Identifiability, Bayesian Learning, and Gibbs Sampler in a Conditionally Autoregressive Model

• Activity Number: 74
• Type: Contributed
• Date/Time: Sunday, July 31, 2011 : 4:00 PM to 5:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Luis M. Castro*+ and Ernesto J. San Martin
• Companies: Universidad de Concepción and Pontificia Universidad Católica de Chile
• Keywords: Bayesian identifiability ; conditionally autoregressive model ; ergodicity ; Gibbs sampler ; measurable separability ; minimal sufficient parameter

Abstract

Bayesian statistics seems to solve a problem which classical statistics fails to do it, namely to estimate unidentified parameters. Following Lindley (1972), it is said that any parameter (in particular, the unidentified ones) having a proper prior distribution also has a proper posterior, and is thus estimable. This explains why Bayesian statistics did not focus its attention on the identification problem. However, recently Bayesian statistics became interesting in identifiability because the Gibbs-sampling procedure seems to be affected when the posterior distribution of an unidentified parameter is computed.

A sampling process can be characterized by a set of minimal sufficient parameter. The posterior distribution of an unidentified parameter is always a function of the posterior distribution of the identified parameter. Consequently, it is the identified parameter which fully characterizes the Bayesian learning process. In this talk we will explain why the Gibbs-sampling procedure seems to fail when computing the posterior distribution of the unidentified parameter, showing that in the CAR model, the ergodicity of the chain seems to fail from a numerical point of view.