Convergence rates of posterior distributions for noniid observations

TL;DR

This paper investigates how quickly Bayesian posterior distributions converge when observations are neither independent nor identically distributed, providing general convergence results and applying them to various complex statistical models.

Contribution

It offers new theoretical results on the convergence rates of posterior distributions for non-i.i.d. data, extending Bayesian asymptotics to a broad class of models.

Findings

Established general convergence rate theorems for non-i.i.d. observations.

Applied results to diverse models including nonparametric regression and time series.

Demonstrated the applicability of the theory to infinite-dimensional statistical models.

Abstract

We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion. We then specialize our results to independent, nonidentically distributed observations, Markov processes, stationary Gaussian time series and the white noise model. We apply our general results to several examples of infinite-dimensional statistical models including nonparametric regression with normal errors, binary regression, Poisson regression, an interval censoring model, Whittle estimation of the spectral density of a time series and a nonlinear autoregressive model.