On consistency of Bayesian parameter estimations for a class of ergodic Markov models
A. I. Nurieva, A. Yu. Veretennikov
TL;DR
This work extends Doob's Bayesian consistency framework from i.i.d. settings to a class of ergodic, discrete-state Markov chains by leveraging a strong LLN for sample distribution functions and an identifiability bridge via a two-dimensional invariant distribution. The Bayesian estimator $\hat{\theta}_n = \mathbb{E}(\theta|X_1,\dots,X_n)$ is shown to converge almost surely to the true parameter value under a product prior, provided the parameter uniquely determines the limiting joint distribution of $(X_n,X_{n+1})$. The key steps combine Lévy–Doob convergence, measurability of the invariant-df mapping $G$, and the one-to-one identifiability of $\theta$ from $\hat{F}^\theta$, with the Prokhorov metric giving the required Polish-space structure. The results have potential applications to reliability theory and risk management where model parameters are unknown and data are generated by ergodic Markov dynamics.
Abstract
The consistency of the Bayesian estimation of a parameter is shown for a class of ergodic discrete Markov chains. J.L. Doob's method was used, offered earlier for the i.i.d. situation. The result may be useful in the reliability theory for models with unknown parameters, in the risk management in financial mathematics, and in other applications.