Approximate Bayesian Computation with composite score functions

Both approximate Bayesian computation (ABC) and composite likelihood methods are useful for Bayesian and frequentist inference, respectively, when the likelihood function is intractable. We propose to use composite likeli- hood score functions as summary statistics in ABC in order to obtain accurate approximations to the posterior distribu- tion. This is motivated by the use of the score function of the full likelihood, and extended to general unbiased estimat- ing functions in complex models. Moreover, we show that if the composite score is suitably standardised, the resulting ABC procedure is invariant to reparameterisations and auto- matically adjusts the curvature of the composite likelihood, and of the corresponding posterior distribution. The method is illustrated through examples with simulated data, and an application to modelling of spatial extreme rainfall data is discussed.