Abstract

When an unbiased estimator of the likelihood is used within an Metropolis-Hastings scheme, it is necessary to tradeoff the number of samples used to evaluate the likelihood against the computing time. Many samples will result in a scheme which has similar properties to the case where the likelihood is exactly known but will be expensive. Few samples will result in faster estimation but at the expense of slower mixing of the Markov chain. We explore the relationship between the number of samples and the efficiency of the resulting Metropolis-Hastings estimates. Under the assumption that the distribution of the additive noise introduced by the log-likelihood estimator is independent of the point at which this log-likelihood is evaluated and other relatively mild assumptions, we provide guidelines on the number of samples to select for a general Metropolis-Hastings proposal. We illustrate on a complex stochastic volatility model that these assumptions are approximately satisfied experimentally and that the theoretical insights with regards to inefficiency and computational time hold true.

Keywords: Bayesian inference; Estimated likelihood; Metropolis-Hastings; Particle filtering.