Open Access
August 2018 Unbiased Monte Carlo: Posterior estimation for intractable/infinite-dimensional models
Sergios Agapiou, Gareth O. Roberts, Sebastian J. Vollmer
Bernoulli 24(3): 1726-1786 (August 2018). DOI: 10.3150/16-BEJ911


We provide a general methodology for unbiased estimation for intractable stochastic models. We consider situations where the target distribution can be written as an appropriate limit of distributions, and where conventional approaches require truncation of such a representation leading to a systematic bias. For example, the target distribution might be representable as the $L^{2}$-limit of a basis expansion in a suitable Hilbert space; or alternatively the distribution of interest might be representable as the weak limit of a sequence of random variables, as in MCMC. Our main motivation comes from infinite-dimensional models which can be parameterised in terms of a series expansion of basis functions (such as that given by a Karhunen–Loeve expansion). We introduce and analyse schemes for direct unbiased estimation along such an expansion. However, a substantial component of our paper is devoted to the study of MCMC schemes which, due to their infinite dimensionality, cannot be directly implemented, but which can be effectively estimated unbiasedly. For all our methods we give theory to justify the numerical stability for robust Monte Carlo implementation, and in some cases we illustrate using simulations. Interestingly the computational efficiency of our methods is usually comparable to simpler methods which are biased. Crucial to the effectiveness of our proposed methodology is the construction of appropriate couplings, many of which resonate strongly with the Monte Carlo constructions used in the coupling from the past algorithm.


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Sergios Agapiou. Gareth O. Roberts. Sebastian J. Vollmer. "Unbiased Monte Carlo: Posterior estimation for intractable/infinite-dimensional models." Bernoulli 24 (3) 1726 - 1786, August 2018.


Received: 1 December 2014; Revised: 1 July 2016; Published: August 2018
First available in Project Euclid: 2 February 2018

zbMATH: 06839251
MathSciNet: MR3757514
Digital Object Identifier: 10.3150/16-BEJ911

Keywords: Bayesian inverse problems , coupling , Markov chain Monte Carlo in infinite dimensions , unbiased estimation

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 3 • August 2018
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