The pseudo-marginal approach for efficient Monte Carlo computations. (English) Zbl 1185.60083

Andrieu, Christophe; Roberts, Gareth O.

Ann. Stat. 37, No. 2, 697-725 (2009).

A powerful and flexible MCMC (Markov chain Monte Carlo) algorithm for stochastic simulation is proposed which builds on pseudo-marginal method originally introduced by M. A. Beaumont [Genetics 164, 1139–1160 (2003)]. In the discrete case, it is the following. Given joint probability distribution \(\pi(\theta,z)\) and conditional distribution \(q_\theta(z)\) the goal is to sample from the marginal distribution \(\pi(\theta)\) (\(\theta \in \Theta\), \(z \in Z\)). If \(\pi(\theta)\) is known analitically or cheap to compute, it would be possible to use Metropolis-Hastings algorithm and to generate samples from a Markov chain with transition probabilities \(p(\theta, \theta')=1 \wedge \frac{\pi(\theta')}{\pi(\theta)}\) if \(\theta' \neq \theta\). If it is not the case, it is proposed in the article to replace probabilities \(\pi(\theta)\), \(\pi(\theta')\) with the estinators \[ \overline{\pi}^N(\theta)=\frac{1}{N} \sum_{k=1}^N \frac{\pi(\theta,z(k))}{q_\theta(z(k))}, \quad \overline{\pi}^N(\theta')=\frac{1}{N} \sum_{k=1}^N \frac{\pi(\theta',z'(k))}{q_{\theta'}(z'(k))}, \] where \(z(k)|\theta \sim q_\theta(\cdot)\), \(z'(k)|\theta' \sim q_{\theta'}(\cdot)\), \(k=1,\dots,N\), the data available during the simulation.
In the article for rather arbitrary spaces \(\Theta\), \(Z\) theoretical results are given describing the convergence properties of the proposed method, and simple numerical examples are given to illustrate the promising empirical characteristics of the technique. Interesting comparisons with a more obvious, but inexact, Monte Carlo approximations to the marginal algorithm are also given.

Reviewer: Alex V. Kolnogorov (Novgorod)

MSC:

60J22	Computational methods in Markov chains
60K35	Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

Markov chain Monte Carlo; auxiliary variable; marginal; convergence

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