Abstract
Pseudo-marginal Markov chain Monte Carlo methods for sampling from intractable distributions have gained recent interest and have been theoretically studied in considerable depth. Their main appeal is that they are exact, in the sense that they target marginally the correct invariant distribution. However, the pseudo-marginal Markov chain can exhibit poor mixing and slow convergence towards its target. As an alternative, a subtly different Markov chain can be simulated, where better mixing is possible but the exactness property is sacrificed. This is the noisy algorithm, initially conceptualised as Monte Carlo within Metropolis, which has also been studied but to a lesser extent. The present article provides a further characterisation of the noisy algorithm, with a focus on fundamental stability properties like positive recurrence and geometric ergodicity. Sufficient conditions for inheriting geometric ergodicity from a standard Metropolis–Hastings chain are given, as well as convergence of the invariant distribution towards the true target distribution.Electronic supplementary materialThe online version of this article (doi:10.1007/s11222-015-9604-3) contains supplementary material, which is available to authorized users.
Highlights
1.1 Intractable target densities and the pseudo-marginal algorithmSuppose our aim is to simulate from an intractable probability distribution π for some random variable X, which takes values in a measurable space (X, B(X ))
The noisy Markov kernels considered are perturbed Metropolis–Hastings kernels defined by a collection of state-dependent distributions for non-negative weights all with expectation 1
Two different sets of sufficient conditions were provided under which the noisy chain inherits geometric ergodicity from the marginal chain
Summary
Suppose our aim is to simulate from an intractable probability distribution π for some random variable X , which takes values in a measurable space (X , B(X )). 2009) falls into this category since it is a Metropolis–Hastings (MH) algorithm targeting a distribution π N , associated to the random vector (X, W ) defined on the product space (X × W, B(X ) × B(W)) where W ⊆ R+0 := [0, ∞). Andrieu and Roberts (2009), Andrieu and Vihola (2014, 2015), Doucet et al (2015), Girolami et al (2013), Maire et al (2014) and Sherlock et al (2015) These studies typically compare the pseudo-marginal Markov chain with a “marginal” Markov chain, arising in the case where all the weights are almost surely equal to 1, and (3) is the standard Metropolis–Hastings acceptance probability associated with the target density π and the proposal q
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