Adaptive Metropolis Algorithm Using Variational Bayesian Adaptive Kalman Filter

Markov chain Monte Carlo (MCMC) methods are powerful computational tools for analysis of complex statistical problems. However, their computational efficiency is highly dependent on the chosen proposal distribution. One way to solve this problem is to use adaptive MCMC algorithms which automatically tune the statistics of a proposal distribution during the MCMC run. In this work, we propose a new adaptive Metropolis-based MCMC algorithm called the variational Bayesian adaptive Metropolis (VBAM) algorithm where the proposal covariance matrix is adapted using the variational Bayesian adaptive Kalman filter (VB-AKF). We prove a strong law of large numbers for the VBAM algorithm. We also provide the empirical convergence results of two simulated examples, where the VBAM results are compared with other existing adaptive Metropolis algorithms.