Unbiased Monte Carlo: posterior estimation for intractable/infinite-dimensional models

TL;DR

This paper introduces a general methodology for unbiased estimation in intractable and infinite-dimensional models, enabling accurate inference without bias from truncation or approximation.

Contribution

It develops unbiased estimation schemes for models represented as limits of distributions, especially in infinite-dimensional settings, with theoretical justification and practical simulations.

Findings

Methods achieve unbiased estimates in complex models.

The computational efficiency is comparable to biased methods.

The approach is supported by theoretical stability analysis.

Abstract

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 parame- terised in terms of a series expansion of basis functions (such as that given by a Karhunen-Loeve expansion). We consider schemes for direct unbiased estimation along such an expansion, as well as those based on MCMC schemes which, due to their dimensionality, cannot be directly imple- mented, 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 and its variants.