  A composite likelihood is a combination of low-dimensional likelihood objects
useful in applications where the data have complex structure. Although
composite likelihood construction is a crucial aspect influencing both
computing and statistical properties of the resulting estimator, currently
there does not seem to exist a universal rule to combine low-dimensional
likelihood objects that is statistically justified and fast in execution. This
paper develops a methodology to select and combine the most informative
low-dimensional likelihoods from a large set of candidates while carrying out
parameter estimation. The new procedure minimizes the distance between
composite likelihood and full likelihood scores subject to a constraint
representing the afforded computing cost. The selected composite likelihood is
sparse in the sense that it contains a relatively small number of informative
sub-likelihoods while the noisy terms are dropped. The resulting estimator is
found to have asymptotic variance close to that of the minimum-variance
estimator constructed using all the low-dimensional likelihoods.
