Auxiliary likelihood-based approximate Bayesian computation in state space models

A new approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics computed from observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a 'match' between observed and simulated summaries are retained, and used to estimate the inaccessible posterior; exact inference being feasible only if the statistics are sufficient. With no reduction to sufficiency being possible in the state space setting, we pursue summaries via the maximization of an auxiliary likelihood function. We derive conditions under which this auxiliary likelihood-based approach achieves Bayesian consistency and show that, in the limit, results yielded by the auxiliary maximum likelihood estimator are replicated by the auxiliary score. In multivariate parameter settings a separate treatment of each parameter dimension, based on integrated likelihood techniques, is advocated as a way of avoiding the curse of dimensionality associated with ABC methods. Three stochastic volatility models for which exact inference is either challenging or infeasible, are used for illustration