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[EQUATION] where [MATH] and [MATH] are non-negative functions. If [MATH] can be factorised as indicated, the summary statistic [MATH] |
will yield the same inference about the parameters [MATH] as the full set of observations [MATH] . When nuisance parameters have to be accounted in the inference procedure, alternate notions of sufficiency are commonly used such as partial or marginal sufficiency \autocites basu2011partialsprott1975marginal. Nonetheles... |
For simplicity, let us consider a problem where we are only interested on statistical inference on a single one-dimensional model parameter |
[MATH] given some observed data. Be given a summary statistic [MATH] and a statistical procedure to obtain an unbiased interval estimate of the parameter of interest which accounts for the effect of nuisance parameters. The resulting interval can be characterised by its width |
[MATH] , defined by some criterion so as to contain on average, upon repeated samping, a given fraction of the probability density, e.g. a central [MATH] |
interval. The expected size of the interval depends on the summary statistic [MATH] chosen: in general, summary statistics that are more informative about the parameters of interest will provide narrower confidence or credible intervals on their value. Under this figure of merit, the problem of choosing an optimal summ... |
[MATH] that minimises the interval width: [EQUATION] The above construction can be extended to several parameters of interest by considering the interval volume or any other function of the resulting confidence or credible regions. |
Method In this section, a machine learning technique to learn non-linear sample summary statistics is described in detail. The method seeks to minimise the expected variance of the parameters of interest obtained via a non-parametric simulation-based synthetic likelihood. A graphical description of the technique is dep... |
The family of summary statistics [MATH] considered in this work is composed by a neural network model applied to each dataset observation |
[MATH] whose parameters [MATH] will be learned during training by means of stochastic gradient descent, as will be discussed later. Therefore, using set-builder notation the family of summary statistics considered can be denoted as: |
[EQUATION] where [MATH] will reduce the dimensionality from the input observations space [MATH] to a lower-dimensional space [MATH] . The next step is to map observation outputs to a dataset summary statistic, which will in turn be calibrated and optimised via a non-parametric likelihood |
[MATH] created using a set of simulated observations [MATH] , generated at a certain instantiation of the simulator parameters [MATH] |
In experimental high energy physics experiments, which are the scientific context that initially motivated this work, histograms of observation counts are the most commonly used non-parametric density estimator because the resulting likelihoods can be expressed as the product of Poisson factors, one for each of the con... |
[MATH] , so each element of the sample summary will correspond to the following sum: [EQUATION] which can in turn be used to build the following likelihood, where the expectation for each bin is taken from the simulated sample [MATH] |
[EQUATION] where the [MATH] factor accounts for the different number of observations in the simulated samples. In cases where the number of observations is itself a random variable providing information about the parameters of interest, or where the simulated observation are weighted, the choice of normalisation of |
[MATH] may be slightly more involved and problem specific, but nevertheless amenable. In the above construction, the chosen family of summary statistics is non-differentiable due to the [MATH] operator, so gradient-based updates for the parameters cannot be computed. To work around this problem, a differentiable approx... |
[MATH] is considered. This function is defined by means of a [MATH] operator: [EQUATION] where the temperature hyper-parameter [MATH] will regulate the softness of the operator. In the limit of [MATH] , the probability of the largest component will tend to 1 while others to 0, and therefore |
[MATH] Similarly, let us denote by [MATH] the differentiable approximation of the non-parametric likelihood obtained by substituting [MATH] with |
[MATH] . Instead of using the observed data [MATH] , the value of [MATH] may be computed when the observation for each bin is equal to its corresponding expectation based on the simulated sample [MATH] , which is commonly denoted as the Asimov likelihood |
[MATH] [EQUATION] for which it can be easily proven that [MATH] so the maximum likelihood estimator (MLE) for the Asimov likelihood is the parameter vector [MATH] used to generate the simulated dataset [MATH] . In Bayesian terms, if the prior over the parameters is flat in the chosen metric, then [MATH] |
is also the maximum a posteriori (MAP) estimator. By taking the negative logarithm and expanding in [MATH] around [MATH] , we can obtain the Fisher information matrix |
for the Asimov likelihood: [EQUATION] which can be computed via automatic differentiation if the simulation is differentiable and included in the computation graph or if the effect of varying [MATH] |
over the simulated dataset [MATH] can be effectively approximated. While this requirement does constrain the applicability of the proposed technique to a subset of likelihood-free inference problems, it is quite common for e.g. physical sciences that the effect of the parameters of interest and the main nuisance parame... |
If [MATH] is an unbiased estimator of the values of [MATH] , the covariance matrix fulfils the Cramér-Rao lower bound \autocites cramer2016mathematicalrao1992information: |
[EQUATION] and the inverse of the Fisher information can be used as an approximate estimator of the expected variance, given that the bound would become an equality in the asymptotic limit for MLE. If some of the parameters [MATH] are constrained by independent measurements characterised by their likelihoods |
[MATH] those constraints can also be easily included in the covariance estimation, simply by considering the augmented likelihood |
[MATH] instead of [MATH] in Eq. [EQUATION] In Bayesian terminology, this approach is referred to as the Laplace approximation where the logarithm of the joint density (including the priors) is expanded around the MAP to a multi-dimensional normal approximation of the posterior density: |
[EQUATION] which has already been approached by automatic differentiation in probabilistic programming frameworks . While a histogram has been used to construct a Poisson count sample likelihood, non-parametric density estimation techniques can be used in its place to construct a product of observation likelihoods base... |
[MATH] instead. For example, an extension of this technique to use kernel density estimation (KDE) should be straightforward, given its intrinsic differentiability. |
The loss function used for stochastic optimisation of the neural network parameters [MATH] can be any function of the inverse of the Fisher information matrix at [MATH] , depending on the ultimate inference aim. The diagonal elements |
[MATH] correspond to the expected variance of each of the [MATH] under the normal approximation mentioned before, so if the aim is efficient inference about one of the parameters [MATH] a candidate loss function is: |
[EQUATION] which corresponds to the expected width of the confidence interval for [MATH] accounting also for the effect of the other nuisance parameters in [MATH] This approach can also be extended when the goal is inference over several parameters of interest |
[MATH] (e.g. when considering a weighted sum of the relevant variances). A simple version of the approach just described to learn a neural-network based summary statistic employing an inference-aware loss is summarised in Algorithm |
Algorithm 1 Inference-Aware Neural Optimisation. Related Work Classification or regression models have been implicitly used to construct summary statistics for inference in several scientific disciplines. For example, in experimental particle physics, the mixture model structure of the problem makes it amenable to supe... |
\autocites hocker2007tmvabaldi2014searching. While a classification objective can be used to learn powerful feature representations and increase the sensitivity of an analysis, it does not take into account the details of the inference procedure or the effect of nuisance parameters like the solution proposed in this wo... |
The first known effort to include the effect of nuisance parameters in classification and explain the relation between classification and the likelihood ratio was by Neal |
. In the mentioned work, Neal proposes training of classifier including a function of nuisance parameter as additional input together with a per-observation regression model of the expectation value for inference. Cranmer et al. |
improved on this concept by using a parametrised classifier to approximate the likelihood ratio which is then calibrated to perform statistical inference. At variance with the mentioned works, we do not consider a classification objective at all and the neural network is directly optimised based on an inference-aware l... |
by a subset of the inference parameters to obtain a parametrised family of summary statistics with a single model. Recently, Brehmer et al. |
\autocites Brehmer:2018hgabrehmer2018constrainingbrehmer2018guide further extended the approach of parametrised classifiers to better exploit the latent-space space structure of generative models from complex scientific simulators. Additionally they propose a family of approaches that include a direct regression of the... |
Within the field of Approximate Bayesian Computation (ABC), there have been some attempts to use neural network as a dimensionality reduction step to generate summary statistics. For example, Jiang et al. |
successfully employ a summary statistic by directly regressing the parameters of interest and therefore approximating the posterior mean given the data, which then can be used directly as a summary statistic. |
A different path is taken by Louppe et al. , where the authors present a adversarial training procedure to enforce a pivotal property on a predictive model. The main concern of this approach is that a classifier which is pivotal with respect to nuisance parameters might not be optimal, neither for classification nor fo... |
Experiments In this section, we first study the effectiveness of the inference-aware optimisation in a synthetic mixture problem where the likelihood is known. We then compare our results with those obtained by standard classification-based summary statistics. All the code needed to reproduce the results presented the ... |
, extensively using TensorFlow and TensorFlow Probability \autocites tran2016edwarddillon2017tensorflow software libraries. 5.1 3D Synthetic Mixture |
In order to exemplify the usage of the proposed approach, evaluate its viability and test its performance by comparing to the use of a classification model proxy, a three-dimensional mixture example with two components is considered. One component will be referred as background |
[MATH] and the other as signal [MATH] ; their probability density functions are taken to correspond respectively to: [EQUATION] [EQUATION] |
so that [MATH] are distributed according to a multivariate normal distribution while [MATH] follows an independent exponential distribution both for background and signal, as shown in Fig. 2a . The signal distribution is fully specified while the background distribution depends on [MATH] , a parameter which shifts the ... |
[EQUATION] where [MATH] is the parameter corresponding to the mixture weight for the signal and consequently [MATH] is the mixture weight for the background. The low-dimensional projections from samples from the mixture distribution for a small [MATH] is shown in Fig. 2b |
Let us assume that we want to carry out inference based on [MATH] i.i.d. observations, such that [MATH] observations of signal and [MATH] observations of background are expected, respectively. While the mixture model parametrisation shown in Eq. 15 is correct as is, the underlying model could also give information on t... |
and [MATH] are the expected number of signal and background observations. Thus the following parametrisation will be more convenient for building sample-based likelihoods: |
[EQUATION] This parametrisation is common for physics analyses at the LHC, because theoretical calculations provide information about the expected number of observations. If the probability density is known, but the expectation for the number of observed events depends on the model parameters, the likelihood can be ext... |
with a Poisson count term as: [EQUATION] which will be used to provide an optimal inference baseline when benchmarking the different approaches. Another quantity of relevance is the conditional density ratio, which would correspond to the optimal classifier (in the Bayes risk sense) separating signal and background eve... |
[EQUATION] noting that this quantity depends on the parameters that define the background distribution [MATH] and [MATH] , but not on [MATH] or [MATH] that are a function of the mixture coefficients. It can be proven (see appendix |
) that [MATH] is a sufficient summary statistic with respect to an arbitrary two-component mixture model if the only unknown parameter is the signal mixture fraction |
[MATH] (or alternatively [MATH] in the chosen parametrisation). In practise, the probability density functions of signal and background are not known analytically, and only forward samples are available through simulation, so alternative approaches are required. |
While the synthetic nature of this example allows to rapidly generate training data on demand, a training dataset of 200,000 simulated observations has been considered, in order to study how the proposed method performs when training data is limited. Half of the simulated observations correspond to the signal component... |
[MATH] . A validation holdout from the training dataset of 200,000 observations is only used for computing relevant metrics during training and to control over-fitting. The final figures of merit that allow to compare different approaches are computed using a larger dataset of 1,000,000 observations. For simplicity, mi... |
An option is to pose the problem as one of classification based on a simulated dataset. A supervised machine learning model such a neural network can be trained to discriminate signal and background observations, considering a fixed parameters [MATH] and [MATH] . The output of such a model typically consist in class pr... |
and [MATH] given an observation [MATH] , which will tend asymptotically to the optimal classifier from Eq. 18 given enough data, a flexible enough model and a powerful learning rule. The conditional class probabilities (or alternatively the likelihood ratio |
[MATH] ) are powerful learned features that can be used as summary statistic; however their construction ignores the effect of the nuisance parameters [MATH] and |
[MATH] on the background distribution. Furthermore, some kind of non-parametric density estimation (e.g. a histogram) has to be considered in order to build a calibrated statistical model using the classification-based learned features, which will in turn smooth and reduce the information available for inference. |
To exemplify the use of this family of classification-based summary statistics, a histogram of a deep neural network classifier output trained on simulated data and its variation computed for different values of [MATH] and [MATH] are shown in Fig. 3a The details of the training procedure will be provided later in this ... |
[MATH] evaluated using the analytical distribution function of signal and background according to Eq. 18 , which is shown in Fig. 3b and corresponds to the optimal classifier. The trained classifier approximates very well the optimal classifier. The summary statistic distribution for background depends considerably on ... |
The statistical model described above has up to four unknown parameters: the expected number of signal observations [MATH] , the background mean shift [MATH] , the background exponential rate in the third dimension |
[MATH] , and the expected number of background observations. The effect of the expected number of signal and background observations |
[MATH] and [MATH] can be easily included in the computation graph by weighting the signal and background observations. This is equivalent to scaling the resulting vector of Poisson counts (or its differentiable approximation) if a non-parametric counting model as the one described in Sec. is used. Instead the effect of... |
[MATH] , both nuisance parameters that will define the background distribution, is more easily modelled as a transformation of the input data [MATH] . In particular, [MATH] is a nuisance parameter that causes a shift on the background along the first dimension and its effect can accounted for in the computation graph b... |
[MATH] to each observation in the mini-batch background distribution. Similarly, the effect of [MATH] can be modelled by multiplying [MATH] by the ratio between the [MATH] |
used for generation and the one being modelled. These transformations are specific for this example, but alternative transformations depending on parameters could also be accounted for as long as they are differentiable or substituted by a differentiable approximation. |
For this problem, we are interested in carrying out statistical inference on the parameter of interest [MATH] . In fact, the performance of inference-aware optimisation as described in Sec. |
will be compared with classification-based summary statistics for a series of inference benchmarks based on the synthetic problem described above that vary in the number of nuisance parameters considered and their constraints: |
Benchmark 0: no nuisance parameters are considered, both signal and background distributions are taken as fully specified ( [MATH] [MATH] |
and [MATH] ). Benchmark 1: [MATH] is considered as an unconstrained nuisance parameter, while [MATH] and [MATH] are fixed. Benchmark 2: |
[MATH] and [MATH] are considered as unconstrained nuisance parameters, while [MATH] is fixed. Benchmark 3: [MATH] and [MATH] are considered as nuisance parameters but with the following constraints: [MATH] |
and [MATH] , while [MATH] is fixed. Benchmark 4: all [MATH] [MATH] and [MATH] are all considered as nuisance parameters with the following constraints: [MATH] |
[MATH] and [MATH] When using classification-based summary statistics, the construction of a summary statistic does depend on the presence of nuisance parameters, so the same model is trained independently of the benchmark considered. In real-world inference scenarios, nuisance parameters have often to be accounted for ... |
will be considered, one for each of the benchmarks, denoted by the same number. The same basic network architecture is used both for cross-entropy and inference-aware training: two hidden layers of 100 nodes followed by ReLU activations. The number of nodes on the output layer is two when classification proxies are use... |
[MATH] , corresponding to the dimensionality of the sample summary statistics. The final layer is followed by a softmax activation function and a temperature [MATH] for inference-aware learning to ensure that the differentiable approximations are closer to the true expectations. Standard mini-batch stochastic gradient ... |
In Fig. 4a , the dynamics of inference-aware optimisation are shown by the validation loss, which corresponds to the approximate expected variance of parameter [MATH] , as a function of the training step for 10 random-initialised instances of the |
INFERNO model corresponding to Benchmark 2. All inference-aware models were trained during 200 epochs with SGD using mini-batches of 2000 observations and a learning rate [MATH] . All the model initialisations converge to summary statistics that provide low variance for the estimator of [MATH] when the nuisance paramet... |
To compare with alternative approaches and verify the validity of the results, the profiled likelihoods obtained for each model are shown in Fig. 4b . The expected uncertainty if the trained models are used for subsequent inference on the value of [MATH] |
can be estimated from the profile width when [MATH] . Hence, the average width for the profile likelihood using inference-aware training, [MATH] , can be compared with the corresponding one obtained by uniformly binning the output of classification-based models in 10 bins, [MATH] . The models based on cross-entropy los... |
A more complete study of the improvement provided by the different INFERNO training procedures is provided in Table 1 where the median and 1-sigma percentiles on the expected uncertainty on |
[MATH] are provided for 100 random-initialised instances of each model. In addition, results for 100 random-initialised cross-entropy trained models and the optimal classifier and likelihood-based inference are also included for comparison. The confidence intervals obtained using INFERNO-based summary statistics are co... |
Given that a certain value of the parameters [MATH] has been used to learn the summary statistics as described in Algorithm while their true value is unknown, the expected uncertainty on [MATH] has also been computed for cases when the true value of the parameters [MATH] |
differs. The variation of the expected uncertainty on [MATH] when either [MATH] or [MATH] is varied for classification and inference-aware summary statistics is shown in Fig. for Benchmark 2. The inference-aware summary statistics learnt for |
[MATH] work well when [MATH] in the range of variation explored. This synthetic example demonstrates that the direct optimisation of inference-aware losses as those described in the Sec. |
is effective. The summary statistics learnt accounting for the effect of nuisance parameters compare very favourably to those obtained by using a classification proxy to approximate the likelihood ratio. Of course, more experiments are needed to benchmark the usefulness of this technique for real-world inference proble... |
Conclusions Classification-based summary statistics for mixture models often suffer from the need of specifying a fixed model of the data, thus neglecting the effect of nuisance parameters in the learning process. The effect of nuisance parameters is only considered downstream of the learning phase, resulting in sub-op... |
In this work we have described a new approach for building non-linear summary statistics for likelihood-free inference that directly minimises the expected variance of the parameters of interest, which is considerably more effective than the use of classification surrogates when nuisance parameters are present. |
The results obtained for the synthetic experiment considered clearly demonstrate that machine learning techniques, in particular neural networks, can be adapted for learning summary statistics that match the particularities of the inference problem at hand, greatly increasing the information available for subsequent in... |
, are left for future studies. Furthermore, the technique presented can be applied to arbitrary likelihood-free problems as long as the effect of parameters over the simulated data can be implemented as a differentiable transformations. As a possible extension, alternative non-parametric density estimation techniques s... |
Acknowledgments Pablo de Castro would like to thank Daniel Whiteson, Peter Sadowski and the other attendants of the ML for HEP meeting at UCI for the initial feedback and support of the idea presented in this paper, as well as Edward Goul for his interest when the project was in early stages. The authors would also lik... |
This work is part of a more general effort to develop new statistical and machine learning techniques to use in High Energy Physics analyses within within the AMVA4NewPhysics project, which is supported by the European Union’s Horizon 2020 research and innovation programme under Grant Agreement number 675440. CloudVene... |
Appendix A Sufficient Statistics for Mixture Models Let us consider the general problem of inference for a two-component mixture problem, which is very common in scientific disciplines such as High Energy Physics. While their functional form will not be explicitly specified to keep the formulation general, one of the c... |
[MATH] is are of all parameters the distributions might depend on. The probability distribution function of the mixture can then be expressed as: |
[EQUATION] where [MATH] is a parameter corresponding to the signal mixture fraction. Dividing and multiplying by [MATH] we have: |
[EQUATION] from which we can already prove that the density ratio [MATH] (or alternatively its inverse) is a sufficient summary statistic for the mixture coefficient parameter [MATH] . This would also be the case for the parametrization using [MATH] and [MATH] if the alternative |
[MATH] formulation presented for the synthetic problem in Sec. 5.1 However, previously in this work (as well as for most studies using classifiers to construct summary statistics) we have been using the summary statistic |
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