  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 paper, 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 SNP data from a case-control study.
