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In Bayesian statistics, many problems can be expressed as the evaluation ofthe expectation of a quantity of interest with respect to the posteriordistribution. Standard Monte Carlo method is often not applicable because theencountered posterior distributions cannot be sampled directly. In this case,the most popular strategies are the importance sampling method, Markov chainMonte Carlo, and annealing. In this paper, we introduce a new scheme forBayesian inference, called Asymptotically Independent Markov Sampling (AIMS),which is based on the above methods. We derive important ergodic properties ofAIMS. In particular, it is shown that, under certain conditions, the AIMSalgorithm produces a uniformly ergodic Markov chain. The choice of the freeparameters of the algorithm is discussed and recommendations are provided forthis choice, both theoretically and heuristically based. The efficiency of AIMSis demonstrated with three numerical examples, which include both multi-modaland higher-dimensional target posterior distributions.

Asymptotically Independent Markov Sampling: a new MCMC scheme for Bayesian Inference