 Pseudo-marginal Metropolis–Hastings algorithm

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 $\mathbb{E}[\hat{L}(\theta ,U)]=L(\theta )$ for $U\sim m_{\theta }$, the extended target is proportional to $p(\theta )\hat{L}(\theta ,u)m_{\theta }(u)$. An independence proposal for new auxiliary randomness cancels the $m_{\theta }$ 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.

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.