A connection between Bayesian and frequentist estimates

For finite sample size, the traditional Bayes estimator depends on the choice of the prior. On the other hand, a frequentist estimator, such as MLE does not use any prior for the parameter(s). In general, a Bayes estimator and a frequentist estimator could be quite different when the sample size is not large. In this article, we provide results showing that the MLE can be obtained as a limiting Bayes estimator by keeping updating the prior. The convergence is independent of the choice of the prior. Some examples using conjugate priors are also provided.