Trajectory averaging for stochastic approximation MCMC algorithms

TL;DR

This paper proves the asymptotic efficiency of trajectory averaging estimators in stochastic approximation MCMC algorithms and demonstrates their application to improve statistical estimation methods.

Contribution

It establishes the asymptotic efficiency of trajectory averaging in SAMCMC and applies it to stochastic approximation MCMC algorithms for statistical estimation.

Findings

Trajectory averaging estimator is asymptotically efficient.

Application to stochastic approximation Monte Carlo improves estimation.

Results are valid under mild conditions.

Abstract

The subject of stochastic approximation was founded by Robbins and Monro [Ann. Math. Statist. 22 (1951) 400--407]. After five decades of continual development, it has developed into an important area in systems control and optimization, and it has also served as a prototype for the development of adaptive algorithms for on-line estimation and control of stochastic systems. Recently, it has been used in statistics with Markov chain Monte Carlo for solving maximum likelihood estimation problems and for general simulation and optimizations. In this paper, we first show that the trajectory averaging estimator is asymptotically efficient for the stochastic approximation MCMC (SAMCMC) algorithm under mild conditions, and then apply this result to the stochastic approximation Monte Carlo algorithm [Liang, Liu and Carroll J. Amer. Statist. Assoc. 102 (2007) 305--320]. The application of the trajectory averaging estimator to other stochastic approximation MCMC algorithms, for example, a stochastic approximation MLE algorithm for missing data problems, is also considered in the paper.