In computational statistics, the pseudo-marginal Metropolis–Hastings algorithm is a Monte Carlo method to sample from a probability distribution. It is an instance of the popular Metropolis–Hastings algorithm that extends its use to cases where the target density is not available analytically. It relies on the fact that the Metropolis–Hastings algorithm can still sample from the correct target distribution if the target density in the acceptance ratio is replaced by an estimate. It is especially popular in Bayesian statistics, where it is applied if the likelihood function is not tractable (see example below).
Algorithm description The aim is to simulate from some probability density function π ( θ ) {\displaystyle \pi (\theta )} . The algorithm follows the same steps as the standard Metropolis–Hastings algorithm except that the evaluation of the target density is replaced by a non-negative and unbiased estimate. For comparison, the main steps of a Metropolis–Hastings algorithm are outlined below.
Metropolis–Hastings algorithm
Given a current state θ n {\displaystyle \theta _{n}} the Metropolis–Hastings algorithm proposes a new state according to some density θ ′ ∼ Q ( ⋅ ∣ θ n ) {\displaystyle \theta '\sim Q(\cdot \mid \theta _{n})} . The algorithm then sets θ n + 1 = θ ′ {\displaystyle \theta _{n+1}=\theta '} with probability
a ( θ n , θ ′ ) = min ( 1 , π ( θ ′ ) π ( θ n ) Q ( θ n ∣ θ ′ ) Q ( θ ′ ∣ θ n ) ) {\displaystyle a(\theta _{n},\theta ')=\min \left(1,{\frac {\pi (\theta ')}{\pi (\theta _{n})}}{\frac {Q(\theta _{n}\mid \theta ')}{Q(\theta '\mid \theta _{n})}}\right)}
otherwise the old state is kept, that is, θ n + 1 = θ n {\displaystyle \theta _{n+1}=\theta _{n}} .
Pseudo-marginal Metropolis–Hastings algorithm If the density π {\displaystyle \pi } is not available analytically the above algorithm cannot be employed. The pseudo-marginal Metropolis–Hastings algorithm in contrast only assumes the existence of an unbiased estimator π ^ θ {\displaystyle {\hat {\pi }}_{\theta }} , i.e. the estimator must satisfy the equation E [ π ^ θ ] = π ( θ ) . {\displaystyle \mathbb {E} [{\hat {\pi }}_{\theta }]=\pi (\theta ).} Now, given θ n {\displaystyle \theta _{n}} and the respective estimate π ^ θ n {\displaystyle {\hat {\pi }}_{\theta _{n}}} the algorithm proposes a new state according to some density θ ′ ∼ Q ( ⋅ ∣ θ n ) {\displaystyle \theta '\sim Q(\cdot \mid \theta _{n})} . Next, compute an estimate π ^ θ ′ {\displaystyle {\hat {\pi }}_{\theta '}} and set θ n + 1 = θ ′ {\displaystyle \theta _{n+1}=\theta '} with probability
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