Bayesian and non-Bayesian evidential updating.

*(English)*Zbl 0622.68069Orthodox probability theory supposes that (1) we commence with known statistical distributions, (2) these distributions give rise to real- valued probabilities, and (3) these probabilities can be updated by the use of Bayes’ theorem. Each of these suppositions has been challenged by recent work in computer science concerning the representation and updating of partial belief. One approach that has found acceptance is the theory of belief functions of Shafer, and the corresponding procedure of updating. This article addresses the relation between Shafer/Dempster theory and a very slight extension of classical Bayesian theory.

There are four main results (not all of which are new): (1) Closed convex sets of classical probability functions provide a representation of belief that includes the theory of belief functions as a proper special case. (2) The impact of ”uncertain evidence” can be formally represented by Dempster conditioning in Shafer’s framework. (3) The impact of ”uncertain evidence” (Jeffrey conditioning) can be formally represented in the convex set framework by classical conditioning. (4) The main theorem: Suppose a distribution of beliefs is given by both \(Bel_ 1\) and the prior set of probability distributions \(S_ P\). Suppose that new evidence is obtained whose impact is given by the simple support function \(Bel_ A\), or by a corresponding shift in the probability of A on each of the distributions in \(S_ P\) (it will not be the same shift for each distribution). Let \(S_{PA}\) be the set of probability functions obtained by updating each member of \(S_ P\) according to Jeffrey’s rule. Then \[ \min_{P\in S_{PA}}P(X)\leq [Bel_ 1\oplus Bel_ A](X)\leq 1- [Bel_ 1\oplus Bel_ A](X)\leq \max_{P\in S_{PA}}P(X). \] Equality holds only in certain very special cases.

There are four main results (not all of which are new): (1) Closed convex sets of classical probability functions provide a representation of belief that includes the theory of belief functions as a proper special case. (2) The impact of ”uncertain evidence” can be formally represented by Dempster conditioning in Shafer’s framework. (3) The impact of ”uncertain evidence” (Jeffrey conditioning) can be formally represented in the convex set framework by classical conditioning. (4) The main theorem: Suppose a distribution of beliefs is given by both \(Bel_ 1\) and the prior set of probability distributions \(S_ P\). Suppose that new evidence is obtained whose impact is given by the simple support function \(Bel_ A\), or by a corresponding shift in the probability of A on each of the distributions in \(S_ P\) (it will not be the same shift for each distribution). Let \(S_{PA}\) be the set of probability functions obtained by updating each member of \(S_ P\) according to Jeffrey’s rule. Then \[ \min_{P\in S_{PA}}P(X)\leq [Bel_ 1\oplus Bel_ A](X)\leq 1- [Bel_ 1\oplus Bel_ A](X)\leq \max_{P\in S_{PA}}P(X). \] Equality holds only in certain very special cases.

##### MSC:

68T05 | Learning and adaptive systems in artificial intelligence |

62A01 | Foundations and philosophical topics in statistics |

60A05 | Axioms; other general questions in probability |

##### Keywords:

confirmation; representation and updating of partial belief; belief functions; Shafer/Dempster theory; extension of classical Bayesian theory; uncertain evidence; conditioning##### Software:

DENDRAL
Full Text:
DOI

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