arXiv subjects
Publications and source records attributed to Edward I. George.
For the normal linear model variable selection problem, we propose selection criteria based on a fully Bayes formulation with a generalization of Zellner's $g$-prior which allows for $p>n$. A special case of the prior formulation is seen to yield tractable closed forms for marginal densities and Bayes factors which reveal new model evaluation characteristics of potential interest.
We develop a Bayesian "sum-of-trees" model where each tree is constrained by a regularization prior to be a weak learner, and fitting and inference are accomplished via an iterative Bayesian backfitting MCMC algorithm that generates samples from a posterior. Effectively, BART is a nonparametric Bayesian regression approach which uses dimensionally adaptive random basis elements. Motivated by ensemble methods in general, and boosting algorithms in particular, BART is defined by a statistical model: a prior and a likelihood. This approach enables full posterior inference including point and interval estimates of the unknown regression function as well as the marginal effects of potential predictors. By keeping track of predictor inclusion frequencies, BART can also be used for model-free variable selection. BART's many features are illustrated with a bake-off against competing methods on 42 different data sets, with a simulation experiment and on a drug discovery classification problem.
Let $X|μ\sim N_p(μ,v_xI)$ and $Y|μ\sim N_p(μ,v_yI)$ be independent $p$-dimensional multivariate normal vectors with common unknown mean $μ$. Based on observing $X=x$, we consider the problem of estimating the true predictive density $p(y|μ)$ of $Y$ under expected Kullback--Leibler loss. Our focus here is the characterization of admissible procedures for this problem. We show that the class of all generalized Bayes rules is a complete class, and that the easily interpretable conditions of Brown and Hwang [Statistical Decision Theory and Related Topics (1982) III 205--230] are sufficient for a formal Bayes rule to be admissible.
It is with pleasure and pride that I introduce this special section in honor of Ingram Olkin. This tribute is especially fitting because, among the many profound and far-reaching contributions that he has made to our profession, Ingram Olkin was the key force behind the genesis of Statistical Science. As put so eloquently by Morrie DeGroot [1], the founding Executive Editor of Statistical Science.
Let $X| μ\sim N_p(μ,v_xI)$ and $Y| μ\sim N_p(μ,v_yI)$ be independent p-dimensional multivariate normal vectors with common unknown mean $μ$. Based on only observing $X=x$, we consider the problem of obtaining a predictive density $\hat{p}(y| x)$ for $Y$ that is close to $p(y| μ)$ as measured by expected Kullback--Leibler loss. A natural procedure for this problem is the (formal) Bayes predictive density $\hat{p}_{\mathrm{U}}(y| x)$ under the uniform prior $π_{\mathrm{U}}(μ)\equiv 1$, which is best invariant and minimax. We show that any Bayes predictive density will be minimax if it is obtained by a prior yielding a marginal that is superharmonic or whose square root is superharmonic. This yields wide classes of minimax procedures that dominate $\hat{p}_{\mathrm{U}}(y| x)$, including Bayes predictive densities under superharmonic priors. Fundamental similarities and differences with the parallel theory of estimating a multivariate normal mean under quadratic loss are described.
In this paper we propose a Bayesian, information theoretic approach to dimensionality reduction. The approach is formulated as a variational principle on mutual information, and seamlessly addresses the notions of sufficiency, relevance, and representation. Maximally informative statistics are shown to minimize a Kullback-Leibler distance between posterior distributions. Illustrating the approach, we derive the maximally informative one dimensional statistic for a random sample from the Cauchy distribution.