APPROACHES FOR BAYESIAN VARIABLE SELECTION

TLDR

The paper discusses variable selection in normal linear regression, noting nonconjugate SSVS and conjugate formulations that enable analytical simplification. The study aims to describe and compare hierarchical mixture prior formulations for variable selection uncertainty in normal linear regression, examining hyperparameter settings based on practical significance and the use of mixtures with point priors. The authors employ hierarchical mixture priors, analyze hyperparameter settings, and use computational methods such as Gray Code sequencing for exhaustive evaluation in moderate problems and fast MCMC for large problems, illustrating procedures on simulated and real financial index tracking data. Rapid updating methods enable feasible exhaustive evaluation and fast MCMC exploration, improving posterior estimates of model probabilities and total visited probability, as demonstrated on simulated and real financial index tracking problems.

Abstract

This paper describes and compares various hierarchical mixture prior formulations of variable selection uncertainty in normal linear regression models. These include the nonconjugate SSVS formulation of George and McCulloch (1993), as well as conjugate formulations which allow for analytical simplification. Hyperpa- rameter settings which base selection on practical significance, and the implications of using mixtures with point priors are discussed. Computational methods for pos- terior evaluation and exploration are considered. Rapid updating methods are seen to provide feasible methods for exhaustive evaluation using Gray Code sequencing in moderately sized problems, and fast Markov Chain Monte Carlo exploration in large problems. Estimation of normalization constants is seen to provide improved posterior estimates of individual model probabilities and the total visited probabil- ity. Various procedures are illustrated on simulated sample problems and on a real problem concerning the construction of financial index tracking portfolios.