An ABC interpretation of the multiple auxiliary variable method

Markov random fields (MRFs) have densities of the form f(y θ) γ(y θ)/Z(θ), (1) where γ(y θ) can be evaluated numerically but Z(θ) cannot in a reasonable time. This makes it challenging to perform inference. This note considers two approaches which both use simulation from f(y θ). The single auxiliary variable (SAV) method (Møller et al., 2006) and the multiple auxiliary variable (MAV) method (Murray et al., 2006) provide unbiased likelihood estimates. Approximate Bayesian computation (Marin et al., 2012) finds parameters which produce simulations similar to the observed data.