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
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.
1989 American Mathematical Society
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