Sequential Monte Carlo and twisted state space models; Twisting models to reduce variance

Modeling time series and systems that exhibit an evolution in time is of interest in several fields, and one class of models that can be used for modeling such systems is state space models. One of the most common tools for inference in non-linear and non-Gaussian state space models is sequential Monte Carlo, also known as particle filters, which uses importance sampling and the sequential structure of the model. When considering state space models, it is also possible to consider twisted state space models, which are defined by a sequence of functions transforming the transitions and emission of the state space model. Several quantities of interest are identical for the original model and the twisted model, thus we can use particle filters to estimate these quantities in the twisted model, and obtain an estimate of the quantities in the original model. The reason the twisted models are of interest is that we can obtain better estimates of these quantities with the twisted models than with the original model, and when considering likelihood estimation there is an optimal sequence that can be used to obtain zero variance estimates of the marginal likelihood. However, this sequence is not obtainable in practice, thus we result in using an approximation. We propose a simple method for creating an approximation of this optimal sequence and see how this method works when considering simulated data. We also demonstrate how the twisted models can be used for both parameter estimation and smoothing and is a viable alternative to using for instance the traditional bootstrap filter on the original model