Parsimonious and Efficient Likelihood Composition by Gibbs Sampling

TL;DR

This paper introduces a Gibbs sampling-based method for selecting and constructing efficient composite likelihood functions, improving inference in high-dimensional complex data by balancing informativeness and sparsity.

Contribution

It proposes a novel Gibbs sampling scheme for optimal sub-likelihood component selection, ensuring convergence to the most informative composite likelihood.

Findings

The method converges to the maximally informative likelihood in equilibrium.

The penalized version produces sparse likelihoods suitable for complex data.

Numerical examples demonstrate effectiveness on simulated and real genotype data.

Abstract

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.