On the usefulness of asymmetric acceptance ratio Metropolis-Hastings update

Consider the standard Metropolis-Hastings (MH) algorithm for a given distribution P on x. This work is on exact-approximate algorithms that expand the scope of MH to situations where its acceptance ratio r(x, x’) is intractable. We present a novel class of exact-approximate MH algorithms. The motivation is the desire to benefit averaging of multiple noisy estimates of r(x, x’) and still preserving detailed balance w.r.t. P. We show that this is indeed possible with the use of a pair of proposal kernels and asymmetric acceptance ratios. Moreover, the steps within one iteration that increase statistical efficiency with the cost of extra computation are parallelizable. One interesting application of the methodology that is discussed is a simple extension of the exchange algorithm of Murray et al (2006) for doubly intractable distributions. Use of the methodology for general latent variable models is also demonstrated with a toy example.