Bayesian exponentially tilted empirical likelihood

While empirical likelihood has been shown to exhibit many of the properties of conventional parametric likelihoods, a formal probabilistic interpretation has so far been lacking. We show that a likelihood function very closely related to empirical likelihood naturally arises from a nonparametric Bayesian procedure which places a type of noninformative prior on the space of distributions. This prior gives preference to distributions having a small support and, among those sharing the same support, it favours entropy-maximising distributions. The resulting nonparametric Bayesian procedure admits a computationally convenient representation as an empirical-likelihood-type likelihood where the probability weights are obtained via exponential tilting. The proposed methodology provides an attractive alternative to the Bayesian bootstrap as a nonparametric limit of a Bayesian procedure for moment condition models.