Monte Carlo Filtering of Piecewise Deterministic Processes

Abstract

We present efficient Monte Carlo algorithms for performing Bayesian inference in a broad class of models: those in which the distributions of interest may be represented by time marginals of continuous-time jump processes conditional on a realization of some noisy observation sequence. The sequential nature of the proposed algorithm makes it particularly suitable for online estimation in time series. We demonstrate that two existing schemes can be interpreted as particular cases of the proposed method. Results are provided which illustrate significant performance improvements relative to existing methods. The Appendix to this document can be found online.

Static-parameter estimation in piecewise deterministic processes using particle Gibbs samplers

“…They are stochastic processes that jump randomly at an almost surely countable number of random times but otherwise evolve deterministically in continuous time. Their description here follows Whiteley et al (2011). We also provide motivating examples.…”

“…t / t 2OE0;t n is completely determined by .K n ; 1WK n ; 1WK n ; 0 /. For simplicity, as in Whiteley et al (2011), we assume the following Markovian prior on the number, times and sizes of jumps in the interval OE0; t n for any n 2 N, p  n .k n ; 1Wk n ; 0Wk n / D S  .t n ; k n /q  0 . 0 / 1 .0;t n .…”

“…A second example of a is a shot-noise-Cox-process model described in Whiteley et al (2011). The model assumes that observations are taken on a Cox process with piecewise deterministic shot-noise intensity, D .…”

“…Our approach is to employ a particle Gibbs sampler as introduced by Andrieu et al (2010), based around the lter for from Whiteley et al (2011), to estimate the static parameters. Our methodological contributions are as follows.…”

“…In order to apply more sophisticated particle ltering techniques to these models, an lter for , based on the -sampler framework by Del Moral et al (2006), was introduced in Whiteley et al (2011).…”