Abstract not found

Zellner’s g-prior remains a popular conventional prior for use in Bayesian variable selection, despite several undesirable consistency issues. In this paper, we study mixtures of g-priors as an alternative to default g-priors that resolve many of the problems with the original formulation, while maintaining the computational tractability that has made the g-prior so popular. We present theoretical properties of the mixture g-priors and provide real and simulated examples to compare the mixture formulation with fixed g-priors, Empirical Bayes approaches and other default procedures.

Abstract not found

For the problem of model choice in linear regression, we introduce a Bayesian adaptive sampling algorithm (BAS), that samples models without replacement from the space of models. For problems that permit enumeration of all models BAS is guaranteed to enumerate the model space in 2 p iterations where p is the number of potential variables under consideration. For larger problems where sampling is required, we provide conditions under which BAS provides perfect samples without replacement. When the sampling probabilities in the algorithm are the marginal variable inclusion probabilities, BAS may be viewed as sampling models “near ” the median probability model of Barbieri and Berger. As marginal inclusion probabilities are not known in advance we discuss several strategies to estimate adaptively the marginal inclusion probabilities within BAS. We illustrate the performance of the algorithm using simulated and real data and show that BAS can outperform Markov chain Monte Carlo methods. The algorithm is implemented in the R package BAS available at CRAN.

Technological advances in genotyping have given rise to hypothesis– based association studies of increasing scope. As a result, the scientific hypotheses addressed by these studies have become more complex and more difficult to address using existing analytic methodologies. Obstacles to analysis include inference in the face of multiple comparisons, complications arising from correlations among the SNPs (single nucleotide polymorphisms), choice of their genetic parametrization and missing data. In this paper we present an efficient Bayesian model search strategy that searches over the space of genetic markers and their genetic parametrization. The resulting method for Multilevel Inference of SNP Associations, MISA, allows computation of multilevel posterior probabilities and Bayes factors at the global, gene and SNP level, with the prior distribution on SNP inclusion in the model providing an intrinsic multiplicity correction. We use simulated data sets to characterize MISA’s statistical power, and show that MISA has higher power to detect association than standard procedures. Using data from the North Carolina Ovarian Cancer Study (NCOCS), MISA identifies variants that were not identified by standard methods and have been externally ’validated ’ in independent studies. We examine sensitivity of the NCOCS results to prior choice and method for imputing missing data. MISA is available in an R package on CRAN.

In variable selection problems that preclude enumeration of models, stochastic search algorithms, often based on Markov Chain Monte Carlo, are commonly used to identify a set of models for model selection or model averaging. Because Monte Carlo frequencies of models are often zero or one in high dimensional problems, posterior probabilities calculated from the observed marginal likelihoods, re-normalized over the sampled models are often employed. Such estimates are the only recourse in the newer generation of stochastic search algorithms. In this paper, we show that the approach of estimating model probabilities based on renormalization of posterior probabilities over the set of sampled models leads to bias in many quantities of interest and may not reduce mean squared error.