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 these priors, the resulting estimator dominates the generalized Bayes estimator with respect to the right invariant prior when alpha=1, i.e., the best (fully) equivariant minimax estimator.
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.
This paper introduces the $(α, Γ)$-descent, an iterative algorithm which operates on measures and performs $α$-divergence minimisation in a Bayesian framework. This gradient-based procedure extends the commonly-used variational approximation by adding a prior on the variational parameters in the form of a measure. We prove that for a rich family of functions $Γ$, this algorithm leads at each step to a systematic decrease in the $α$-divergence and derive convergence results. Our framework recovers the Entropic Mirror Descent algorithm and provides an alternative algorithm that we call the Power…
Romanian Institute of Science and Technology（罗马尼亚科学技术研究院）
Bayesian Neural Networks (BNNs) often result uncalibrated after training, usually tending towards overconfidence. Devising effective calibration methods with low impact in terms of computational complexity is thus of central interest. In this paper we present calibration methods for BNNs based on the alpha divergences from Information Geometry. We compare the use of alpha divergence in training and in calibration, and we show how the use in calibration provides better calibrated uncertainty estimates for specific choices of alpha and is more efficient especially for complex network architectur…
University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）
To obtain uncertainty estimates with real-world Bayesian deep learning models, practical inference approximations are needed. Dropout variational inference (VI) for example has been used for machine vision and medical applications, but VI can severely underestimates model uncertainty. Alpha-divergences are alternative divergences to VI's KL objective, which are able to avoid VI's uncertainty underestimation. But these are hard to use in practice: existing techniques can only use Gaussian approximating distributions, and require existing models to be changed radically, thus are of limited use f…
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.
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…
In this paper, we study the approximation and estimation of $s$-concave densities via Rényi divergence. We first show that the approximation of a probability measure $Q$ by an $s$-concave densities exists and is unique via the procedure of minimizing a divergence functional proposed by Koenker and Mizera (2010) if and only if $Q$ admits full-dimensional support and a first moment. We also show continuity of the divergence functional in $Q$: if $Q_n \to Q$ in the Wasserstein metric, then the projected densities converge in weighted $L_1$ metrics and uniformly on closed subsets of the continuity…
We describe the underlying probabilistic interpretation of alpha and beta divergences. We first show that beta divergences are inherently tied to Tweedie distributions, a particular type of exponential family, known as exponential dispersion models. Starting from the variance function of a Tweedie model, we outline how to get alpha and beta divergences as special cases of Csiszár's $f$ and Bregman divergences. This result directly generalizes the well-known relationship between the Gaussian distribution and least squares estimation to Tweedie models and beta divergence minimization.
Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantification
Department of Applied Mechanics（应用力学系）；Indian Institute of Technology Delhi（印度理工学院德里）；Department of Mechanical Engineering（机械工程系）；Northern Arizona University（北亚利桑那大学）
We show that the $f$-divergence between any two densities of potentially different location-scale families can be reduced to the calculation of the $f$-divergence between one standard density with another location-scale density. It follows that the $f$-divergence between two scale densities depends only on the scale ratio. We then report conditions on the standard distribution to get symmetric $f$-divergences: First, we prove that all $f$-divergences between densities of a location family are symmetric whenever the standard density is even, and second, we illustrate a generic symmetric propert…
LTCI, Télécom Paris（巴黎电信学院LTCI实验室）；Institut Polytechnique de Paris（巴黎理工学院）；SAMOVAR, Télécom SudParis（南巴黎电信学院SAMOVAR实验室）
This paper focuses on $α$-divergence minimisation methods for Variational Inference. More precisely, we are interested in algorithms optimising the mixture weights of any given mixture model, without any information on the underlying distribution of its mixture components parameters. The Power Descent, defined for all $α\neq 1$, is one such algorithm and we establish in our work the full proof of its convergence towards the optimal mixture weights when $α<1$. Since the $α$-divergence recovers the widely-used forward Kullback-Leibler when $α\to 1$, we then extend the Power Descent to the case $…
We study the problem of nonparametric estimation of density functions with a product form on the domain $\triangle=\{( x_1, \ldots, x_d)\in \mathbb{R}^d, 0\leq x_1\leq \dots \leq x_d \leq 1\}$. Such densities appear in the random truncation model as the joint density function of observations. They are also obtained as maximum entropy distributions of order statistics with given marginals. We propose an estimation method based on the approximation of the logarithm of the density by a carefully chosen family of basis functions. We show that the method achieves a fast convergence rate in probabil…
Minimization of suitable statistical distances~(between the data and model densities) has proved to be a very useful technique in the field of robust inference. Apart from the class of $ϕ$-divergences of \cite{a} and \cite{b}, the Bregman divergence (\cite{c}) has been extensively used for this purpose. However, since the data density must have a linear presence in the cross product term of the Bregman divergence involving both the data and model densities, several useful divergences cannot be captured by the usual Bregman form. In this respect, we provide an extension of the ordinary Bregman …
The Jensen-Shannon divergence is a renown bounded symmetrization of the Kullback-Leibler divergence which does not require probability densities to have matching supports. In this paper, we introduce a vector-skew generalization of the scalar $α$-Jensen-Bregman divergences and derive thereof the vector-skew $α$-Jensen-Shannon divergences. We study the properties of these novel divergences and show how to build parametric families of symmetric Jensen-Shannon-type divergences. Finally, we report an iterative algorithm to numerically compute the Jensen-Shannon-type centroids for a set of probabil…
Based on $X \sim N_d(θ, σ^2_X I_d)$, we study the efficiency of predictive densities under $α-$divergence loss $L_α$ for estimating the density of $Y \sim N_d(θ, σ^2_Y I_d)$. We identify a large number of cases where improvement on a plug-in density are obtainable by expanding the variance, thus extending earlier findings applicable to Kullback-Leibler loss. The results and proofs are unified with respect to the dimension $d$, the variances $σ^2_X$ and $σ^2_Y$, the choice of loss $L_α$; $α\in (-1,1)$. The findings also apply to a large number of plug-in densities, as well as for restricted par…
We consider nonparametric estimation of $L_2$, Renyi-$α$ and Tsallis-$α$ divergences between continuous distributions. Our approach is to construct estimators for particular integral functionals of two densities and translate them into divergence estimators. For the integral functionals, our estimators are based on corrections of a preliminary plug-in estimator. We show that these estimators achieve the parametric convergence rate of $n^{-1/2}$ when the densities' smoothness, $s$, are both at least $d/4$ where $d$ is the dimension. We also derive minimax lower bounds for this problem which con…
Robust inference based on the minimization of statistical divergences has proved to be a useful alternative to the classical maximum likelihood based techniques. Recently Ghosh et al. (2013) proposed a general class of divergence measures for robust statistical inference, named the S-Divergence Family. Ghosh (2014) discussed its asymptotic properties for the discrete model of densities. In the present paper, we develop the asymptotic properties of the proposed minimum S-Divergence estimators under continuous models. Here we use the Basu-Lindsay approach (1994) of smoothing the model densities …
In this paper, we introduce new classes of divergences by extending the definitions of the Bregman divergence and the skew Jensen divergence. These new divergence classes (g-Bregman divergence and skew g-Jensen divergence) satisfy some properties similar to the Bregman or skew Jensen divergence. We show these g-divergences include divergences which belong to a class of f-divergence (the Hellinger distance, the chi-square divergence and the alpha-divergence in addition to the Kullback-Leibler divergence). Moreover, we derive an inequality between the g-Bregman divergence and the skew g-Jensen d…
The squared Wasserstein distance is a natural quantity to compare probability distributions in a non-parametric setting. This quantity is usually estimated with the plug-in estimator, defined via a discrete optimal transport problem which can be solved to $ε$-accuracy by adding an entropic regularization of order $ε$ and using for instance Sinkhorn's algorithm. In this work, we propose instead to estimate it with the Sinkhorn divergence, which is also built on entropic regularization but includes debiasing terms. We show that, for smooth densities, this estimator has a comparable sample comple…
Sony Computer Science Laboratories, Inc.（索尼计算机科学实验室公司）；CSIRO Data61（澳大利亚联邦科学与工业研究组织Data61）；Australian National University（澳大利亚国立大学）
It is well-known that the Bhattacharyya, Hellinger, Kullback-Leibler, $α$-divergences, and Jeffreys' divergences between densities belonging to a same exponential family have generic closed-form formulas relying on the strictly convex and real-analytic cumulant function characterizing the exponential family. In this work, we report (dis)similarity formulas which bypass the explicit use of the cumulant function and highlight the role of quasi-arithmetic means and their multivariate mean operator extensions. In practice, these cumulant-free formulas are handy when implementing these (dis)similar…
Estimating divergences in a consistent way is of great importance in many machine learning tasks. Although this is a fundamental problem in nonparametric statistics, to the best of our knowledge there has been no finite sample exponential inequality convergence bound derived for any divergence estimators. The main contribution of our work is to provide such a bound for an estimator of Rényi-$α$ divergence for a smooth Hölder class of densities on the $d$-dimensional unit cube $[0, 1]^d$. We also illustrate our theoretical results with a numerical experiment.
We develop and analyze $M$-estimation methods for divergence functionals and the likelihood ratios of two probability distributions. Our method is based on a non-asymptotic variational characterization of $f$-divergences, which allows the problem of estimating divergences to be tackled via convex empirical risk optimization. The resulting estimators are simple to implement, requiring only the solution of standard convex programs. We present an analysis of consistency and convergence for these estimators. Given conditions only on the ratios of densities, we show that our estimators can achieve …
This paper addresses an estimation problem of an additive functional of $ϕ$, which is defined as $θ(P;ϕ)=\sum_{i=1}^kϕ(p_i)$, given $n$ i.i.d. random samples drawn from a discrete distribution $P=(p_1,...,p_k)$ with alphabet size $k$. We have revealed in the previous paper that the minimax optimal rate of this problem is characterized by the divergence speed of the fourth derivative of $ϕ$ in a range of fast divergence speed. In this paper, we prove this fact for a more general range of the divergence speed. As a result, we show the minimax optimal rate of the additive functional estimation fo…
This paper studies forward and reverse projections for the Rényi divergence of order $α\in (0, \infty)$ on $α$-convex sets. The forward projection on such a set is motivated by some works of Tsallis {\em et al.} in statistical physics, and the reverse projection is motivated by robust statistics. In a recent work, van Erven and Harremoës proved a Pythagorean inequality for Rényi divergences on $α$-convex sets under the assumption that the forward projection exists. Continuing this study, a sufficient condition for the existence of forward projection is proved for probability measures on a gene…
