  Our article deals with Bayesian inference for a general state space model
with the simulated likelihood computed by the particle filter. We show
empirically that the partially or fully adapted particle filters can be much
more efficient than the standard particle, especially when the signal to noise
ratio is high. This is especially important because using the particle filter
within MCMC sampling is O(T^2), where T is the sample size. We also show that
an adaptive independent proposal for the unknown parameters based on a mixture
of normals can be much more efficient than the usual optimal random walk
methods because the simulated likelihood is not continuous in the parameters
and the cost of constructing a good adaptive proposal is negligible compared to
the cost of evaluating the simulated likelihood. Independent \MH proposals are
also attractive because they are easy to run in parallel on multiple
processors. The article also shows that the proposed \aimh sampler converges to
the posterior distribution. We also show that the marginal likelihood of any
state space model can be obtained in an efficient and unbiased manner by using
the \pf making model comparison straightforward. Obtaining the marginal
likelihood is often difficult using other methods. Finally, we prove that the
simulated likelihood obtained by the auxiliary particle filter is unbiased.
This result is fundamental to using the particle for MCMC sampling and is first
obtained in a more abstract and difficult setting by Del Moral (2004). However,
our proof is direct and will make the result accessible to readers.
