Quasi-stationary Monte Carlo and the ScaLE Algorithm

TL;DR

This paper presents a novel Monte Carlo algorithm based on quasi-stationary distributions, combining sequential Monte Carlo and diffusion simulation, offering an exact, efficient alternative to traditional MCMC methods especially suited for large datasets.

Contribution

It introduces a new class of Monte Carlo algorithms that avoid Metropolis-Hastings steps, ensuring exactness and scalability for big data applications.

Findings

Algorithm has theoretical guarantees of correct limiting distribution.

Applicable to large-scale data with sub-linear cost.

Circumvents accept/reject steps of traditional MCMC.

Abstract

This paper introduces a class of Monte Carlo algorithms which are based upon the simulation of a Markov process whose quasi-stationary distribution coincides with a distribution of interest. This differs fundamentally from, say, current Markov chain Monte Carlo methods which simulate a Markov chain whose stationary distribution is the target. We show how to approximate distributions of interest by carefully combining sequential Monte Carlo methods with methodology for the exact simulation of diffusions. The methodology introduced here is particularly promising in that it is applicable to the same class of problems as gradient based Markov chain Monte Carlo algorithms but entirely circumvents the need to conduct Metropolis-Hastings type accept/reject steps whilst retaining exactness: the paper gives theoretical guarantees ensuring the algorithm has the correct limiting target distribution. Furthermore, this methodology is highly amenable to big data problems. By employing a modification to existing na{\"\i}ve sub-sampling and control variate techniques it is possible to obtain an algorithm which is still exact but has sub-linear iterative cost as a function of data size.