Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models

TL;DR

This paper introduces a simple ABC-based inference method for state space models that uses auxiliary likelihoods to improve accuracy and computational efficiency, especially in complex multi-parameter scenarios.

Contribution

It proposes a novel auxiliary likelihood-based ABC approach with score-based summaries and parameter-wise treatment, enhancing Bayesian inference in challenging state space models.

Findings

Method achieves Bayesian consistency under certain conditions.

Approach performs well on stochastic volatility models.

Outperforms existing approximate and exact inference methods.

Abstract

A computationally simple 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 for the 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. With no reduction to a low-dimensional set of sufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach - including a computationally efficient version based on the auxiliary score - achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multi-parameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally challenging, or infeasible, are used for illustration. We demonstrate that our approach compares favorably against an extensive set of approximate and exact comparators. An empirical illustration completes the paper.