Revisiting identification concepts in Bayesian analysis

TL;DR

This paper explores how Bayesian methods handle identification issues in statistical models, showing that priors can sometimes turn nonidentified parameters into identified ones and proposing techniques for partially identified models.

Contribution

It introduces new Bayesian techniques for dealing with unidentified and partially identified models, including prior construction and marginalization methods.

Findings

Prior distributions can identify nonidentified parameters in Bayesian analysis.

Methods for constructing priors and posteriors in partially identified models.

Marginalization helps extract information about nonidentified parameters.

Abstract

This paper studies the role played by identification in the Bayesian analysis of statistical and econometric models. First, for unidentified models we demonstrate that there are situations where the introduction of a non-degenerate prior distribution can make a parameter that is nonidentified in frequentist theory identified in Bayesian theory. In other situations, it is preferable to work with the unidentified model and construct a Markov Chain Monte Carlo (MCMC) algorithms for it instead of introducing identifying assumptions. Second, for partially identified models we demonstrate how to construct the prior and posterior distributions for the identified set parameter and how to conduct Bayesian analysis. Finally, for models that contain some parameters that are identified and others that are not we show that marginalizing out the identified parameter from the likelihood with respect to its conditional prior, given the nonidentified parameter, allows the data to be informative about the nonidentified and partially identified parameter. The paper provides examples and simulations that illustrate how to implement our techniques.