Twisting targets in Sequential Monte Carlo

Abstract

Sequential Monte Carlo methods are often used for inference in state space models that are nonlinear and non-Gaussian. Inference about the latent variables of a state space model can be performed in an online setting by using Sequential Monte Carlo methods. It is also possible to obtain estimates of the marginal likelihood of the observations from a state space model in an online setting. There are different ways of introducing flexibility in Sequential Monte Carlo methods. One way is by considering the target distributions where one may alter all the intermediate target distributions except the final target distribution. Altering or twisting the intermediate target distributions can be done in different ways. One possibility is to alter the transition density and the observation density of the state space model by utilising a set of functions. These functions have few requirements and they may utilise observations from the state space model. There exists a specific set of functions which utilise all the observations, which implies an offline setting. This specific set of functions can also be used to alter the transition density and the observation density of the state space model. If Sequential Monte Carlo methods now are used to estimate the likelihood it can be shown that the variance of the likelihood estimates is minimised. The functions in the specific set are in general intractable and additionally defined in an offline setting. We therefore need to approximate these functions in an offline setting, subsequently we can use these to alter the transition density and the observation density. The motivation being that we now may obtain likelihood estimates with lower variance. Having the final target distribution unaltered implies that the marginal likelihood obtained by using intermediate twisting target distributions is equal to the marginal likelihood obtained by using unaltered target distributions. We will consider a modified setup for utilising the twisting target framework in a batch setting, that is we assume that observations become available in batches. We also consider numerical experiments to check the effect of the batch setting on the variance of the likelihood estimates.