SMC^2: A sequential Monte Carlo algorithm with particle Markov chain Monte Carlo updates

We consider the generic problem of performing sequential Bayesian inference in a state-space model with observation process $(y_{t})$, state process $(x_{t})$ and fixed parameter $\theta$. An idealized approach would be to apply the \emph{iterated batch importance sampling} (IBIS) algorithm of \citet{Chopin:IBIS}. This is a sequential Monte Carlo algorithm \emph{in the $\theta-$dimension}, that samples values of $\theta,$ reweights iteratively these values using the likelihood increments $p(y_{t}|y_{1:t-1},\theta)$, and rejuvenates the $\theta-$particles through a resampling step and a MCMC update step. In state-space models these likelihood increments are intractable in most cases, but they may be unbiasedly estimated by a particle filter \emph{in the $x-$dimension,}for any fixed $\theta$. This motivates the \SMCSQ algorithm proposed in this article: a sequential Monte Carlo algorithm, defined in the $\theta-$dimension, which propagates and resamples many particle filters in the $x-$dimension. The filters in the $x$-dimension are an example of the random weight particle filter as in \citet{high}. On the other hand, the particle Markov chain Monte Carlo (PMCMC) framework developed in \citet{PMCMC} allows us to design appropriate MCMC rejuvenation steps. Thus, the $\theta$-particles target the correct posterior distribution at each iteration $t$, despite the intractability of the likelihood increments. We explore the applicability of our algorithm in both sequential and non-sequential applications and consider various degrees of freedom, as for example increasing dynamically the number of $x$-particles. We contrast our approach to various competing methods, both conceptually and empirically through a detailed simulation study.