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When dealing with Bayesian inference the choice of the prior often remains adebatable question. Empirical Bayes methods offer a data-driven solution tothis problem by estimating the prior itself from an ensemble of data. In thenonparametric case, the maximum likelihood estimate is known to overfit thedata, an issue that is commonly tackled by regularization. However, themajority of regularizations are ad hoc choices which lack invariance underreparametrization of the model and result in inconsistent estimates forequivalent models. We introduce a non-parametric, transformation invariantestimator for the prior distribution. Being defined in terms of the missinginformation similar to the reference prior, it can be seen as an extension ofthe latter to the data-driven setting. This implies a natural interpretation asa trade-off between choosing the least informative prior and incorporating theinformation provided by the data, a symbiosis between the objective andempirical Bayes methodologies.

Objective Priors in the Empirical Bayes Framework