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Bayesian Indifference Procedures

Novick, Melvin R.; Hall, W. J.
Publication Year:
Report Number:
ETS Research Bulletin
Document Type:
Page Count:
Subject/Key Words:
Bayesian Statistics, Indifference, Statistical Inference


In a logical probability approach to inference, distributions on a parameter space are interpreted as representing states of knowledge, and any prevailing state of knowledge is taken to have been arrived at from a previous state of ignorance (indifference) followed by the accumulation of prior data. Previous attempts at describing indifference and states of knowledge by use of the principle of insufficient reason and by use of natural conjugate distributions are discussed, and it is shown that they are not sufficient in themselves. Specifically, if one permits order-preserving reparametrization, any prior distribution whatsoever for one parameter implies that another parameter is uniformly distributed and yet another has a natural conjugate distribution relative to an arbitrary sample. An indifference principle that requires postulating what size and what kind of samples will permit statistical inference is then introduced--e.g., one observation from a two-parameter normal model is not sufficient to permit inference about the variance but two (noncoinciding) observations are. With some limitation on the class of priors considered, it is then shown that this principle permits unique specification of indifference for the more commonly encountered statistical models. It is also shown that this specification does not depend on the scale of measurement of the random variable. The consistency of this principle with recent work of the authors on prediction theory is also demonstrated.

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