Optimal combination of composite likelihoods using approximate Bayesian computation with application to state-space models

Composite likelihood provides approximate inference when the full likelihood is intractable and sub-likelihood functions of marginal events can be evaluated relatively easily. It has been successfully applied for many complex models. However, its wider application is limited by two issues. First, weight selection of marginal likelihood can have a significant impact on the information efficiency and is currently an open question. Second, calibrated Bayesian inference with composite likelihood requires curvature adjustment which is difficult for dependent data. This work shows that approximate Bayesian computation (ABC) can properly address these two issues by using multiple composite score functions as summary statistics. First, the summary-based posterior distribution gives the optimal Godambe information among a wide class of estimators defined by linear combinations of estimating functions. Second, to make ABC computationally feasible for models where marginal likelihoods have no closed form, a novel approach is proposed to estimate all simulated marginal scores using a Monte Carlo sample with size N. Sufficient conditions are given for the additional noise to be negligible with N fixed as the data size n goes to infinity, and the computational cost is O(n). Third, asymptotic properties of ABC with summary statistics having heterogeneous convergence rates is derived, and an adaptive scheme to choose the component composite scores is proposed. Numerical studies show that the new method significantly outperforms the existing Bayesian composite likelihood methods, and the efficiency of adaptively combined composite scores well approximates the efficiency of particle MCMC using the full likelihood.