Robust Particle Density Tempering for State Space Models

Density tempering (also called density annealing) for state space models is a sequential Monte Carlo (SMC) approach to Bayesian inference for general state models, that provides an alternative to MCMC. It moves a collection of parameters and latent states (which we call particles) through a number of stages, with each stage having its own target density. Initially, the particles are generated from a distribution that is easy to sample from, e.g. the prior; the target density at the final stage is the posterior density of interest. Tempering is usually carried out either in batch mode, involving all of the data at each stage, or in sequential mode, where the tempering involves adding observations at each stage; we call this data tempering. Our article proposes two innovations for particle based density tempering. First, data tempering is made more robust to outliers and structural changes by adding batch tempering at each stage. Second, we propose generating the parameters and states at each stage using two Gibbs type Markov moves, where the parameters are generated conditional on the states and conversely. We explain how this allows the tempering to scale up in terms of the number parameters and states it can handle. Most of the current literature uses a pseudo-marginal Markov move step with the states integrated out and the parameters generated by a random walk proposal; this strategy is inefficient when the states or parameters are high dimensional. The article demonstrates the performance of the proposed methods using univariate stochastic volatility models with outliers and structural breaks and high dimensional factor stochastic volatility models having both a large number of parameters and a large number of latent state.