  We consider the generic problem of performing sequential Bayesian inference
in a state-space model with observation process y, state process x and fixed
parameter theta. An idealized approach would be to apply the iterated batch
importance sampling (IBIS) algorithm of Chopin (2002). This is a sequential
Monte Carlo algorithm 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 in the x-dimension, for any fixed theta. This motivates the
SMC^2 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 Fearnhead et al. (2010). On the other
hand, the particle Markov chain Monte Carlo (PMCMC) framework developed in
Andrieu et al. (2010) 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, included here and in a supplement, and
based on particularly challenging examples.
