  This article addresses the problem of efficient Bayesian inference in dynamic
systems using particle methods and makes a number of contributions. First, we
develop a correlated pseudo-marginal (CPM) approach for Bayesian inference in
state space (SS) models that is based on filtering the disturbances, rather
than the states. This approach is useful when the state transition density is
intractable or inefficient to compute, and also when the dimension of the
disturbance is lower than the dimension of the state. Second, we propose a
block pseudo-marginal (BPM) method that uses as the estimate of the likelihood
the average of G independent unbiased estimates of the likelihood. We associate
a set of underlying uniform of standard normal random numbers used to construct
each of the individual unbiased likelihood estimates and then use
component-wise Markov Chain Monte Carlo to update the parameter vector jointly
with one set of these random numbers at a time. This induces a correlation of
approximately 1-1/G between the logs of the estimated likelihood at the
proposed and current values of the model parameters. Third, we show for some
non-stationary state space models that the BPM approach is much more efficient
than the CPM approach, because it is difficult to translate the high
correlation in the underlying random numbers to high correlation between the
logs of the likelihood estimates. Although our focus has been on applying the
BPM method to state space models, our results and approach can be used in a
wide range of applications of the PM method, such as panel data models,
subsampling problems and approximate Bayesian computation.
