Higher-order properties of Bayesian empirical likelihood

Abstract

Empirical likelihood serves as a good nonparametric alternative to the traditional parametric likelihood. The former involves much less assumptions than the latter, but very often gets the same asymptotic inferential efficiency. While empirical likelihood has been studied quite extensively in the frequentist literature, the corresponding Bayesian literature is somewhat sparse. Bayesian methods hold promise, however, especially with the availability of historical information, which often can be used successfully for the construction of priors. In addition, Bayesian methods very often overcome the curse of dimensionality by providing suitable dimension reduction through judicious use of priors and analyzing data with the resultant posteriors. In this paper, we provide asymptotic expansion of posteriors for a very general class of priors along with the empirical likelihood and its variations, such as the exponentially tilted empirical likelihood and the Cressie–Read version of the empirical likelihood. Other than obtaining the celebrated Bernstein–von Mises theorem as a special case, our approach also aids in finding non-subjective priors based on empirical likelihood and its variations as mentioned above.