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There is a growing interest in the literature for adaptive Markov chain MonteCarlo methods based on sequences of random transition kernels $\{P_n\}$ wherethe kernel $P_n$ is allowed to have an invariant distribution $\pi_n$ notnecessarily equal to the distribution of interest $\pi$ (target distribution).These algorithms are designed such that as $n\to\infty$, $P_n$ converges to$P$, a kernel that has the correct invariant distribution $\pi$. Typically, $P$is a kernel with good convergence properties, but one that cannot be directlyimplemented. It is then expected that the algorithm will inherit the goodconvergence properties of $P$. The equi-energy sampler of [Ann. Statist. 34(2006) 1581--1619] is an example of this type of adaptive MCMC. We show in thispaper that the asymptotic variance of this type of adaptive MCMC is always atleast as large as the asymptotic variance of the Markov chain with transitionkernel $P$. We also show by simulation that the difference can be substantial.

A cautionary tale on the efficiency of some adaptive Monte Carlo schemes