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An Information Theoretic Approach to Incorporating Prior Information in Binomial Sampling
Dyer, Danny D.
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The incorporation of prior information about [see pdf for notation], where [see pdf for notation] is the success probability in a binomial sampling model, is an essential feature of Bayesian statistics. Methodology based on information theoretic concepts is introduced which (a) quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and (b) allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information about [see pdf for notation] which is embedded in any prior distribution. In effect,, the most conservative prior distribution from a specified class (each member of which carries the available prior information about [see pdf for notation]) is that prior distribution within the class over which the likelihood function has the greatest average domination. The most conservative prior distributions from nine different families of prior distributions over the interval (0,1) including the beta distribution are determined and compared for three situations: (1) no prior specification for 9 is available, (2) a prior point estimate of [see pdf for notation] is available, and (3) a prior interval estimate of [see pdf for notation] is available. The results of the comparisons not only advocate the use of the beta prior distribution in binomial sampling but also indicate which particular one to use in the three aforementioned situations.