Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms

By Christophe Andrieu, Matti Vihola. 2012

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We study convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms (Andrieu and Roberts [Ann. Statist. 37 (2009) 697-725]). We find that the asymptotic variance of the pseudo-marginal algorithm is always at least as large as that of the marginal algorithm. We show that if the marginal chain admits a (right) spectral gap and the weights (normalised estimates of the target density) are uniformly bounded, then the pseudo-marginal chain has a spectral gap. In many cases, a similar result holds for the absolute spectral gap, which is equivalent to geometric ergodicity. We consider also unbounded weight distributions and recover polynomial convergence rates in more specific cases, when the marginal algorithm is uniformly ergodic or an independent Metropolis-Hastings or a random-walk Metropolis targeting a super-exponential density with regular contours. Our results on geometric and polynomial convergence rates imply central limit theorems. We also prove that under general conditions, the asymptotic variance of the pseudo-marginal algorithm converges to the asymptotic variance of the marginal algorithm if the accuracy of the estimators is increased.

Subfields: Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9fb09f9302dd57f16bd3451a0817474d2a41abd7

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Citations in the corpus

The correlated pseudomarginal method (2015, methodology): “Qualitative convergence results for this procedure have been obtained by Andrieu and Roberts (2009) and Andrieu and Vihola (2015). Assuming that the likelihood estimator is evaluated using importance sampling or particle filters for state-space models with N particles, it has also been shown under v”
Toward a general, scaleable framework for Bayesian teaching with applications to topic models (2016, methodology): “To generate documents from the teaching distribution, we employ pseudo-marginal sampling [18, 19], which allows exact Metropolis-Hasting to be performed using approximated functions.”
Bayesian computation: a summary of the current state, and samples backwards and forwards (2015, background): “It is not difficult to verify that the bivariate chain on extended state spaceΘ×R+ enjoys the correct π(θ |y)marginal onΘ and the approach is valid, seeAndrieu andRoberts (2009) for details (and also Andrieu and Vihola 2015 for an abstracted account). One specific instance of constructing unbiased e”
On Markov chain Monte Carlo methods for tall data (2015, background): “The possibility to use an unbiased estimator of γ comes at a price: ﬁrst, the asymptotic variance σ 2 lim in (4) of an MCMC estimator based on a pseudo-marginal chain will always be larger than that of an estimator based on the underlying “marginal” MH (Andrieu & Vihola, 2015).”
Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers (2013, methodology): “For PMMH, [5] show that if the essential supremum of the relative normalizing constant estimate γ̂ Nθ,T (Z1:T )/γθ,T is moreover bounded essentially uniformly in θ then the existence of a spectral gap of the idealized Metropolis–Hastings Markov kernel it approximates is inherited by PMMH.”
Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, background): “In particular, the improvements from using multiple samples of z to estimate π can be concluded from Andrieu and Vihola (2012) where the eﬃciency of the algorithm is studied in terms of its spectral gap and CLT asymptotic variance.”
On the efficiency of pseudo-marginal random walk Metropolis algorithms (2013, background): “C O ] 1 6 Ju n 20 14 2 KdV12]) because it requires no further information about the target, such as the local gradient or Hessian, which are generally more computationally expensive to approximate than the target itself [PDS11].”