Updating Subjective Probability

Diaconis, Persi; Zabell, Sandy L.

doi:10.1080/01621459.1982.10477893

Cited by 236 publications

(133 citation statements)

References 29 publications

Supporting

Mentioning

128

Contrasting

Unclassified

Order By: Relevance

“…Moreover, among all probability measures R on A satisfying R E 1;j 1 \ E 2;j 2 À \ Á Á Á \E k;j k Þ ¼ l E 1;j 1 \ E 2;j 2 \ Á Á Á \ E k;j k À Á , and hence preserving the independence of the partitions E (1) ,…,E (k) , Q is nearest to P on several measures of closeness, including the variation distance, the Hellinger distance, and Kullback-Leibler divergence, in the latter two cases, uniquely so (see Diaconis and Zabell 1982).…”

Section: Independence Preservation 283mentioning

confidence: 96%

Peer Disagreement and Independence Preservation

Wagner

2010

Erkenn

View full text Add to dashboard Cite

It has often been recommended that the differing probability distributions of a group of experts should be reconciled in such a way as to preserve each instance of independence common to all of their distributions. When probability pooling is subject to a universal domain condition, along with state-wise aggregation, there are severe limitations on implementing this recommendation. In particular, when the individuals are epistemic peers whose probability assessments are to be accorded equal weight, universal preservation of independence is, with a few exceptions, impossible. Under more reasonable restrictions on pooling, however, there is a natural method of preserving the independence of any fixed finite family of countable partitions, and hence of any fixed finite family of discrete random variables.

show abstract

Section: Independence Preservation 283mentioning

confidence: 96%

Peer Disagreement and Independence Preservation

Wagner

2010

Erkenn

View full text Add to dashboard Cite

show abstract

“…This property, however, is not unique to cross-entropy minimization (Diaconis and Zabell 1982). Justifications that identify cross-entropy minimization as the unique method satisfying certain desirable properties can be brought forward along two distinct lines: the first type of argument consists of formal conditions on the input/output relation defined by a method, and a proof that cross-entropy minimization is the only rule that will satisfy these conditions.…”

Section: Inductive Reasoning By Cross-entropy Minimizationmentioning

confidence: 98%

A Logic For Inductive Probabilistic Reasoning

Jaeger¹

2005

Synthese

View full text Add to dashboard Cite

Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from "70% of As are Bs" and "a is an A" infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey's rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an autonomous system acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation |= that models strict, probabilistically valid, inferences, and a relation |≈ that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone.

show abstract

“…Furthermore, an iterated proportional fitting procedure may be devised to determine the weights [17]; this is a maximum entropy procedure with many variants and extensions [7]. Another advantage is that the computation of the function f remains a simple summation procedure, the same number of weights are used, and the weights can be made to desirable constraints such as falling with certain pre-specified intervals.…”

Section: Extensionsmentioning

confidence: 99%

Aspects of the Binary CMAC: Unimodularity and Probabilistic Reconstruction

Menon¹

2005

Neural Process Lett

View full text Add to dashboard Cite

The CMAC is a neural net for computing real-valued functions. Conceptually, the CMAC maps a point in the function's domain to a set of locations in an associative memory. Each location "contains" a weight, and the sum of the weights is taken to be the function's value at that point. The overall process may be modeled as the multiplication of an input vector by an "association matrix." This paper highlights some aspects of the CMAC's mapping and function computation procedures. Regarding the mapping procedure, it is shown that the associative matrix of a univariate CMAC has the consecutive-ones property; this implies that the matrix is totally-unimodular, a property of great importance in integer optimization. For a multivariate CMAC, the association matrix can be partitioned into sub-matrices, each with the consecutive-ones property. Regarding the function computation procedure, it is shown that a univariate CMAC can compute a function exactly iff a certain distribution is reconstructible from its one dimensional marginals. A CMAC extension, free of this limitation and derived from the theory of balanced matrices, is briefly discussed. These results are generalizable to multivariate CMACs in a relatively straightforward manner.

show abstract

Updating Subjective Probability

Cited by 236 publications

References 29 publications

Peer Disagreement and Independence Preservation

Peer Disagreement and Independence Preservation

A Logic For Inductive Probabilistic Reasoning

Aspects of the Binary CMAC: Unimodularity and Probabilistic Reconstruction

Contact Info

Product

Resources

About