Quasi-stationary Monte Carlo and the ScaLE algorithm

Abstract

This paper introduces a class of Monte Carlo algorithms which are based on 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 while retaining exactness: the paper gives theoretical guarantees ensuring that the algorithm has the correct limiting target distribution. Furthermore, this methodology is highly amenable to ‘big data’ problems. By employing a modification to existing naive subsampling and control variate techniques it is possible to obtain an algorithm which is still exact but has sublinear iterative cost as a function of data size.