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The class of Cressie-Read empirical likelihoods are constructed with weightsderived at a minimum distance from the empirical distribution in theCressie-Read family of divergences indexed by $\gamma$ under the constraint ofan unbiased set of $M$-estimating equations. At first order, they provide validposterior probability statements for any given prior, but the bias in coverageof the resulting empirical quantile is inversely proportional to the asymptoticefficiency of the corresponding $M$-estimator. The Cressie-Read empiricallikelihoods based on the maximum likelihood estimating equations bring aboutquantiles covering with $O(n^{-1})$ accuracy at the underlying posteriordistribution. The choice of $\gamma$ has an impact on the variance in smallsamples of the posterior quantile function. Examples are given for the $M$-typeestimating equations of location and for the quasi-likelihood functions in thegeneralized linear models.

Accuracy and validity of posterior distributions using the Cressie-Read empirical likelihoods