Bayesian filtering with intractable likelihood using sequential MCMC

We develop a sequential estimation methodology for a class of nonlinear, non-Gaussian state space models in which the observation process is intractable to express in closed form, but trivial to simulate. In addition we consider models in which the latent state vector and the observation vector are very high dimensional. To overcome these two difficulties we propose the class of Sequential Markov chain Monte Carlo (SMCMC) algorithms in which we incorporate a component of Approximate Bayesian Computation (ABC). In doing so we tackle both the curse of dimensionality via the SMCMC and the intractability of the likelihood via the ABC component. We demonstrate how the proposed algorithm outperforms alternative approaches in two challenging state space model examples.