Moving Target Monte Carlo

The Markov Chain Monte Carlo (MCMC) methods are popular when considering sampling from a high-dimensional random variable x with possibly unnormalised probability density p and observed data d. However, MCMC requires evaluating the posterior distribution p(x|d) of the proposed candidate x at each iteration when constructing the acceptance rate. This is costly when such evaluations are intractable. In this paper, we introduce a new non-Markovian sampling algorithm called Moving Target Monte Carlo (MTMC). The acceptance rate at n-th iteration is constructed using an iteratively updated approximation of the posterior distribution an(x) instead of p(x|d). The true value of the posterior p(x|d) is only calculated if the candidate x is accepted. The approximation an utilises these evaluations and converges to p as n rightarrow ∞. A proof of convergence and estimation of convergence rate in different situations are given.