 
Summary: On Adaptive Markov Chain Monte Carlo Algorithms
Yves F. Atchadé1 and Jerey S. Rosenthal2
(July, 2003; revised March 2005)
Abstract
We look at adaptive MCMC algorithms that generate stochastic processes based on se
quences of transition kernels, where each transition kernel is allowed to depend on the past
of the process. We show under certain conditions that the generated stochastic process
is ergodic, with appropriate stationary distribution. We then consider the Random Walk
Metropolis (RWM) algorithm with normal proposal and scale parameter . We propose an
adaptive version of this algorithm that sequentially adjusts using a RobbinsMonro type
algorithm in order to nd the optimal scale parameter opt as in Roberts et al. (1997). We
show, under some additional conditions that this adaptive algorithm is ergodic and that n,
the sequence of scale parameter obtained converges almost surely to opt. Our algorithm
thus automatically determines and runs the optimal RWM scaling, with no manual tuning
required. We close with a simulation example.
Key words: Adaptive Markov Chain Monte Carlo, Metropolis algorithm, Parameter Tuning,
RobbinsMonro Algorithm, Mixingales.
MSC Numbers: 65C05, 65C40, 60J27, 60J35
1 Introduction
Markov Chain Monte Carlo (MCMC) methods have become an important numerical tool in statistics
