F. Javier Girón

CONSISTENCY OF BAYESIAN PROCEDURES FOR VARIABLE SELECTION

It has long been known that for the comparison of pairwise nested models, a decision based on the Bayes factor produces a consistent model selector (in the frequentist sense). Here we go beyond the usual consistency for nested pairwise models, and show that for a wide class of prior distributions, including intrinsic priors, the corresponding Bayesian procedure for variable selection in normal regression is consistent in the entire class of normal linear models. We find that the asymptotics of the Bayes factors for intrinsic priors are equivalent to those of the Schwarz (BIC) criterion. Also, recall that the Jeffreys–Lindley paradox refers to the well-known fact that a point null hypothesis on the normal mean parameter is always accepted when the variance of the conjugate prior goes to infinity. This implies that some limiting forms of proper prior distributions are not necessarily suitable for testing problems. Intrinsic priors are limits of proper prior distributions, and for finite sample sizes they have been proved to behave extremely well for variable selection in regression; a consequence of our results is that for intrinsic priors Lindley’s paradox does not arise.

Cluster Analysis, Model Selection, and Prior Distributions on Models

Clustering is an important and challenging statistical problem for which there is an extensive literature. Modeling approaches include mixture models and product partition models. Here we develop a product partition model and a model selection procedure based on Bayes factors from intrinsic priors. We also find that the choice of the prior on model space is of utmost importance, almost overshadowing the other parts of the clustering problem, and we examine the behavior of posterior odds based on different model space priors. We find, somewhat surprisingly, that procedures based on the often-used uniform prior (in which all models are given the same prior probability) lead to inconsistent model selection procedures. We examine other priors, and find that a new prior, the hierarchical uniform prior leads to consistent model selection procedures and has other desirable properties. Lastly, we examine our procedures, and competitors, on a range of examples.

An Objective Bayesian Procedure for Variable Selection in Regression

The Bayesian analysis of the variable selection problem in linear regression when using objective priors needs some form of encompassing the class of all submodels of the full linear model as they are nonnested models. After we provide a nested setting, objective intrinsic priors suitable for computing model posterior probabilities, on which the selection is based, can be derived.

A characterization of the Behrens-Fisher distribution with applications to bayesian inference

The Behrens-Fisher distribution is generally defined as the convolution of two Student t distributions and it is well known that it can be represented as a scale mixture of normals. By extending this standardized distribution to a location-scale family in the usual way we prove that this generalised Behrens--Fisher distribution can also be represented as a location mixture of t distributions when the mixing distribution is, in turn, a Student t. This characterization is applied to the computation of certain predictive distributions appearing in the Bayesian analysis of two sample problems. 0 AcadCmie des Sciences/Elsevier, Paris

Independence issues in imprecise data models: a Bayesian approach

A non-precise data problem is defined as a two-stage hierarchical model. The question considered in this Note is to explore some theoretical and practical problems involving independence and/or conditional independence relations among the parameters and the data. We present some results in this direction.

Automatic Diagnostic of Breast Cancer: A Case Study

The examination of mammograms along with some (historical) information on the patients lead physicians, in some informal way, to declare a patient as having or not breast cancer. This diagnostic is usually based on historical variables, such as age, familiar antecedents, etc. and other semiologic variables derived from the analysis of the mammogram. Most of these variables are usually of an ordinal nature.

Asymptotic Behavior of Fractional Blending Systems

After a brief description of the dynamic systems of ageing wines and spirits known as “fractional blending systems” or “criaderas and solera” systems, a general mathematical model is presented to determine, on the one hand, the distribution of the age of liquids of all the scales of the system and, on the other hand, the mean or average age of the liquids as the system is run. A theorem on the existence of an asymptotic equilibrium distribution of the “fractional blending systems” is given. This result refers to the existence of a unique asymptotic distribution of the ages which turns out to be a generalization of the Pascal distribution. This, in turn, implies the existence and uniqueness of an equilibrium mean or average of the ages.