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

There is a growing interest in the literature for adaptive Markov chain Monte Carlo methods based on sequences of random transition kernels $\{P_n\}$ where the kernel $P_n$ is allowed to have an invariant distribution $\pi_n$ not necessarily 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 directly implemented. It is then expected that the algorithm will inherit the good convergence 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 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.