Convergence rates of posterior distributions for observations without the iid structure

TL;DR

This paper develops new prior-dependent conditions for Bayesian posterior convergence rates applicable to non-i.i.d. data, including Markov processes and nonlinear autoregressive models, using Hausdorff α-entropy.

Contribution

It introduces a prior-dependent integration condition and employs Hausdorff α-entropy to extend posterior convergence theorems to non-i.i.d. observations.

Findings

Established a general posterior convergence rate theorem for Markov processes.

Improved known convergence rates for nonlinear autoregressive models.

Extended existing theorems to broader classes of dependent data.

Abstract

The classical condition on the existence of uniformly exponentially consistent tests for testing the true density against the complement of its arbitrary neighborhood has been widely adopted in study of asymptotics of Bayesian nonparametric procedures. Because we follow a Bayesian approach, it seems to be more natural to explore alternative and appropriate conditions which incorporate the prior distribution. In this paper we supply a new prior-dependent integration condition to establish general posterior convergence rate theorems for observations which may not be independent and identically distributed. The posterior convergence rates for such observations have recently studied by Ghosal and van der Vaart \cite{ghv1}. We moreover adopt the Hausdorff $\alpha$-entropy given by Xing and Ranneby \cite{xir1}\cite{xi1}, which is also prior-dependent and smaller than the widely used metric entropies. These lead to extensions of several existing theorems. In particular, we establish a posterior convergence rate theorem for general Markov processes and as its application we improve on the currently known posterior rate of convergence for a nonlinear autoregressive model.