Bayesian model criticism is an important part of the practice of Bayesian statistics. Traditionally, model criticism methods have been based on the predictive check, an adaptation of goodness-of-fit testing to Bayesian modeling and an effective method to understand how well a model captures the distribution of the data. In modern practice, however, researchers iteratively build and develop many models, exploring a space of models to help solve the problem at hand. While classical predictive checks can help assess each one, they cannot help the researcher understand how the models relate to each other. This paper introduces the posterior predictive null check (PPN), a method for Bayesian model criticism that helps characterize the relationships between models. The idea behind the PPN is to check whether data from one model's predictive distribution can pass a predictive check designed for another model. This form of criticism complements the classical predictive check by providing a comparative tool. A collection of PPNs, which we call a PPN study, can help us understand which models are equivalent and which models provide different perspectives on the data. With mixture models, we demonstrate how a PPN study, along with traditional predictive checks, can help select the number of components by the principle of parsimony. With probabilistic factor models, we demonstrate how a PPN study can help understand relationships between different classes of models, such as linear models and models based on neural networks. Finally, we analyze data from the literature on predictive checks to show how a PPN study can improve the practice of Bayesian model criticism. Code to replicate the results in this paper is available at \url{https://github.com/gemoran/ppn-code}.
University College London（伦敦大学学院）；CNRS, Sorbonne Université（法国国家科学研究中心，索邦大学）；Department of Econometrics and Business Statistics, Monash University（蒙纳士大学计量经济学与商业统计系）
Predictive variational inference: Learn the predictively optimal posterior distribution
College of Information and Computer Sciences, University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校信息与计算机科学学院）；Department of Statistics and Data Sciences, University of Texas, Austin（德克萨斯大学奥斯汀分校统计与数据科学系）
In the hypothesis testing framework, p-value is often computed to determine rejection of the null hypothesis or not. On the other hand, Bayesian approaches typically compute the posterior probability of the null hypothesis to evaluate its plausibility. We revisit Lindley's paradox (Lindley, 1957) and demystify the conflicting results between Bayesian and frequentist hypothesis testing procedures by casting a two-sided hypothesis as a combination of two one-sided hypotheses along the opposite directions. This can naturally circumvent the ambiguities of assigning a point mass to the null and cho…
In randomized experiments with noncompliance, tests may focus on compliers rather than on the overall sample. Rubin (1998) put forth such a method, and argued that testing for the complier average causal effect and averaging permutation based p-values over the posterior distribution of the compliance status could increase power, as compared to general intent-to-treat tests. The general scheme is to repeatedly do a two-step process of imputing missing compliance statuses and conducting a permutation test with the completed data. In this paper, we explore this idea further, comparing the use of …
Posterior uncertainty for kernel density estimates
Norwegian Defence Research Establishment (FFI)（挪威国防研究所）；Department of Mathematics, University of Oslo（奥斯陆大学数学系）；BI Norwegian Business School, Oslo（BI挪威商学院）
Recent decades have seen an interest in prediction problems for which Bayesian methodology has been used ubiquitously. Sampling from or approximating the posterior predictive distribution in a Bayesian model allows one to make inferential statements about potentially observable random quantities given observed data. The purpose of this note is to use statistical decision theory as a basis to justify the use of a posterior predictive distribution for making a point prediction.
Variable selection for Gaussian process models is often done using automatic relevance determination, which uses the inverse length-scale parameter of each input variable as a proxy for variable relevance. This implicitly determined relevance has several drawbacks that prevent the selection of optimal input variables in terms of predictive performance. To improve on this, we propose two novel variable selection methods for Gaussian process models that utilize the predictions of a full model in the vicinity of the training points and thereby rank the variables based on their predictive relevanc…
When researchers carry out a null hypothesis significance test, it is tempting to assume that a statistically significant result lowers Prob(H0), the probability of the null hypothesis being true. Technically, such a statement is meaningless for various reasons: e.g., the null hypothesis does not have a probability associated with it. However, it is possible to relax certain assumptions to compute the posterior probability Prob(H0) under repeated sampling. We show in a step-by-step guide that the intuitively appealing belief, that Prob(H0) is low when significant results have been obtained und…
As a convention, p-value is often computed in frequentist hypothesis testing and compared with the nominal significance level of 0.05 to determine whether or not to reject the null hypothesis. The smaller the p-value, the more significant the statistical test. We consider both one-sided and two-sided hypotheses in the composite hypothesis setting. For one-sided hypothesis tests, we establish the equivalence of p-value and the Bayesian posterior probability of the null hypothesis, which renders p-value an explicit interpretation of how strong the data support the null. For two-sided hypothesis …
Frequentist-style large-sample properties of Bayesian posterior distributions, such as consistency and convergence rates, are important considerations in nonparametric problems. In this paper we give an analysis of Bayesian asymptotics based primarily on predictive densities. Our analysis is unified in the sense that essentially the same approach can be taken to develop convergence rate results in iid, mis-specified iid, independent non-iid, and dependent data cases.
The prior distribution on parameters of a sampling distribution is the usual starting point for Bayesian uncertainty quantification. In this paper, we present a different perspective which focuses on missing observations as the source of statistical uncertainty, with the parameter of interest being known precisely given the entire population. We argue that the foundation of Bayesian inference is to assign a distribution on missing observations conditional on what has been observed. In the conditionally i.i.d. setting with an observed sample of size $n$, the Bayesian would thus assign a predict…
Minimum Bayes factors are commonly used to transform two-sided p-values to lower bounds on the posterior probability of the null hypothesis, as in Pericchi et al. (2017). In this article, we show posterior probabilities of hypothesis by transforming the commonly used -eplog(p), proposed by Vovk (1993) and Sellke et al. (2001). This is achieved after adjusting this minimum Bayes factor with the information to approximate it to an exact Bayes factor, not only when p is a p-value but also when p is a pseudo p-value in the sense of Casella and Berger (2001). Additionally, we show the fit to a refi…
We give a decomposition of the posterior predictive variance using the law of total variance and conditioning on a finite dimensional discrete random variable. This random variable summarizes various features of modeling that are used to form the prediction for a future outcome. Then, we test which terms in this decomposition are small enough to ignore. This allows us identify which of the discrete random variables are most important to prediction intervals. The terms in the decomposition admit interpretations based on conditional means and variances and are analogous to the terms in a Cochran…
Optimality results for two outstanding Bayesian estimation problems are given in this paper: the estimation of the sampling distribution for the squared total variation function and the estimation of the density for the $L^1$-squared loss function. The posterior predictive distribution provides the solution to these problems. Some examples are presented to illustrate it. The Bayesian estimation problem of a distribution function is also addressed. Consistency of the estimator of the density is proved.
University of Oxford（牛津大学）；Novo Nordisk（诺和诺德）；University College London（伦敦大学学院）
Bayesian methods are a popular choice for statistical inference in small-data regimes due to the regularization effect induced by the prior. In the context of density estimation, the standard nonparametric Bayesian approach is to target the posterior predictive of the Dirichlet process mixture model. In general, direct estimation of the posterior predictive is intractable and so methods typically resort to approximating the posterior distribution as an intermediate step. The recent development of quasi-Bayesian predictive copula updates, however, has made it possible to perform tractable predi…
When using complex Bayesian models to combine information, the checking for consistency of the information being combined is good statistical practice. Here a new method is developed for detecting prior-data conflicts in Bayesian models based on comparing the observed value of a prior to posterior divergence to its distribution under the prior predictive distribution for the data. The divergence measure used in our model check is a measure of how much beliefs have changed from prior to posterior, and can be thought of as a measure of the overall size of a relative belief function. It is shown …
Surprise Reduction and Nullification in Bayesian and Inverse Bayesian Inference under Ambiguous Prediction-Error Attribution
Tokyo Denki University（东京都立大学）；Osaka Metropolitan University（大阪公立大学）；The University of Tokyo（东京大学）；Utsunomiya University（宇都宫大学）；Waseda University（早稻田大学）
Posterior predictive p-values are a common approach to Bayesian model-checking. This article analyses their frequency behaviour, that is, their distribution when the parameters and the data are drawn from the prior and the model respectively. We show that the family of possible distributions is exactly described as the distributions that are less variable than uniform on [0,1], in the convex order. In general, p-values with such a property are not conservative, and we illustrate how the theoretical worst-case error rate for false rejection can occur in practice. We describe how to correct the …
Two procedures for checking Bayesian models are compared using a simple test problem based on the local Hubble expansion. Over four orders of magnitude, p-values derived from a global goodness-of-fit criterion for posterior probability density functions (Lucy 2017) agree closely with posterior predictive p-values. The former can therefore serve as an effective proxy for the difficult-to-calculate posterior predictive p-values.
The interdependence between nodes in graphs is key to improve class predictions on nodes and utilized in approaches like Label Propagation (LP) or in Graph Neural Networks (GNN). Nonetheless, uncertainty estimation for non-independent node-level predictions is under-explored. In this work, we explore uncertainty quantification for node classification in three ways: (1) We derive three axioms explicitly characterizing the expected predictive uncertainty behavior in homophilic attributed graphs. (2) We propose a new model Graph Posterior Network (GPN) which explicitly performs Bayesian posterior…
In this study, we present a multi-class graphical Bayesian predictive classifier that incorporates the uncertainty in the model selection into the standard Bayesian formalism. For each class, the dependence structure underlying the observed features is represented by a set of decomposable Gaussian graphical models. Emphasis is then placed on the Bayesian model averaging which takes full account of the class-specific model uncertainty by averaging over the posterior graph model probabilities. An explicit evaluation of the model probabilities is well known to be infeasible. To address this issue…
A natural method for approximating out-of-sample predictive evaluation is leave-one-out cross-validation (LOOCV) --- we alternately hold out each case from a full data set and then train a Bayesian model using Markov chain Monte Carlo (MCMC) without the held-out; at last we evaluate the posterior predictive distribution of all cases with their actual observations. However, actual LOOCV is time-consuming. This paper introduces two methods, namely iIS and iWAIC, for approximating LOOCV with only Markov chain samples simulated from a posterior based on a \textit{full} data set. iIS and iWAIC aim …
Bayesian interpretations of neural processing require that biological mechanisms represent and operate upon probability distributions in accordance with Bayes' theorem. Many have speculated that synaptic failure constitutes a mechanism of variational, i.e., approximate, Bayesian inference in the brain. Whereas models have previously used synaptic failure to sample over uncertainty in model parameters, we demonstrate that by adapting transmission probabilities to learned network weights, synaptic failure can sample not only over model uncertainty, but complete posterior predictive distributions…
Admixture models are a ubiquitous approach to capture latent population structure in genetic samples. Despite the widespread application of admixture models, little thought has been devoted to the quality of the model fit or the accuracy of the estimates of parameters of interest for a particular study. Here we develop methods for validating admixture models based on posterior predictive checks (PPCs), a Bayesian method for assessing the quality of a statistical model. We develop PPCs for five population-level statistics of interest: within-population genetic variation, background linkage dise…
University of Chicago（芝加哥大学）；University of Toronto（多伦多大学）；Vector Institute（矢量研究所）
While uncertainty estimation is a well-studied topic in deep learning, most such work focuses on marginal uncertainty estimates, i.e. the predictive mean and variance at individual input locations. But it is often more useful to estimate predictive correlations between the function values at different input locations. In this paper, we consider the problem of benchmarking how accurately Bayesian models can estimate predictive correlations. We first consider a downstream task which depends on posterior predictive correlations: transductive active learning (TAL). We find that TAL makes better us…
University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）；Huawei Noah’s Ark Lab（华为诺亚方舟实验室）
Making predictions robust is an important challenge. A separate challenge in federated learning (FL) is to reduce the number of communication rounds, particularly since doing so reduces performance in heterogeneous data settings. To tackle both issues, we take a Bayesian perspective on the problem of learning a global model. We show how the global predictive posterior can be approximated using client predictive posteriors. This is unlike other works which aggregate the local model space posteriors into the global model space posterior, and are susceptible to high approximation errors due to th…
Within the last few years, there has been a move towards using statistical models in conjunction with neural networks with the end goal of being able to better answer the question, "what do our models know?". From this trend, classical metrics such as Prediction Interval Coverage Probability (PICP) and new metrics such as calibration error have entered the general repertoire of model evaluation in order to gain better insight into how the uncertainty of our model compares to reality. One important component of uncertainty modeling is model uncertainty (epistemic uncertainty), a measurement of …
For predictive evaluation based on quasi-posterior distributions, we develop a new information criterion, the posterior covariance information criterion (PCIC. PCIC generalises the widely applicable information criterion WAIC so as to effectively handle predictive scenarios where likelihoods for the estimation and the evaluation of the model may be different. A typical example of such scenarios is the weighted likelihood inference, including prediction under covariate shift and counterfactual prediction. The proposed criterion utilises a posterior covariance form and is computed by using only …
