Approximate Bayesian Computation in State Space Models

TL;DR

This paper introduces an approximate Bayesian computation (ABC) method for inference in state space models, leveraging auxiliary models and the unscented Kalman filter to achieve consistent parameter estimation without likelihood evaluation.

Contribution

It proposes a novel ABC approach using auxiliary models and the MLE to attain asymptotic sufficiency and Bayesian consistency in state space models.

Findings

Achieves Bayesian consistency with auxiliary model-based ABC.

Demonstrates effectiveness using a stochastic volatility model.

Provides a fast approximation method with the unscented Kalman filter.

Abstract

A new approach to inference in state space models is proposed, based on approximate Bayesian computation (ABC). ABC avoids evaluation of the likelihood function by matching observed summary statistics with statistics computed from data simulated from the true process; exact inference being feasible only if the statistics are sufficient. With finite sample sufficiency unattainable in the state space setting, we seek asymptotic sufficiency via the maximum likelihood estimator (MLE) of the parameters of an auxiliary model. We prove that this auxiliary model-based approach achieves Bayesian consistency, and that - in a precise limiting sense - the proximity to (asymptotic) sufficiency yielded by the MLE is replicated by the score. In multiple parameter settings a separate treatment of scalar parameters, based on integrated likelihood techniques, is advocated as a way of avoiding the curse of dimensionality. Some 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 as a result of intractable transitions. The ABC method is demonstrated using the unscented Kalman filter as a fast and simple way of producing an approximation in this setting, with a stochastic volatility model for financial returns used for illustration.