A pseudo-marginal Metropolis–Hastings algorithm replaces an intractable likelihood function by a nonnegative unbiased likelihood estimator and includes its auxiliary randomness in the state. If for , the extended target is proportional to . An independence proposal for new auxiliary randomness cancels the factors in the Metropolis–Hastings acceptance probability. On rejection the old estimate must be retained. Integrating the extended target gives the desired marginal posterior distribution exactly.
An unbiased likelihood estimator has expectation equal to the likelihood function at each parameter. A pseudo-marginal algorithm requires this estimator to be nonnegative. Importance sampling provides for independent random variables , provided the proposal covers the target support. Relative variance affects the mixing time even though unbiasedness guarantees the correct marginal target.
Articles by others on the same topic
The Pseudo-marginal Metropolis-Hastings (PMMH) algorithm is a Markov Chain Monte Carlo (MCMC) method used for sampling from complex posterior distributions, particularly in Bayesian inference settings. It is especially useful when the likelihood function is intractable or computationally expensive to evaluate directly. ### Overview In standard MCMC methods, a proposal distribution is used to explore the parameter space, and the acceptance criterion is based on the ratio of the posterior probabilities.