Of late, the Bayesian approach is being widely used in many applications especially when the priors are either neutral or uninformative. Many Markov Chain Monte Carlo (MCMC) algorithms are developed to solve the estimation problems related to this. In the absence of relevant prior, an uninformative prior is assigned and the parameter of interest is estimated. The interest then is to know the accuracy of the estimate. This paper proposes frequentist accuracy for the Bayesian estimate of the parameter. The estimate is considered as a function of the data and its frequentist variability is computed. The main result of this study is a general accuracy formula for the delta method standard deviation of the estimated parameter that can be applied to all prior distributions. Even in complicated situations the formula is computationally inexpensive. The MCMC calculations that give the Bayes estimate also provide its frequentist standard deviation. Many of the examples will demonstrate near equality between Bayesian and frequentist standard deviations. However a class of reasonable examples where the frequentist accuracy can be less than half of its Bayesian counterpart are also provided. The general accuracy formula takes on a particularly simple form from the exponential family. Exponential family structure also allows substituting parametric bootstrap sampling for MCMC calculations, at least for uninformative priors. This has computational advantages that it helps to connect Bayesian inference with the bootstrap approach. (27 refs.)
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