The computation of two Bayesian predictive distributions which are discrete mixtures of incomplete beta functions is considered. The number of iterations can easily become large for these distributions and thus, the accuracy of the result can be questionable. Therefore, existing algorithms for that class of mixtures are improved by introducing round-off error calculation into the stopping rule. A further simple modification is proposed to deal with possible underflows that may prevent recurrence to work properly.
One-step ahead prediction for the multinomial model is considered. The performance of a predictive density is evaluated by the average Kullback-Leibler divergence from the true density to the predictive density. Asymptotic approximations of risk functions of Bayesian predictive densities based on Dirichlet priors are obtained. It is shown that a Bayesian predictive density based on a specific Dirichlet prior is asymptotically minimax. The asymptotically minimax prior is different from known objective priors such as the Jeffreys prior or the uniform prior.
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.
Construction methods for prior densities are investigated from a predictive viewpoint. Predictive densities for future observables are constructed by using observed data. The simultaneous distribution of future observables and observed data is assumed to belong to a parametric submodel of a multinomial model. Future observables and data are possibly dependent. The discrepancy of a predictive density to the true conditional density of future observables given observed data is evaluated by the Kullback-Leibler divergence. It is proved that limits of Bayesian predictive densities form an essentia…
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.
We investigate shrinkage priors for constructing Bayesian predictive distributions. It is shown that there exist shrinkage predictive distributions asymptotically dominating Bayesian predictive distributions based on the Jeffreys prior or other vague priors if the model manifold satisfies some differential geometric conditions. Kullback--Leibler divergence from the true distribution to a predictive distribution is adopted as a loss function. Conformal transformations of model manifolds corresponding to vague priors are introduced. We show several examples where shrinkage predictive distributio…
We investigate Bayesian shrinkage methods for constructing predictive distributions. We consider the multivariate Normal model with a known covariance matrix and show that the Bayesian predictive density with respect to Stein's harmonic prior dominates the best invariant Bayesian predictive density, when the dimension is greater than three. Alpha-divergence from the true distribution to a predictive distribution is adopted as a loss function.
Bayesian predictive densities when the observed data $x$ and the target variable $y$ to be predicted have different distributions are investigated by using the framework of information geometry. The performance of predictive densities is evaluated by the Kullback--Leibler divergence. The parametric models are formulated as Riemannian manifolds. In the conventional setting in which $x$ and $y$ have the same distribution, the Fisher--Rao metric and the Jeffreys prior play essential roles. In the present setting in which $x$ and $y$ have different distributions, a new metric, which we call the pr…
A Bayesian framework is attractive in the context of prediction, but a fast recursive update of the predictive distribution has apparently been out of reach, in part because Monte Carlo methods are generally used to compute the predictive. This paper shows that online Bayesian prediction is possible by characterizing the Bayesian predictive update in terms of a bivariate copula, making it unnecessary to pass through the posterior to update the predictive. In standard models, the Bayesian predictive update corresponds to familiar choices of copula but, in nonparametric problems, the appropriate…
We reveal the relations between the conditional normalized maximum likelihood (CNML) distributions and Bayesian predictive densities based on the latent information priors (LIPs). In particular, CNML3, which is one type of CNML distributions, is investigated. The Bayes projection of a predictive density, which is an information projection of the predictive density on a set of Bayesian predictive densities, is considered. We prove that the sum of the Bayes projection divergence of CNML3 and the conditional mutual information is asymptotically constant. This result implies that the Bayes project…
This paper considers estimation of the predictive density for a normal linear model with unknown variance under alpha-divergence loss for -1 <= alpha <= 1. We first give a general canonical form for the problem, and then give general expressions for the generalized Bayes solution under the above loss for each alpha. For a particular class of hierarchical generalized priors studied in Maruyama and Strawderman (2005, 2006) for the problems of estimating the mean vector and the variance respectively, we give the generalized Bayes predictive density. Additionally, we show that, for a subclass of t…
We consider Bayesian shrinkage predictions for the Normal regression problem under the frequentist Kullback-Leibler risk function. Firstly, we consider the multivariate Normal model with an unknown mean and a known covariance. While the unknown mean is fixed, the covariance of future samples can be different from training samples. We show that the Bayesian predictive distribution based on the uniform prior is dominated by that based on a class of priors if the prior distributions for the covariance and future covariance matrices are rotation invariant. Then, we consider a class of priors for t…
Simultaneous predictive distributions for independent Poisson observables are investigated. A class of improper prior distributions for Poisson means is introduced. The Bayesian predictive distributions based on priors from the introduced class are shown to be admissible under the Kullback-Leibler loss. A Bayesian predictive distribution based on a prior in this class dominates the Bayesian predictive distribution based on the Jeffreys prior.
We study asymptotically minimax predictive distributions in an infinite sequence model. First, we discuss the connection between the prediction in the infinite sequence model and the prediction in the function model. Second, we construct an asymptotically minimax predictive distribution when the parameter space is a known ellipsoid. We show that the Bayesian predictive distribution based on the Gaussian prior distribution is asymptotically minimax in the ellipsoid. Third, we construct an asymptotically minimax predictive distribution for any Sobolev ellipsoid. We show that the Bayesian predict…
In this paper, we treat estimation and prediction problems where negative multinomial variables are observed and in particular consider unbalanced settings. First, the problem of estimating multiple negative multinomial parameter vectors under the standardized squared error loss is treated and a new empirical Bayes estimator which dominates the UMVU estimator under suitable conditions is derived. Second, we consider estimation of the joint predictive density of several multinomial tables under the Kullback-Leibler divergence and obtain a sufficient condition under which the Bayesian predictive…
The prediction of the variance-covariance matrix of the multivariate normal distribution is important in the multivariate analysis. We investigated Bayesian predictive distributions for Wishart distributions under the Kullback-Leibler divergence. The conditional reducibility of the family of Wishart distributions enables us to decompose the risk of a Bayesian predictive distribution. We considered a recently introduced class of prior distributions, which is called the family of enriched standard conjugate prior distributions, and compared the Bayesian predictive distributions based on these pr…
Robust Predictive Uncertainty and Double Descent in Contaminated Bayesian Random Features
The University of Manchester, UK（英国曼彻斯特大学）；University of Pisa, Italy（意大利比萨大学）；Nanyang Technological University, Singapore（新加坡南洋理工大学）；Max Planck Institute for Mathematics in the Sciences, Germany（德国马克斯·普朗克数学研究所）
Let $(X_n:n\ge 1)$ be a sequence of random observations. Let $σ_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the $n$-th predictive distribution and $σ_0(\cdot)=P(X_1\in\cdot)$ the marginal distribution of $X_1$. In a Bayesian framework, to make predictions on $(X_n)$, one only needs the collection $σ=(σ_n:n\ge 0)$. Because of the Ionescu-Tulcea theorem, $σ$ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability has to be selected. In this paper, $σ$ is subjected to two requirements: (i) The resulting s…
We derive a novel sensitivity analysis of input variables for predictive epistemic and aleatoric uncertainty. We use Bayesian neural networks with latent variables as a model class and illustrate the usefulness of our sensitivity analysis on real-world datasets. Our method increases the interpretability of complex black-box probabilistic models.
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…
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；Aalto University（阿尔托大学）
We introduce a method, KL-LIME, for explaining predictions of Bayesian predictive models by projecting the information in the predictive distribution locally to a simpler, interpretable explanation model. The proposed approach combines the recent Local Interpretable Model-agnostic Explanations (LIME) method with ideas from Bayesian projection predictive variable selection methods. The information theoretic basis helps in navigating the trade-off between explanation fidelity and complexity. We demonstrate the method in explaining MNIST digit classifications made by a Bayesian deep convolutional…
Inria Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧中心）；University College London（伦敦大学学院）
Generalised Bayesian learning algorithms are increasingly popular in machine learning, due to their PAC generalisation properties and flexibility. The present paper aims at providing a self-contained survey on the resulting PAC-Bayes framework and some of its main theoretical and algorithmic developments.
Given a sequence $X=(X_1,X_2,\ldots)$ of random observations, a Bayesian forecaster aims to predict $X_{n+1}$ based on $(X_1,\ldots,X_n)$ for each $n\ge 0$. To this end, in principle, she only needs to select a collection $σ=(σ_0,σ_1,\ldots)$, called ``strategy" in what follows, where $σ_0(\cdot)=P(X_1\in\cdot)$ is the marginal distribution of $X_1$ and $σ_n(\cdot)=P(X_{n+1}\in\cdot\mid X_1,\ldots,X_n)$ the $n$-th predictive distribution. Because of the Ionescu-Tulcea theorem, $σ$ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no …
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…
We investigate predictive densities for multivariate normal models with unknown mean vectors and known covariance matrices. Bayesian predictive densities based on shrinkage priors often have complex representations, although they are effective in various problems. We consider extended normal models with mean vectors and covariance matrices as parameters, and adopt predictive densities that belong to the extended models including the original normal model. We adopt predictive densities that are optimal with respect to the posterior Bayes risk in the extended models. The proposed predictive dens…
Entropic tilting (ET) is a Bayesian decision-analytic method for constraining distributions to satisfy defined targets or bounds for sets of expectations. This report recapitulates the foundations and basic theory of ET for conditioning predictive distributions on such constraints, recognising the increasing interest in ET in several application areas. Contributions include new results related to connections with regular exponential families of distributions, and the extension of ET to relaxed entropic tilting (RET) where specified values for expectations define bounds rather than exact target…
We review the Akaike, deviance, and Watanabe-Akaike information criteria from a Bayesian perspective, where the goal is to estimate expected out-of-sample-prediction error using a biascorrected adjustment of within-sample error. We focus on the choices involved in setting up these measures, and we compare them in three simple examples, one theoretical and two applied. The contribution of this review is to put all these information criteria into a Bayesian predictive context and to better understand, through small examples, how these methods can apply in practice.
The implementation of Bayesian predictive procedures under standard normal models is considered. Two distributions are of particular interest, the K-prime and K-square distributions. They also give exact inferences for simple and multiple correlation coefficients. Their cumulative distribution functions can be expressed in terms of infinite series of multiples of incomplete beta function ratios, thus adequate for recursive calculations. Efficient algorithms are provided. To deal with special cases where possible underflows may prevent a recurrence to work properly, a simple solution is propose…
The widely recommended procedure of Bayesian model averaging is flawed in the M-open setting in which the true data-generating process is not one of the candidate models being fit. We take the idea of stacking from the point estimation literature and generalize to the combination of predictive distributions, extending the utility function to any proper scoring rule, using Pareto smoothed importance sampling to efficiently compute the required leave-one-out posterior distributions and regularization to get more stability. We compare stacking of predictive distributions to several alternatives: …
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…
I begin by summarizing key ideas of the paper under discussion. Then I will talk about a graphical modeling perspective, posterior contraction rates and alternative methods of aggregation. Moreover, I will also discuss possible applications of the stacking method to other problems, in particular, aggregating (sub)posterior distributions in distributed computing.
This paper describes a new Bayesian interpretation of a class of skew--Student $t$ distributions. We consider a hierarchical normal model with unknown covariance matrix and show that by imposing different restrictions on the parameter space, corresponding Bayes predictive density estimators under Kullback-Leibler loss function embrace some well-known skew--Student $t$ distributions. We show that obtained estimators perform better in terms of frequentist risk function over regular Bayes predictive density estimators. We apply our proposed methods to estimate future densities of medical data: th…
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…
Principled Bayesian deep learning (BDL) does not live up to its potential when we only focus on marginal predictive distributions (marginal predictives). Recent works have highlighted the importance of joint predictives for (Bayesian) sequential decision making from a theoretical and synthetic perspective. We provide additional practical arguments grounded in real-world applications for focusing on joint predictives: we discuss online Bayesian inference, which would allow us to make predictions while taking into account additional data without retraining, and we propose new challenging evaluat…
We develop scalable methods for producing conformal Bayesian predictive intervals with finite sample calibration guarantees. Bayesian posterior predictive distributions, $p(y \mid x)$, characterize subjective beliefs on outcomes of interest, $y$, conditional on predictors, $x$. Bayesian prediction is well-calibrated when the model is true, but the predictive intervals may exhibit poor empirical coverage when the model is misspecified, under the so called ${\cal{M}}$-open perspective. In contrast, conformal inference provides finite sample frequentist guarantees on predictive confidence interva…
We show that regularizing Bayesian predictive regressions provides a framework for prior sensitivity analysis. We develop a procedure that jointly regularizes expectations and variance-covariance matrices using a pair of shrinkage priors. Our methodology applies directly to vector autoregressions (VAR) and seemingly unrelated regressions (SUR). The regularization path provides a prior sensitivity diagnostic. By exploiting a duality between regularization penalties and predictive prior distributions, we reinterpret two classic Bayesian analyses of macro-finance studies: equity premium predictab…
