Local Scoring Rules: A Versatile Tool for Inference

In many applications of highly structured statistical models the likelihood function is intractable; in particular, finding the normalisation constant of the distribution can be demanding. One way to sidestep this problem is to to adopt composite likelihood methods, such as the pseudo-likelihood approach. In this paper we display composite likelihood as a special case of a general estimation technique based on proper scoring rules, which supply an unbiased estimating equation for any statistical model. The important class of key local scoring rules avoids the need to compute normalising constants. Another application arises in Bayesian model selection. The log Bayes factor measures by how much the predictive log score for one model improves on that for another. However, Bayes factors are not well-defined when improper prior distributions are used. If we replace the log score by a suitable local proper scoring rule, these problems are avoided.