A Bayesian Variation of Basu’s Theorem and its Ramification in Statistical Inference

One of the celebrated results of Professor D. Basu is his 1955 paper on ancillary statistics, which established the well known Basu’s Theorem. A Bayesian version of this result, where the parameter Θ is treated as a random variable, is developed in this note, along with other extensions of the related classical results, such as Rao-Blackwell and Lehmann-Scheffé theorems and the relation between complete sufficiency and minimal sufficiency. These extensions shed new light on these fundamental theorems for frequentist statistical inference in the context Bayesian inference.