A Note on Compatible Prior Distributions in Univariate Finite Mixture and Markov-Switching Models

Abstract

Finite mixture and Markov-switching models generalize and, therefore, nest
specifications featuring only one component. While specifying priors in the
general (mixture) model and its special (single-component) case, it may be
desirable to ensure that the prior assumptions introduced into both structures
are compatible in the sense that the prior distribution in the nested model
amounts to the conditional prior in the mixture model under relevant parametric
restriction. The study provides the rudiments of setting compatible priors in
Bayesian univariate finite mixture and Markov-switching models. Once some
primary results are delivered, we derive specific conditions for compatibility
in the case of three types of continuous priors commonly engaged in Bayesian
modeling: the normal, inverse gamma, and gamma distributions. Further, we
study the consequences of introducing additional constraints into the mixture
model’s prior on the conditions. Finally, the methodology is illustrated through
a discussion of setting compatible priors for Markov-switching AR(2) models.