Contemporary Mathematics

Volume 94, 1989

Invariance of Full Conditional

Probabilities under Group Actions

THOMAS E. ARMSTRONG

ABsTRACT. In previous articles the author and William Sudderth introduced

locally coherent rates of exchange as assessments of relative likelihood for

events capable of encompassing improper priors and in a sense equivalent

with Renyi's theory of conditional probability. One manner in which im-

proper priors frequently arise is under group invariance of the statistical

problem.

In this article assumptions of translation invariance for a locally coherent

exchange rate are shown to be closely related to nonexistence ofTarski para-

doxical decompositions or supra-amenability. In the next section questions

of countable additivity or properness for invariant locally coherent exchange

rates are examined. It is shown in particular that a translation invariant full

conditional probability exists on Rn agreeing with conditional probabilities

induced by the usual Hausdorff measures. The last section deals with ex-

changeable full conditional probabilities which are shown to always exist.

1. Paradoxical Coverings and Supra-amenability. Of primary importance

to statisticians particularly Bayesian ones, is the notion of a prior distribution

which is a probability on the algebra of events involved in a particular statisti-

cal experiment before statistical procedures based on sampling are performed.

A prior is used to sum up pre-existing knowledge relevant to the particular

problem. If one uses not a probability but rather an infinite positive measure

one has an improper prior. Improper priors have become a common tool for

certain Bayesian statisticians.

The primary example of an improper prior is Lebesgue measure on Rn.

The events of primary concern for statistical problems with such a prior are

those bounded in Rn or at least bounded in measure. In such a situation the

statistical problem using such a prior is invariant under translations or a larger

group of symmetries. This, using Laplacian dicta, means that an event is to

1980 Mathematics Subject Classification (1985 Revision). Primary 60A05, 60A10, 60815,

62A05, 28H12, 28A35, 28A60, 28C10.

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1989 American Mathematical Society

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http://dx.doi.org/10.1090/conm/094/1012973