A Multi-Index Markov Chain Monte Carlo Method

Abstract

In this article we consider computing expectations w.r.t. probability laws associated to a certain class of stochastic systems. In order to achieve such a task, one must not only resort to numerical approximation of the expectation, but also to a biased discretization of the associated probability. We are concerned with the situation for which the discretization is required in multiple dimensions, for instance in space-time.In such contexts, it is known that the multi-index Monte Carlo (MIMC) method of [7] can improve upon i.i.d. sampling from the most accurate approximation of the probability law. Through a non-trivial modification of the multilevel Monte Carlo (MLMC) method, this method can reduce the work to obtain a given level of error, relative to i.i.d. sampling and relative even to MLMC. In this article we consider the case when such probability laws are too complex to be sampled independently, for example a Bayesian inverse problem where evaluation of the likelihood requires solution of a partial differential equation (PDE) model which needs to be approximated at finite resolution. We develop a modification of the MIMC method which allows one to use standard Markov chain Monte Carlo (MCMC) algorithms to replace independent and coupled sampling, in certain contexts. We prove a variance theorem for a simplified estimator which shows that using our MIMCMC method is preferable, in the sense above, to i.i.d. sampling from the most accurate approximation, under appropriate assumptions. The method is numerically illustrated on a Bayesian inverse problem associated to a stochastic partial differential equation (SPDE), where the path measure is conditioned on some observations.

Unbiased multi-index Monte Carlo

“…The analysis in such a scenario is of interest as is its application, to enhance the range of examples where our approach can be implemented. This is being conducted in [10].…”

Advanced Multilevel Monte Carlo Methods

“…We remark that the method that will be described below has use outside the case of Section 2.2.2 (see e.g. Jasra et al , 2018). The main utility of the method to be described is in the case that sampling from (the distributions associated to) coupled pairs of discretizations in a meaningful way (e.g.…”

“…The motivation for the idea will become clear as it is described in more detail. The approach is considered in Jasra et al (2018) (see also Jasra et al , 2018).…”

“…One strand consists of considering multidimension discretizations, such as in Haji‐Ali et al (2016). There are a small number of papers on this topic, such as Crisan et al (2018) and Jasra et al (2018), but there seem to be many possible avenues for future work. Some of the theoretical results of Jasra et al .…”

“…Finally, it is commented that it would be of interest to perform a systematic comparison on some benchmark problems such as the Bayesian inverse problem in Section 2.1 across the various existing MLMC methods for this type of problem, to asses their benefits and drawbacks in practice. For example, such a comparison would include at least the three distinct multilevel MCMC (the third is a generalisation of the method Jasra et al , 2018, introduced in 2018 but not presented in this work), as well as MLSMC.…”