The iterated auxiliary particle filter and applications to state space models and diffusion processes.

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Abstract

The novel research work presented in this thesis consists of an offline, iterated particle filter to facilitate statistical inference in general state space hidden Markov models. Given a model and a sequence of observations, the associated marginal likelihood L is central to likelihood-based inference for unknown statistical parameters. We define a class of “twisted” models: each member is specified by a sequence of positive functions ψ and has an associated ψ - auxiliary particle filter that provides unbiased estimates of L. We identify a sequence ψ* that is optimal in the sense that the ψ* -auxiliary particle filter’s estimate of L has zero variance. In practical applications, ψ* is unknown so the ψ* - auxiliary particle filter cannot straightforwardly be implemented. We use an iterative scheme to approximate ψ*, and demonstrate empirically that the resulting iterated auxiliary particle filter significantly outperforms the most popular competitors in some challenging settings. Applications include parameter estimation using a particle Markov chain Monte Carlo algorithm. An adaptation of the iAPF for statistical inference in the context of diffusion processes along with a number of examples and applications in this setting is provided

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