4,025 research outputs found
Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem
This paper studies the multiplicity-correction effect of standard Bayesian
variable-selection priors in linear regression. Our first goal is to clarify
when, and how, multiplicity correction happens automatically in Bayesian
analysis, and to distinguish this correction from the Bayesian Ockham's-razor
effect. Our second goal is to contrast empirical-Bayes and fully Bayesian
approaches to variable selection through examples, theoretical results and
simulations. Considerable differences between the two approaches are found. In
particular, we prove a theorem that characterizes a surprising aymptotic
discrepancy between fully Bayes and empirical Bayes. This discrepancy arises
from a different source than the failure to account for hyperparameter
uncertainty in the empirical-Bayes estimate. Indeed, even at the extreme, when
the empirical-Bayes estimate converges asymptotically to the true
variable-inclusion probability, the potential for a serious difference remains.Comment: Published in at http://dx.doi.org/10.1214/10-AOS792 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Training samples in objective Bayesian model selection
Central to several objective approaches to Bayesian model selection is the
use of training samples (subsets of the data), so as to allow utilization of
improper objective priors. The most common prescription for choosing training
samples is to choose them to be as small as possible, subject to yielding
proper posteriors; these are called minimal training samples.
When data can vary widely in terms of either information content or impact on
the improper priors, use of minimal training samples can be inadequate.
Important examples include certain cases of discrete data, the presence of
censored observations, and certain situations involving linear models and
explanatory variables. Such situations require more sophisticated methods of
choosing training samples. A variety of such methods are developed in this
paper, and successfully applied in challenging situations
Optimal predictive model selection
Often the goal of model selection is to choose a model for future prediction,
and it is natural to measure the accuracy of a future prediction by squared
error loss. Under the Bayesian approach, it is commonly perceived that the
optimal predictive model is the model with highest posterior probability, but
this is not necessarily the case. In this paper we show that, for selection
among normal linear models, the optimal predictive model is often the median
probability model, which is defined as the model consisting of those variables
which have overall posterior probability greater than or equal to 1/2 of being
in a model. The median probability model often differs from the highest
probability model
Posterior propriety and admissibility of hyperpriors in normal hierarchical models
Hierarchical modeling is wonderful and here to stay, but hyperparameter
priors are often chosen in a casual fashion. Unfortunately, as the number of
hyperparameters grows, the effects of casual choices can multiply, leading to
considerably inferior performance. As an extreme, but not uncommon, example use
of the wrong hyperparameter priors can even lead to impropriety of the
posterior. For exchangeable hierarchical multivariate normal models, we first
determine when a standard class of hierarchical priors results in proper or
improper posteriors. We next determine which elements of this class lead to
admissible estimators of the mean under quadratic loss; such considerations
provide one useful guideline for choice among hierarchical priors. Finally,
computational issues with the resulting posterior distributions are addressed.Comment: Published at http://dx.doi.org/10.1214/009053605000000075 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Objective Bayes testing of Poisson versus inflated Poisson models
The Poisson distribution is often used as a standard model for count data.
Quite often, however, such data sets are not well fit by a Poisson model
because they have more zeros than are compatible with this model. For these
situations, a zero-inflated Poisson (ZIP) distribution is often proposed. This
article addresses testing a Poisson versus a ZIP model, using Bayesian
methodology based on suitable objective priors. Specific choices of objective
priors are justified and their properties investigated. The methodology is
extended to include covariates in regression models. Several applications are
given.Comment: Published in at http://dx.doi.org/10.1214/074921708000000093 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
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