Pseudo-marginal Metropolis–Hastings algorithm

The Pseudo-Marginal Metropolis-Hastings algorithm is a Monte Carlo method to sample from probability distribution. It is an instance of the popular Metropolis-Hastings algorithm that extends its use to cases where the density of the target function is not available analytically. It is especially popular in Bayesian statistics, where it is applied if the likelihood function is not tractable (see example below). The main observation

History

The use of an unbiased estimator in the Metropolis-Hastings algorithm appeared first in and was introduced.

Algorithm Description

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.

Proof of correctness

The algorithm can be written as

Example: Latent variable model

Consider a model with consisting of i.i.d. latent real-valued random variables $Z_{1},\ldots ,Z_{n}$ with $Z_{i}\sim f_{\theta }(\cdot )$ and suppose one can only observe these variables through some additional noise $Y_{i}\mid Z_{i}=z\sim g(\cdot \mid z)$ for some conditional density $g$. We are interested in Bayesian analysis of this model based on some observed data $y_{1},\ldots ,y_{n}$. Therefore, we introduce some prior distribution $p(\theta )$ on the parameter. In order to compute the posterior distribution

${\textstyle p(\theta \mid y_{1},\ldots ,y_{n})\propto f_{\theta }(y_{1},\ldots ,y_{n})p(\theta )}$

we need to find the likelihood function $f_{\theta }(y_{1},\ldots ,y_{n})$. The likelihood contribution of any observed data point $y$ is then

$p_{\theta }(y)=\int g(y\mid z)f_{\theta }(z)dz$

and the joint likelihood of the observed data $y_{1},\ldots ,y_{n}$ is

$f_{\theta }(y_{1},\ldots ,y_{n})=\prod _{i=1}^{n}p_{\theta }(y_{i})=\prod _{i=1}^{n}\int g(y_{i}\mid z_{i})f_{\theta }(z_{i})dz_{i}.$

If the integral on the right-hand side is not analytically available.