Ranges of Prior Measures.
Abstract
Partial prior knowledge is quantified by a range R(L,U) of sigma-finite prior measures Q satisfying L(A) < or = Q(A) < or = U(A) for all measurable sets A, and is interpreted as a probability statement in a betting framework. The concept of conditional probability distributions is generalized to that of conditional measures, and Bayes theorem is extended to accommodate unbounded priors. According to Bayes theorem, the range R(L,U) of prior measures is transformed upon observing X into a similar range R(LX,UX) of posterior measures. Upper and lower expectations and variances induced by such ranges of measures are obtained. Under weak regularity conditions, these upper and lower posterior expectations are strongly consistent estimators. The range of posterior expectations of an arbitrary function b on the parameter space is asymptotically b sub N + or - alpha sigma sub N + o(sigma sub N) where b sub N and sigma-squared sub N are the posterior mean and variance of b induced by the upper prior measure U, and where alpha is a constant reflecting the uncertainty about the prior in terms of the derivative of L with respect to U. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1979
- Accession Number
- ADA068937
Entities
People
- J. A. Hartigan
- Lorraine Derobertis
Organizations
- University of Wisconsin–Madison