External Seminar: Chris Sherlock (Lancaster University)

[Statistics and Probability Seminar]

Optimisation of delayed acceptance particle marginal random walk Metropolis algorithms

Delayed-acceptance Metropolis-Hastings (DA-MH) and delayed-acceptance pseudo-marginal Metropolis-Hastings (DA-PsMMH) algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased stochastic approximation thereof, but a computationally cheap deterministic approximation is available. An initial accept-reject stage uses the cheap approximation for computing the Metropolis-Hastings ratio; proposals which are accepted at this stage are then subjected to a further accept-reject step which corrects for the error in the approximation. Since the expensive posterior, or the approximation thereof, is only evaluated for proposals which are accepted at the first stage, the cost of the algorithm is reduced.

We focus on the random walk Metropolis (RWM) and consider the DA-PsMRWM, of which the DA-RWM is a special case. Justified by a limiting diffusion argument, we develop theoretical expressions for limiting efficiency and acceptance rates in high dimension which inform a practical strategy for algorithm tuning. A simulation study verifies a number of heuristic properties suggested by our theory. For the example considered, our optimised DA-PsMRWM algorithm is nearly an order of magnitude more efficient than the equivalent optimised PsMRWM algorithm.