  We show that it is feasible to carry out exact Bayesian inference for
non-Gaussian state space models using an adaptive Metropolis Hastings sampling
scheme with the likelihood approximated by the particle filter. Furthermore, an
adapyive independent Metropolis Hastings sampler based on a mixture of normals
proposal is computationally much more efficient than an adaptive random walk
proposal because the cost of constructing a good adaptive proposal is
negligible compared to the cost of approximating the likelihood. Independent
Metropolis Hastings proposals are also attractive because they are easy to run
in parallel on multiple processors. We also show that when the particle filter
is used, the marginal likelihood of any model is obtained in an efficient and
unbiased manner, making model comparison straightforward.
