Asymptotic Bayesian analysis based on a limited information estimator

We study asymptotic Bayesian analysis based on a limited information estimator, with an unknown partial likelihood function. It is found that the asymptotic distribution of the estimator approximates the posterior distribution, provided that the estimator’s distribution converges uniformly in local neigborhoods around the true parameter value. This provides a Bayesian interpretation to classical limited information procedures, making them available for a semi-parametric Bayesian analysis. Uniform convergence in distribution is essential for such result, as illustrated by examples in which the estimators are pointwise asymptotically normal at every parameter value, but the posterior distributions display discontinuous and possibly non-normal behaviors.