Parsimonious and Efficient Likelihood Composition by Gibbs Sampling

The traditional maximum likelihood estimator (MLE) is often of limited use in complex high-dimensional data due to the intractability of the underlying likelihood function. Maximum composite likelihood estimation (McLE) avoids full likelihood specification by combining a number of partial likelihood objects depending on small data subsets, thus enabling inference for complex data. A fundamental difficulty in making the McLE approach practicable is the selection from numerous candidate likelihood objects for constructing the composite likelihood function. In this article, we propose a flexible Gibbs sampling scheme for optimal selection of sub-likelihood components. The sampled composite likelihood functions are shown to converge to the one maximally informative on the unknown parameters in equilibrium, since sub-likelihood objects are chosen with probability depending on the variance of the corresponding McLE. A penalized version of our method generates sparse likelihoods with a relatively small number of components when the data complexity is intense. Our algorithms are illustrated through numerical examples on simulated data as well as real genotype single nucleotide polymorphism (SNP) data from a case–control study.