Optimal Combination of Composite Likelihoods Using Approximate Bayesian Computation with Application to State-Space Models

Composite likelihood (CL) combines tractable low-dimensional marginal or conditional likelihoods, but its use is hindered by arbitrary sub-likelihood weights and lack of calibration of CL-based posterior uncertainty. For identifiable, low to moderate-dimensional parametric models, this work addresses both issues using approximate Bayesian computation (ABC) with multiple composite score summaries. First, identically distributed sub-scores are batched to control summary dimension, and equal weighting within a batch is supported by an asymptotic optimality result. The summary-based posterior distribution gives optimal Godambe information among a wide class of estimators defined by estimating functions. Second, when marginal likelihoods have no closed form, ABC is computationally feasible using a novel approach to estimate marginal scores of all pseudo datasets using a Monte Carlo sample with size N. Theoretical results show that the additional noise can be negligible with fixed N, therefore the computational cost is linear in the data size. Third, an adaptive scheme is proposed to choose the component composite scores, supported by results on asymptotic behaviour of ABC with summary statistics having heterogeneous convergence rates. Numerical studies show that the new method substantially improves on existing Bayesian composite-likelihood methods and can approach full-likelihood efficiency when the selected components capture the dominant dependence structure.