 
Summary: A cautionary tale on the efficiency of some adaptive
Monte Carlo Schemes
Yves F. Atchad´e
(First version Jul. 2007; Rev. Jan. 09 and Aug. 09)
Abstract: There is a growing interest in the literature for adaptive Markov Chain Monte
Carlo methods based on sequences of random transition kernels {Pn} where the kernel Pn
is allowed to have an invariant distribution n not necessarily equal to the distribution of
interest (target distribution). These algorithms are designed such that as n , Pn
converges to P, a kernel that has the correct invariant distribution . Typically, P is a
kernel with good convergence properties, but one that cannot be directly implemented. It
is then expected that the algorithm will inherit the good convergence properties of P. The
equienergy sampler of [15] is an example of this type of adaptive MCMC. We show in this
paper, that the asymptotic variance of this type of adaptive MCMC is always at least as
large as the asymptotic variance of the Markov chain with transition kernel P. We also show
by simulation that the difference can be substantial.
AMS 2000 subject classifications: Primary 60C05, 60J27, 60J35, 65C40.
Keywords and phrases: Monte Carlo methods, Adaptive MCMC, EquiEnergy sampler,
Martingale approximation, Central limit theorems, Importance resampling.
1. Introduction
Adaptive Markov Chain Monte Carlo (AMCMC) is an approach to Markov Chain Monte Carlo
