Flexible Particle Markov chain Monte Carlo methods with an application to a factor stochastic volatility model

Particle Markov Chain Monte Carlo methods are used to carry out inference in non-linear and non-Gaussian state space models, where the posterior density of the states is approximated using particles. Current approaches usually perform Bayesian inference using a particle Marginal Metropolis-Hastings algorithm, a particle Gibbs sampler, or a particle Metropolis within Gibbs sampler. This paper shows how the three ways of generating variables mentioned above can be combined in a flexible manner to give sampling schemes that converge to a desired target distribution. The advantage of our approach is that the sampling scheme can be tailored to obtain good results for different applications, for example when some parameters and the states are highly correlated. We investigate the properties of this flexible sampling scheme, including conditions for uniform convergence to the posterior. We illustrate our methods with a factor stochastic volatility state space model where one group of parameters can be generated in a straightforward manner in a particle Gibbs step by conditioning on the states, and a second group of parameters are generated without conditioning on the states because of the high dependence between such parameters and the states.