A note on optimal allocation and an ordering of the family of beta prior distributions

Abstract

Because the uniform (0,1) distribution is flat our intuition often leads us to consider it as the least informative in the family of beta distributions as a prior for the parameter P. This thinking derives from what is often called Bayes' Postulate. A simple concept, also based on intuition, is introduced which leads to a simple way of determining which of two beta priors gives "more information" in the context of estimating the difference between two proportions. For the new concept, the amount of prior information on a parameter depends not only on the dispersion of the prior distribution, but also on the expected dispersion of the sample distribution. One consequence of this new concept is that the uniform (0,1) distribution is shown to give more prior information in some cases than some other distributions in the family of beta (α, β) priors. Another is that for a given sample size n , we are able to define a linear ordering on this family of probability distributions. With the linear ordering for a given n, it appears that there is no least informative beta prior.

Keywords:

• beta family
• difference of two proportions
• more in,formative prior
• optimal allocation