Abstract

Co-authors: Andrew Golightly (Newcastle University) and Daniel Henderson (Newcastle University)

Delayed-acceptance (DA) pseudo-marginal Metropolis-Hastings (MH) algorithms can be applied when it is computationally expensive to calculate an unbiased estimate of the true posterior, but a computationally cheap approximation is available. A first accept-reject stage is applied, with the cheap approximation substituted for the true posterior in the MH acceptance ratio. Only for those proposals which pass through the first stage is the computationally expensive true posterior (or unbiased estimate thereof) evaluated, with a second accept-reject stage ensuring that detailed balance is satisfied with respect to the intended posterior. A weighted average of all previous unbiased estimates of the true posterior provides a generic approximation to the true posterior. If only the k-nearest neighbours are used in the average then evaluation of the approximate posterior can be made computationally cheap provided that the points at which the posterior has been estimated unbiasedly are stored in a multi-dimensional binary tree, similar to a KD-tree.