In discriminative settings such as regression and classification there are two random variables at play, the inputs X and the targets Y. Here, we demonstrate that the Variational Information Bottleneck can be viewed as a compromise between fully empirical and fully Bayesian objectives, attempting to minimize the risks due to finite sampling of Y only. We argue that this approach provides some of the benefits of Bayes while requiring only some of the work.
We offer a novel view of AdaBoost in a statistical setting. We propose a Bayesian model for binary classification in which label noise is modeled hierarchically. Using variational inference to optimize a dynamic evidence lower bound, we derive a new boosting-like algorithm called VIBoost. We show its close connections to AdaBoost and give experimental results from four datasets.
Bayesian Inference and Information Bottleneck are the two most popular objectives for neural networks, but they can be optimised only via a variational lower bound: the Variational Information Bottleneck (VIB). In this manuscript we show that the two objectives are actually equivalent to the InfoMax: maximise the information between the data and the labels. The InfoMax representation of the two objectives is not relevant only per se, since it helps to understand the role of the network capacity, but also because it allows us to derive a variational objective, the Variational InfoMax (VIM), tha…
This paper argues that the half-Cauchy distribution should replace the inverse-Gamma distribution as a default prior for a top-level scale parameter in Bayesian hierarchical models, at least for cases where a proper prior is necessary. Our arguments involve a blend of Bayesian and frequentist reasoning, and are intended to complement the original case made by Gelman (2006) in support of the folded-t family of priors. First, we generalize the half-Cauchy prior to the wider class of hypergeometric inverted-beta priors. We derive expressions for posterior moments and marginal densities when these…
A statistical model or a learning machine is called regular if the map taking a parameter to a probability distribution is one-to-one and if its Fisher information matrix is always positive definite. If otherwise, it is called singular. In regular statistical models, the Bayes free energy, which is defined by the minus logarithm of Bayes marginal likelihood, can be asymptotically approximated by the Schwarz Bayes information criterion (BIC), whereas in singular models such approximation does not hold. Recently, it was proved that the Bayes free energy of a singular model is asymptotically give…
Uladzimir Karatkevich National University of Belarus（白俄罗斯乌拉济米尔·卡拉特克维奇国立大学）
Independent sociological polls are forbidden in Belarus. Online polls performed without sound scientific rigour do not yield representative results. Yet, both inside and outside Belarus it is of great importance to obtain precise estimates of the ratings of all candidates. These ratings could function as reliable proxies for the election's outcomes. We conduct an independent poll based on the combination of the data collected via Viber and on the streets of Belarus. The Viber and the street data samples consist of almost 45000 and 1150 unique observations respectively. Bayesian regressions wit…
In this work, we delve into the nonparametric empirical Bayes theory and approximate the classical Bayes estimator by a truncation of the generalized Laguerre series and then estimate its coefficients by minimizing the prior risk of the estimator. The minimization process yields a system of linear equations the size of which is equal to the truncation level. We focus on the empirical Bayes estimation problem when the mixing distribution, and therefore the prior distribution, has a support on the positive real half-line or a subinterval of it. By investigating several common mixing distribution…
The Bayesian statistical paradigm uses the language of probability to express uncertainty about the phenomena that generate observed data. Probability distributions thus characterize Bayesian analysis, with the rules of probability used to transform prior probability distributions for all unknowns - parameters, latent variables, models - into posterior distributions, subsequent to the observation of data. Conducting Bayesian analysis requires the evaluation of integrals in which these probability distributions appear. Bayesian computation is all about evaluating such integrals in the typical c…
While the importance of prior selection is well understood, establishing guidelines for selecting priors in hierarchical models has remained an active, and sometimes contentious, area of Bayesian methodology research. Choices of hyperparameters for individual families of priors are often discussed in the literature, but rarely are different families of priors compared under similar models and hyperparameters. Using simulated data, we evaluate the performance of inverse gamma and half-$t$ priors for estimating the standard deviation of random effects in three hierarchical models: the 8-schools …
Sparsity via Hyperpriors: A Theoretical and Algorithmic Study under Empirical Bayes Framework
School of Mathematical Sciences, Ocean University of China（中国海洋大学数学科学学院）；Department of Applied Mathematics and Computer Science, Technical University of Denmark（丹麦技术大学应用数学与计算机科学系）；Laboratory of Marine Mathematics, Ocean University of China（中国海洋大学海洋数学实验室）
