The Iterated Auxiliary Particle Filter

We present 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${\bm \psi }$ and has an associated ψ${\bm \psi }$-auxiliary particle filter that provides unbiased estimates of L. We identify a sequence ψ*${\bm \psi }^{*}$ that is optimal in the sense that the ψ*${\bm \psi }^{*}$-auxiliary particle filter’s estimate of L has zero variance. In practical applications, ψ*${\bm \psi }^{*}$ is unknown so the ψ*${\bm \psi }^{*}$-auxiliary particle filter cannot straightforwardly be implemented. We use an iterative scheme to approximate ψ*${\bm \psi }^{*}$ and demonstrate empirically that the resulting iterated auxiliary particle filter significantly outperforms the bootstrap particle filter in challenging settings. Applications include parameter estimation using a particle Markov chain Monte Carlo algorithm.