Exact and Approximate Bayesian Inference for Low Integer-Valued Time Series Models with Intractable Likelihoods

In this paper we develop a likelihood-free simulation methodology in order to obtain Bayesian inference for models for low integer-valued time series data that have computationally demanding likelihood functions. The algorithm fits within the framework of particle Markov chain Monte Carlo (PMCMC) methods and uses a so-called alive particle filter. The particle filter requires only model simulations and, in this regard, our approach has connections with approximate Bayesian computation (ABC). However, an advantage of using the PMCMC approach in this setting is that simulated data can be matched with data observed one-at-a-time, rather than attempting to match on the full dataset simultaneously or on a low-dimensional non-sufficient summary statistic, which is common practice in ABC. For low integer-valued time series data, we find that it is often computationally feasible to match simulated data with observed data exactly. The alive particle filter uses negative binomial sampling in order to maintain a fixed number of particles. The algorithm creates an unbiased estimate of the likelihood, resulting in exact posterior inferences when included in an MCMC algorithm. In cases where exact matching is computationally prohibitive, a tolerance is introduced as in ABC. This paper further develops the alive particle filter by introducing auxiliary variables so that partially observed and/or non-Markovian models can be accommodated. We demonstrate that Bayesian model choice problems involving such models can be handled with this approach. The methodology is illustrated on a wide variety of models for simulated and real low-count time series data involving a rich set of applications.