A Bayesian construction of asymptotically unbiased estimators

A differential geometric framework to construct an asymptotically unbiased estimator of a function of a parameter is presented. The derived estimator asymptotically coincides with the uniformly minimum variance unbiased estimator, if a complete sufficient statistic exists. The framework is based on the maximum a posteriori estimation, where the prior is chosen such that the estimator is unbiased. The framework is demonstrated for the second-order asymptotic unbiasedness (unbiased up to $O(n^{-1})$ for a sample of size $n$). The condition of the asymptotic unbiasedness leads the choice of the prior such that the departure from a kind of harmonicity of the estimand is canceled out at each point of the model manifold. For a given estimand, the prior is given as an integral. On the other hand, for a given prior, we can address the bias of what estimator can be reduced by solving an elliptic partial differential equation. A family of invariant priors, which generalizes the Jeffreys prior, is mentioned as a specific example. Some illustrative examples of applications of the proposed framework are provided.