Regularization of Bayesian quasi-likelihoods constructed from complex estimating functions

Bayesian quasi-likelihoods constructed from estimating functions extend the scope of Bayesian inference to a wide range of semi-parametric problems. Nonetheless, when the estimating functions possess complex structure, like containing highly irregular weighting matrices, the quasi-likelihoods constructed from those estimating functions may deform with multiple modes, and thus lose their nice statistical properties. This makes incorporating the information from complex estimating functions, such as composite score functions, into Bayesian inference infeasible. This paper makes two contributions. First, we demonstrate and explain theoretically why the quasi-likelihoods constructed from complex estimating functions deform. Second, we introduce a method of quasi-likelihood regularization which effectively handles the deformation and restores the nice statistical properties of the quasi-likelihoods. The regularization can be easily implemented and asymptotically preserves all information from the original estimating functions, ensuring good estimation accuracy. Numerical studies are conducted to verify the effectiveness of the quasi-likelihood regularization on GARCH(1,1) model and Gaussian random field examples. Direct incorporation of the information from composite likelihoods into the regularized quasi-likelihoods is a highlighted application demonstrated in the numerical studies. We further apply the regularized quasi-likelihood to the spatio-temporal modeling of the ground-level ozone concentration in Hong Kong, and illustrate how our regularized quasi-likelihoods facilitate the simultaneous inference from multiple sources of complex semi-parametric information.