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 $\mathit{}\mathbf{\psi }$ and has an associated $\mathit{}\mathbf{\psi }$-auxiliary particle filter that provides unbiased estimates of L. We identify a sequence ${\mathit{}\mathbf{\psi }}^{*}$ that is optimal in the sense that the ${\mathit{}\mathbf{\psi }}^{*}$-auxiliary particle filter’s estimate of L has zero variance. In practical applications, ${\mathit{}\mathbf{\psi }}^{*}$ is unknown so the ${\mathit{}\mathbf{\psi }}^{*}$-auxiliary particle filter cannot straightforwardly be implemented. We use an iterative scheme to approximate ${\mathit{}\mathbf{\psi }}^{*}$ 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.