftig X12 j−1 2/parenrightig >0,1√mm/summationdisplay j=1/parenleftig X2... | https://arxiv.org/abs/2503.10990v1 |
Theorem 3.1 Proof of Theorem 3.1. First, suppose that there exists a Nash equilibrium in pure strategies that choosesy⋆. Then, using Lemma B.6, we obtain that (y⋆,y⋆)is a Nash equilibrium. For any y, we have P(y⋆≻y)⩾minσP(y⋆≻σ) =P(y⋆≻y⋆) =1 2. Therefore, for any y̸=y⋆, we haveP(y⋆≻y)>1 2, as we have assumed that P(y≻y′... | https://arxiv.org/abs/2503.10990v1 |
statistics X(j)with X(1)⩾X(2)⩾...⩾X(m). Particularly, for any i= 1,...,n + 1andj= 1,...,m, we define U(j) ias the score of the individual corresponding to X(j)for thei-th response. Let l=⌈m 2⌉. Asm⩾2, we havel⩾1. Step 1 WhenX(1)...X (l)⩾nl−1(logn)2, we claim that P/parenleftbigg ̸∃i∈{2,···,n+ 1}s.t./braceleftig U(j) i... | https://arxiv.org/abs/2503.10990v1 |
1for all 1⩽j⩽l, then there exists isuch that 1 m/summationtextm j=11{Uj i>Uj 1}⩾1 2. Therefore, we have P/parenleftbigg ∃i∈{2,···,n+ 1}s.t./braceleftig U(j) i>U(j) 1/bracerightig 1⩽j⩽l/parenrightbigg ⩽P ∃i∈{2,···,n+ 1}s.t.1 mm/summationdisplay j=11{Uj i>Uj 1}⩾1 2 , which leads to the following inequality: P ∀i∈... | https://arxiv.org/abs/2503.10990v1 |
P(˜π≻π) =/summationdisplay ij˜πiπjP(yi≻yj) =/summationdisplay yi,yj∈S1˜πiπjP(yi≻yj) +/summationdisplay yi∈S1,yj∈Sc 1˜πiπjP(yi≻yj) =1 2+/summationdisplay yi,yj∈S1˜πiπj/parenleftbigg P(yi≻yj)−1 2/parenrightbigg +/summationdisplay yi∈S1,yj∈Sc 1˜πiπj/parenleftbigg P(yi≻yj)−1 2/parenrightbigg . Noting that/summationtext yi,... | https://arxiv.org/abs/2503.10990v1 |
4.3 The policy gradient of Z(x,πϕ)is given by ∇ϕZ(x,πϕ) =1 τ·Ey′∼πref(·|x)/bracketleftbig Ey∼πϕ(·|x)[∇ϕlogπϕ(y|x)·P(y≻y′|x)] exp/parenleftbig −1 τ·Ey∼πϕ(·|x)[P(y≻y′|x)]/parenrightbig/bracketrightbig Ey′∼πref(·|x)/bracketleftbig exp/parenleftbig −1 τ·Ey∼πϕ(·|x)[P(y≻y′|x)]/parenrightbig/bracketrightbig =1 τ·Ey∼πϕ(·|x),y′... | https://arxiv.org/abs/2503.10990v1 |
0.2 0.4 0.6 0.8 1.0 preference ratio a−102−101−1000100101102π1=0.6,τ=0.01 RLHFr1 RLHFr2 NashRSr1 NashRSr2 0.0 0.2 0.4 0.6 0.8 1.0 preference ratio a−103−102−101−1000100101102103π1=0.6,τ=0.001 RLHFr1 RLHFr2 NashRSr1 NashRSr2 0.0 0.2 0.4 0.6 0.8 1.0 preference ratio a−104−103−102−101−1000100101102103104π1=0.6,τ=0.0001 RL... | https://arxiv.org/abs/2503.10990v1 |
Page 1 of 41 THE PUSHED BETA DISTRIBUTION AND CONTAMINATED BINARY SAMPLING BEN O’NEILL*, ACIL Allen** WRITTEN 14 MARCH 2025 Abstract We examine a generalisation of the beta distribution that we call the pushed beta distribution. This is a continuous univariate distribution on the unit interval which generalises the bet... | https://arxiv.org/abs/2503.11128v1 |
a continuous univariate distribution with support inside the unit interval. It has probability density function over 0 ≤𝑥≤ 1 given by:1 PushBeta (𝑥|𝛼,𝛽,𝛾,𝜙,𝜅)=𝑥ఈିଵ(1−𝑥)ఉିଵ(1−𝑥𝜙)(ଵି)ఊ(1−𝜙+𝑥𝜙)ఊ B(𝛼,𝛽)𝐹ଵଶ(−𝛾,𝛼ଵି𝛽,𝛼+𝛽;𝜙). where 𝛼> 0 and 𝛽> 0 are the shape parameters , 𝛾≥ 0 is the push-intensit... | https://arxiv.org/abs/2503.11128v1 |
cases respectively) in terms of first-order stochastic dominance. THEOREM 1A (Stochastic dominance for left-pushed distribution): Take push-proportions 0 ≤𝜙<𝜙ଵ≤ 1 and any positive push-intensity 𝛾> 0 and define the random variables: 𝑋 ~ LPushBeta (𝛼,𝛽,𝛾,𝜙)𝑋ଵ ~ LPushBeta (𝛼,𝛽,𝛾,𝜙ଵ). Then 𝑋 stochastical... | https://arxiv.org/abs/2503.11128v1 |
. The assumption that we know the probability parameter 𝜙 for the contaminating sequence is crucial to the model because the product 𝜃𝜙 is the minimal sufficient parameter (O’Neill 2005) — it is not possible to identify either 𝜃 or 𝜙 if both are unknown. The left-pushed beta distribution arises from this model usi... | https://arxiv.org/abs/2503.11128v1 |
right-pushed beta distribution arises from this model in an analogous way to the previous case. This alternative contaminated binary model gives us the sampling density: 𝑓(𝒙|𝜃)=ෑ((1−𝜃)𝜙)௫(1−𝜙+𝜃𝜙)ଵି௫ ୀଵ=((1−𝜃)𝜙)௫̇(1−𝜙+𝜃𝜙)ି௫̇. Page 7 of 41 Using the prior 𝜃 ~ RPushBeta (𝛼,𝛽,𝛾,𝜙) for the unknown ... | https://arxiv.org/abs/2503.11128v1 |
means that the left-push makes the density more decreasing (less increasing) and the right-push makes the density more increasing (less decreasing). The push contribution in the second derivative is negative for both types of push, which means that both types of push make the density more concave (less convex) than it ... | https://arxiv.org/abs/2503.11128v1 |
conditions are sufficient for the right-pushed beta density to be monotonic over its support: (a) If 𝛼= 1, 𝛽= 1, 0 <𝜙≤ 1 and 𝛾> 0 (strictly increasing); (b) If 𝛽≤ 1 and 𝛼> 1 (strictly increasing); (c) If 𝛽< 1 and 𝛼≥ 1 (strictly decreasing); (d) If 𝛽≤ 1 and 𝛼+𝛾𝜙ଶ> 1 (strictly decreasing); (e) If 𝛽≥ 1, 𝛼< 1... | https://arxiv.org/abs/2503.11128v1 |
can examine its moments. In Theorem 6 below we show the raw moments of the pushed beta distribution. As with the density function these do not have a closed form and they involve integration of the density kernel, which is equivalent to computation of the beta and hypergeometric functions. The mean and variance formula... | https://arxiv.org/abs/2503.11128v1 |
we get zero KL divergence when 𝜃=𝜃. This is a comforting property and it shows that correct specification of the contaminated binomial model leads to sensible properties for the KL divergence. THEOREM 7A (KL divergence for contaminated binomial model): For the contaminated binomial model in Table 1 the Kullback-Leib... | https://arxiv.org/abs/2503.11128v1 |
sidesteps the particular form of the moments for the pushed beta distribution shown in Theorem 6 above. To augment these results, the reader may find it edifying to see a simple heuristic demonstration of why the form of the moments in Theorem 6 leads to the asserted posterior converge — in particular, why these moment... | https://arxiv.org/abs/2503.11128v1 |
to the true value of the parameter 𝜃 (i.e., the model displays posterior consistency). The theorem also shows the consequences of mis-specification, which is posterior inconsistency in which the posterior converges to a point mass on the broader value 𝜃∗. It should be unsurprising that the contaminated binomial model... | https://arxiv.org/abs/2503.11128v1 |
left-pushed case, taking any fixed value for 𝜙, the other distribution parameters can be maximised (conditional on 𝜙) by setting the first three partial derivatives in the score function to zero, which is equivalent to solving the method-of-moments equations: 1 𝑛log(𝑥) ୀଵ=𝔼(log(𝑋)), 1 𝑛log(1−𝑥) ୀଵ=𝔼(l... | https://arxiv.org/abs/2503.11128v1 |
get around this problem, we use a midpoint quadrature method based on sample points taken from the quantile function for the beta distribution, which concentrate near to the peak of the kernel function, reducing arithmetic underflow in the integral approximation. To do this we first define the function: 𝑆(𝑥|𝜙)≡ log(... | https://arxiv.org/abs/2503.11128v1 |
write the various probability functions for the pushed beta distribution. The density function, cumulative distribution function and raw moments are given on the log-scale as: logPushBeta (𝑥|𝛼,𝛽,𝛾,𝜙,𝜅)=(𝛼−1)log(𝑥)+(𝛽−1)log(1−𝑥)−𝐻(1|𝛼,𝛽,𝛾,𝜙) + (1−𝜅)𝛾log(1−𝑥𝜙)+𝜅𝛾log(1−𝜙+𝑥𝜙), logCDF ୳ୱ୦ୣ୲ୟ(𝑥|𝛼... | https://arxiv.org/abs/2503.11128v1 |
bias. Page 26 of 41 In such cases, one potentially valuable experimental protocol is for the researcher to introduce “controlled contamination” of the survey, whereby the response of the participant to a sensitive question is contaminated with an event with known probability (but unknown outcome). This can occur by ask... | https://arxiv.org/abs/2503.11128v1 |
a “No” answer for participant 𝑖. Then the answers to the survey question for the 𝑛 participants in the survey are contaminated Bernoulli values:8 𝑋ଵ,𝑋ଶ,…,𝑋 ~ IID Bern ((1−𝜃)𝜙), where 𝜃 is the true prevalence of tax cheating in the population and 𝜙= ⅓ is the probability of a non-contaminated answer. Suppose th... | https://arxiv.org/abs/2503.11128v1 |
reason, we recommend that contaminated binomial sampling should only be used when the contamination is under the control of the experimenter, such that it is known. References BAILEY, W.N. (1935) Generalised Hypergeometric Series . University Press: Cambridge. BERK, R.H. (1966) Limiting behavior of posterior distributi... | https://arxiv.org/abs/2503.11128v1 |
OF THEOREM 2B: Analogous to the proof of Theorem 2A. ■ PROOF OF THEOREM 3A: To establish monotonicity (either constant, increasing or decreasing) we need to show the appropriate sign for the first derivative of the log-density over the interior of the support. By way of reminder, for all 0 <𝑥< 1 this first derivative ... | https://arxiv.org/abs/2503.11128v1 |
small 𝛾. Specifically, taking 𝛽+𝛾𝜙ଶ< 1 gives: 𝛽−1 (1−𝑥)ଶ−𝛾𝜙ଶ (1−𝑥𝜙)ଶ>𝛽−𝛾𝜙ଶ−1 (1−𝑥𝜙)ଶ> 0. Under this condition the first term is non-negative and the sum of the second and third terms is positive, so the total is positive (yielding a strictly quasi-convex function function). This establishes each of the q... | https://arxiv.org/abs/2503.11128v1 |
then we also have a positive discriminant which then yields the result of the theorem (i.e., this is the boundary where 𝛾 is “sufficiently small” for the result). ■ PROOF OF THEOREM 5B: Analogous to the proof of Theorem 5A. ■ PROOF OF THEOREM 6A: We first establish the alternative form as: 𝔼(𝑋)=න𝑥PushBeta (𝑥|𝛼,... | https://arxiv.org/abs/2503.11128v1 |
to meet two conditions — the “prior support” condition and the “testing condition”. We now show that our model meets both of these conditions: (a) Prior support condition: The present model uses the fixed parameter space Θ =[0,1] and a prior that encompasses the interior of the parameter space in its support. From Theo... | https://arxiv.org/abs/2503.11128v1 |
derivatives: 𝜕ℓ௫ 𝜕𝛼(𝛼,𝛽,𝛾,𝜙,1)= log(𝑥)−𝜕 𝜕𝛼logቌන𝑥ఈିଵ(1−𝑥)ఉିଵ(1−𝜙+𝑥𝜙)ఊ𝑑𝑥ଵ ቍ = log (𝑥)−∫log(𝑥)𝑥ఈିଵ(1−𝑥)ఉିଵ(1−𝜙+𝑥𝜙)ఊ𝑑𝑥ଵ ∫𝑥ఈିଵ(1−𝑥)ఉିଵ(1−𝜙+𝑥𝜙)ఊ𝑑𝑥ଵ = log (𝑥)−නlog(𝑥)RPushBeta (𝑥|𝛼,𝛽,𝛾,𝜙)𝑑𝑥ଵ = log(𝑥)−𝑅ଵ(𝛼,𝛽,𝛾,𝜙), 𝜕ℓ௫ 𝜕𝛽(𝛼,𝛽,𝛾,𝜙,1)= log(1−𝑥)−𝜕 𝜕𝛽logቌන𝑥ఈିଵ(1−�... | https://arxiv.org/abs/2503.11128v1 |
On continuity of Chatterjee’s rank correlation and related dependence measures Jonathan Ansari1and Sebastian Fuchs1 1University of Salzburg, Austria 2025-03-17 Abstract While measures of concordance—such as Spearman’s rho, Kendall’s tau, and Blomqvist’s beta— are continuous with respect to weak convergence, Chatterjee’... | https://arxiv.org/abs/2503.11390v1 |
alln∈N. Thusξfails to be continuous with respect to weak convergence1. The lack of weak continuity illustrated in Example 1.1 has some drawbacks, particularly when testing independence between XandYresulting in trivial power against certain alternatives, see Bücher and Dette [7]. However, as we study in Section 2, Chat... | https://arxiv.org/abs/2503.11390v1 |
RP(Y <y )2dPY(y),both depending only on Ran(FY). IfFYis continuous, then Ran(FY) = [0,1],and the constants in (4) take the values a= 6andb= 2. Further, note that for distribution functions FandG,we have the identity Ran(F) =Ran(G)⇐⇒F◦F−1(t) =G◦G−1(t)forλ-almost all t∈(0,1), (5) see [1, Proposition 2.14]. Hence, to obta... | https://arxiv.org/abs/2503.11390v1 |
to prove lim n→∞/integraldisplay RP(Yn<y,Y′ n<y) dPYn(y) =/integraldisplay RP(Y <y,Y′<y) dPY(y). (8) Then the statement follows from lim n→∞ξ(Yn,Xn) = lim n→∞/parenleftbigg an/integraldisplay RP(Yn<y,Y′ n<y) dPYn(y)−bn/parenrightbigg =a/integraldisplay RP(Y <y,Y′<y) dPY(y)−b=ξ(Y,X) applying (7) and (8) as well as the r... | https://arxiv.org/abs/2503.11390v1 |
related dependence measures are defined by comparing conditional distributions. Hence, studying continuity of these measures requires to study continuity of conditional distributions. To this end, we make use of the concept of conditional weak convergence for which we use the following concepts. 5 Ford∈N,denote byU(Rd)... | https://arxiv.org/abs/2503.11390v1 |
variables and assume that Xorεhas a continuous distribution function. Consider the model Y=X+σεfor some parameter σ >0.Then the Markov product (Y,Y′) of(X,Y )is weakly continuous in σ,andξ(Y,X)is continuous in σ. Proof:Letε′be a copy of εthat is independent from (X,ε).Letσnbe a non-negative sequence of real numbers con... | https://arxiv.org/abs/2503.11390v1 |
X(t)for almost all t∈(0,1). Then we have CYn,Y′n→CY,Y′uniformly. If additionally (iii)FYn◦F−1 Yn(t)→FY◦F−1 Y(t)for almost all t∈(0,1), it follows that ξ(Yn,Xn)→ξ(Y,X). Proof:Define the rank transformed random variables Zn:=FYn(Yn)andZ:=FY(Y).Since Chatter- jee’s rank correlation is invariant under distributional transf... | https://arxiv.org/abs/2503.11390v1 |
a continuity condition in the conditioning variable nor weak convergence of the marginal distributions. (b) Condition (ii)in Theorem 3.5 on range convergence of FXnis used for the pointwise convergence of the copulas CYn,Y′n. Note that the copula CYn,Y′nin(25)is given through a generalization of the ∗- product in (19)a... | https://arxiv.org/abs/2503.11390v1 |
X. The following continuity result for the measure Tdefined by (26) is an immediate consequence of Theorem 2.2 and Theorem 3.1. Corollary 4.1 (Continuity of T)Let(X,Y)be(p+q)-dimensionalrandom vectorand let (Xn,Yn)n∈N be a sequence of (p+q)-dimensional random vectors. Let V1⊂Rpbe open such that P(X∈V1) = 1and, for alli... | https://arxiv.org/abs/2503.11390v1 |
particularly Theorem 3.1, also apply to the kernel partial correlation. Theoptimaltransport-basedWassersteincorrelationcoefficientstudiedinWiesel[39]underliessimilar modes of convergence. Due to Wiesel [39, Theorem 4.1], it is continuous with respect to the adapted Wassersteindistance,whichalsoaccountsforthedistancebet... | https://arxiv.org/abs/2503.11390v1 |
(a)(fn)n∈Nis asymptotically uniformly equicontinuous on (0,∞),and (b)(fn)n∈Nis pointwise bounded, i.e., M(x) := supn∈Nfn(x)<∞for allx∈(0,∞), then conditions (i)–(iv)in Corollary 4.1 are satisfied and thus T(Yn,Xn)→T(Y,X). As an application of Theorem 5.1(i), we immediately obtain the following continuity result for the... | https://arxiv.org/abs/2503.11390v1 |
From [2, Example A.7], we obtain in the case of a normal distribution the closed-form expression T(Y,X) = 1−3−3 π/bracketleftig arcsin/parenleftig 1 2+ρ2 XY 1+ρX/parenrightig + arcsin/parenleftig 1 2+(1+ρX)ρ2 Y−2(2ρY−1)ρ2 XY 2(1+ρX)−4ρ2 XY/parenrightig/bracketrightig 5 2−3 πarcsin/parenleftig1+ρ2 Y 2/parenright... | https://arxiv.org/abs/2503.11390v1 |
(Xn,Yn)d−−→(X,Y). In order to verify condition (ii) of Corollary 4.1, we restrict to the case q= 1. As in the proof of Theorem 3.1, we use Sethuraman [31, Lemma 3 & Lemma 4] in combination with the characterization of uniform conditional convergence in Sweeting [36, Theorem 4]. Let Vbe the column space of Σ11in (30) wi... | https://arxiv.org/abs/2503.11390v1 |
we denote byf.We denote the denominator of (42) as G(a2) :=/integraldisplay (a,∞)(s2−a2)−1/2s−(p−1)dFR(s) =/integraldisplay (0,∞)(z2+a2)−p/2f(/radicalbig z2+a2) dz. 17 Similarly, the modified numerators of (41) and (42) are Hn(a2) :=/integraldisplay (a,√ r2+a2)(s2−a2)−1/2s−(p−1)dFRn(s) =/integraldisplay (0,r)(z2+a2)−p/... | https://arxiv.org/abs/2503.11390v1 |
Theorem 5.7. Sinceℓ1-norm symmetric distributions are closed under marginalization Mc- Neil and Neslehová [25, Theorem 3.1], we may restrict to the case q= 1.Let(Xn,Yn)d=RnSp+1 be a sequence of ℓ1-norm symmetric random vectors and let (X,Y)d=RSp+1such thatFRnandFR are continuous with FRn(0) =FR(0) = 0 for alln∈N.Then, ... | https://arxiv.org/abs/2503.11390v1 |
Azadkia and S. Chatterjee. A simple measure of conditional dependence. Ann. Stat. , 49(6): 3070–3102, 2021. [6] P. Billingsley. Convergence of Probability Measures . John Wiley & Sons, second edition, 1999. [7] A. Bücher and H. Dette. On the lack of weak continuity of Chatterjee’s correlation coefficient. Available at ... | https://arxiv.org/abs/2503.11390v1 |
R. B. Nelsen. An Introduction to Copulas. Springer, New York, 2nd edition, 2006. [30] W. Rudin. Real and Complex Analysis. McGraw-Hill, New York, 3rd edition, 1987. [31] J. Sethuraman. Some limit theorems for joint distributions. Sankhy¯ a, Ser. A , 23:379–386, 1961. [32] M. Shaked and J. G. Shantikumar. Stochastic Ord... | https://arxiv.org/abs/2503.11390v1 |
Towards practical PDMP sampling: Metropolis adjustments, locally adaptive step-sizes, and NUTS-based time lengths A. Chevallier1, S. Power2, and M. Sutton3 1Universit´ e de Strasbourg 2University of Bristol 3Queensland University of Technology March 17, 2025 Abstract Piecewise-Deterministic Markov Processes (PDMPs) hol... | https://arxiv.org/abs/2503.11479v1 |
preserving the Metropolis correction. One approach incorporates the step size into the state space of the Markov chain itself [14, 15], while another leverages delayed rejection techniques [16, 17, 18]. In all of these methods, the step size remains constant throughout a single HMC iteration but is adapted between iter... | https://arxiv.org/abs/2503.11479v1 |
simulation of PDMPs then reduces to i) solving the ODE corresponding to the determin- istic motion, ii) simulating event times, and iii) simulating the jumps which occur at the jump times. For most PDMPs of practical interest, the process is designed so that i) and iii) are trivially solvable, and so the main challenge... | https://arxiv.org/abs/2503.11479v1 |
No U-Turn Criterion The pre-eminent example of a kinetic MCMC procedure is arguably the Hamiltonian Monte Carlo (HMC) algorithm [10]. Given a kinetic particle ( x, v), HMC generates proposal moves by numerically simulating Hamiltonian dynamics with respect to the Hamiltonian function H(x, v) =−logπ(x) + 1 2|v|2, and th... | https://arxiv.org/abs/2503.11479v1 |
T] is defined as ZT−tfort∈[0, T]. It has been shown [23] that the time-reversed process over [0 , T] is also a PDMP, whose characteristics can be identified more-or-less explicitly; we write λrandQrfor its jump rate and jump kernel. The general construction of the reverse process, that is λrandQr, can be found in [23].... | https://arxiv.org/abs/2503.11479v1 |
base measure. To that end, it is crucial to precisely define what the path space is, and the probability measure with which it is equipped. The path of a piecewise deterministic Markov process (PDMP) ( Zt)t∈[0,T]is fully determined by itsskeleton , which consists of the initial state Z0, the event times τi, and the sta... | https://arxiv.org/abs/2503.11479v1 |
whose density with respect to the reference measure on path space is readily available. We then simulate the corresponding process for a fixed time T. By construction, we can compare this proposed path to its time-reversal (whose density is similarly available), and then use this as the basis for a Metropolis-adjustmen... | https://arxiv.org/abs/2503.11479v1 |
computed exactly. The first condition ensures that the kernel that associates (˜ z0, γ) to (˜ zT, γ) is a Markov kernel, the second condition ensures that we can simulate the process in practice, and the third condition allows for the Metropolis adjustment. Piecewise constant approximation The simplest approximation is... | https://arxiv.org/abs/2503.11479v1 |
tries to find a hsuch that |λ(t)−λ(t+h)|<tol is not scale invariant. 3.4 Metropolis-Adjustment of PDMP Discretisations We assume a similar assumption to Assumption 1 for the backward process. Assumption 3 (Reverse jump kernel parametrisation) .There exists a space Vrwith a canonical measure λVrsuch that for each x∈E,Qr... | https://arxiv.org/abs/2503.11479v1 |
kernel does not need to be parameterized, and therefore V={0} 11 is a singleton (where 0 is meaningless). The jump is then defined as ξ(x,v):V→E 07→(x, v−2(∇π(x)·v)∇π(x)). The reverse process is similar, and Vr=V, ξr z=ξ−1 z. The application Rcan be then written as: R(z0, t0, v0, ..., t k, vk) =R(zT, T−tk, vk, ..., T −... | https://arxiv.org/abs/2503.11479v1 |
reduce the distance with any other point of the trajectory, while the second indicates the same when building the trajectory backward in time. The two conditions can be rewritten as (xt2−xt1)·v2>= 0, (xt2−xt1)·v1>= 0. We call a stopping criterion consistent if it satisfies the following assumption: Assumption 5 (consis... | https://arxiv.org/abs/2503.11479v1 |
density ν(t) =T−t T2if the criterion was hit going forward, and ν(t) =t T2if the criterion was hit backward. 14 Proof. We give here the intuition of the proof; Section 4.3 provided a more complete rigorous proof. In principle, when looking at a path, it is impossible to distinguish whether the process was built forward... | https://arxiv.org/abs/2503.11479v1 |
1, vb 1..., tb kb, vb kb, tf 1, vf 1, ..., tf kf, vf kf). 15 Lemma 1 (probability density on Fg).LetSg:Fg7→ {0,1}be the function that is 1 if the path was actually stopped on the left, and is 0otherwise. Then, the stopped PDMP induces a probability density onFgwith density: pFg(z, tb k1, vb k1, ..., tf k2, vf k2, b) =p... | https://arxiv.org/abs/2503.11479v1 |
Fgto a path in FX. In other words, the pushforward measure of pFgbygis the law of ( X, l). The application gapplies Rtk1to the reversed part of the path and shifts the rest by time −tk1: g:Fg7→E× ∪k {0} ×V×(R+×V)k−1 ×R2⊂FX (z, tb k1, vb k1, ..., tf k2, vf k2, b)7→(R−tk1(z, tb k1, vb k1, ..., tb 1, vb 1), tf 1−tb k1, ... | https://arxiv.org/abs/2503.11479v1 |
reversal Rtand its volume change Ψtare continuous in t. Using the same algorithm to build (X, l)as in Section 4 but using an approximate process yields the following νon[0, T]for the law of l|X: ν(l)∝π(Xl)qr [−l,0]((Xt+l)t∈[−l,0]|Xl)q[0,T−l]((Xt+l)t∈[T−l,T]|Xl) T−l T2 Ψl(X[0,l]), if the process was stopped forward, or ... | https://arxiv.org/abs/2503.11479v1 |
of gradient evaluations. 6.1.1 Exact BPS-NUTS In the case of a multivariate normal distribution N(0d, Id), BPS is expected to require O(√ d) events per independent sample (see [25] for a detailed analysis). We expect the O(√ d) scaling to hold for the NUTS version of exact BPS. Remark 5. For Gaussian targets, the order... | https://arxiv.org/abs/2503.11479v1 |
divided by the computational complexity. Figure 2(a) clearly indicates that the value of hshould not be scaled with dimension. Moreover, the acceptance rate does not appear to converge to a non-zero value when using this optimal h, as shown in Figure 2(b). An optimal acceptance rate converging to 0 would represent a si... | https://arxiv.org/abs/2503.11479v1 |
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