We study a class of adaptive Markov Chain Monte Carlo (MCMC) processes which aim at behaving as an "optimal" target process via a learning procedure. We show, under appropriate conditions, that the adaptive process and "optimal" (nonadaptive) MCMC algorithm share identical asymptotic properties. The special case of adaptive MCMC algorithms governed by stochastic approximation is considered in details and we apply our results to the adaptive Metropolis algorithm of [1]. We also propose a new class of adaptive MCMC algorithms, called quasi-perfect adaptive MCMC which possesses appealing theoretical and practical properties, as demonstrated through numerical simulations.
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