Rules having rare exceptions are best interpreted as assertions of high conditional probability. In other words, a rule \emph{If $X$ then $Y$} is interpreted as meaning that $\Pr(Y|X) \approx 1$. In this paper, such rules are regarded as statements about imprecise probabilities, and imprecise probabilities are identified with convex sets of precise probabilities. A general approach to reasoning with such rules, based on second-order probability, is advocated. Within this general approach, different reasoning methods are needed, with the selection of a specific method being dependent upon what knowledge is available about the relative tightness of the approximation $\Pr(Y|X) \approx 1$ across rules. A method of reasoning, entailment with universal near surety, is formulated for the case when \emph{no} knowledge is available concerning these relative tightnesses. Finally, it is shown that reasoning via entailment with universal near surety is equivalent to carrying out a particular test on a directed graph.
Keywords. Conditional probability, second-order probability, Bayesian inference, nonmonotonic logic, rule-based systems, threshold knowledge, informant, robustness, directed graph.
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Authors addresses:
Donald Bamber
SPAWARSYSCEN D44215
53345 RYNE ROAD
SAN DIEGO CA 92152-7251
USA
I.R. Goodman
Code D44215,Topside, Bldg. A33,
SPAWAR Systems Center,
San Diego, CA 92152
E-mail addresses:
| Donald Bamber | bamber@spawar.navy.mil |
| I.R. Goodman | goodman@spawar.navy.mil |