Flexible Density Tempering Approaches for State Space Models with an Application to Factor Stochastic Volatility Models

Duan (2015) propose a tempering or annealing approach to Bayesian inference for time series state space models. In such models the likelihood is often analytically and computationally intractable. Their approach generalizes the annealed importance sampling (AIS) approach of Neal (2001) and DelMoral (2006) when the likelihood can be computed analytically. Annealing is a sequential Monte Carlo approach that moves a collection of parameters and latent state variables through a number of levels, with each level having its own target density, in such a way that it is easy to generate both the parameters and latent state variables at the initial level while the target density at the final level is the posterior density of interest. A critical component of the annealing or density tempering method is the Markov move component that is implemented at every stage of the annealing process. The Markov move component effectively runs a small number of Markov chain Monte Carlo iterations for each combination of parameters and latent variables so that they are better approximations to that level of the tempered target density. Duan (2015) used a pseudo marginal Metropolis-Hastings approach with the likelihood estimated unbiasedly in the Markov move component. One of the drawbacks of this approach, however, is that it is difficult to obtain good proposals when the parameter space is high dimensional, such as for a high dimensional factor stochastic volatility models. We propose using instead more flexible Markov move steps that are based on particle Gibbs and Hamiltonian Monte Carlo and demonstrate the proposed methods using a high dimensional stochastic volatility factor model. An estimate of the marginal likelihood is obtained as a byproduct of the estimation procedure.