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 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 a precise limiting sense – results yielded by the auxiliary maximum likelihood estimator are replicated by the auxiliary score. Particular attention is given to a structure in which the state variable is driven by a continuous time process, with exact inference typically infeasible in this case due to intractable transitions Two models for continuous time stochastic volatility are used for illustration, with auxiliary likelihoods constructed by applying computationally efficient filtering methods to discrete time approximations. The extent to which the conditions for consistency are satisfied is demonstrated in both cases, and the accuracy of the proposed technique when applied to a square root volatility model also demonstrated numerically. In multiple parameter settings a separate treatment of each parameter, based on integrated likelihood techniques, is advocated as a way of avoiding the curse of dimensionality associated with ABC methods.