## Bernoulli

- Bernoulli
- Volume 24, Number 3 (2018), 1726-1786.

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

Sergios Agapiou, Gareth O. Roberts, and Sebastian J. Vollmer

#### 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 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.

#### Article information

**Source**

Bernoulli Volume 24, Number 3 (2018), 1726-1786.

**Dates**

Received: December 2014

Revised: July 2016

First available in Project Euclid: 2 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1517540459

**Digital Object Identifier**

doi:10.3150/16-BEJ911

**Keywords**

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

#### Citation

Agapiou, Sergios; Roberts, Gareth O.; Vollmer, Sebastian J. Unbiased Monte Carlo: Posterior estimation for intractable/infinite-dimensional models. Bernoulli 24 (2018), no. 3, 1726--1786. doi:10.3150/16-BEJ911. https://projecteuclid.org/euclid.bj/1517540459

#### Supplemental materials

- Supplement to “Unbiased Monte Carlo: posterior estimation for intractable/infinite-dimensional models”. We provide detailed proofs, further consideration on unbiased estimators for Bayesian linear inverse problems and an elliptic inverse problem as detailed example.Digital Object Identifier: doi:10.3150/16-BEJ911SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.