  Finite mixture and Markov-switching models generalize and, therefore, nest
specifications featuring only one component. While specifying priors in the
two: 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 coherent 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 coherent
priors in Bayesian univariate finite mixture and Markov-switching models. Once
some primary results are delivered, we derive specific conditions for coherence
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 (such as the ones enforcing identifiability or some sort of regularity,
e.g. second-order stationarity) on the coherence conditions. Finally, the
methodology is illustrated through a discussion of setting coherent priors for
a class of Markov-switching AR(2) models.
