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From Statistics to Beliefs
, 1992
"... An intelligent agent uses known facts, including statistical knowledge, to assign degrees of belief to assertions it is uncertain about. We investigate three principled techniques for doing this. All three are applications of the principle of indifference, because they assign equal degree of belief ..."
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Cited by 44 (12 self)
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An intelligent agent uses known facts, including statistical knowledge, to assign degrees of belief to assertions it is uncertain about. We investigate three principled techniques for doing this. All three are applications of the principle of indifference, because they assign equal degree of belief to all basic "situations " consistent with the knowledge base. They differ because there are competing intuitions about what the basic situations are. Various natural patterns of reasoning, such as the preference for the most specific statistical data available, turn out to follow from some or all of the techniques. This is an improvement over earlier theories, such as work on direct inference and reference classes, which arbitrarily postulate these patterns without offering any deeper explanations or guarantees of consistency. The three methods we investigate have surprising characterizations: there are connections to the principle of maximum entropy, a principle of maximal independence, an...
Asymptotic Conditional Probabilities for FirstOrder Logic
 In Proc. 24th ACM Symp. on Theory of Computing
, 1992
"... Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for firstorder formulas. That is, given firstorder formulas ' and `, we consider the number of structures with domain f1; : : : ; Ng that satisfy `, and c ..."
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Cited by 13 (7 self)
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Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for firstorder formulas. That is, given firstorder formulas ' and `, we consider the number of structures with domain f1; : : : ; Ng that satisfy `, and compute the fraction of them in which ' is true. We then consider what happens to this probability as N gets large. This is closely connected to the work on 01 laws that considers the limiting probability of firstorder formulas, except that now we are considering asymptotic conditional probabilities. Although work has been done on special cases of asymptotic conditional probabilities, no general theory has been developed. This is probably due in part to the fact that it has been known that, if there is a binary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We show that in this general case, almost all the questions one might want to ask (such as d...
Asymptotic Conditional Probabilities: The Nonunary Case
 J. SYMBOLIC LOGIC
, 1993
"... Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for firstorder sentences. Given firstorder sentences ' and `, we consider the structures with domain f1; : : : ; Ng that satisfy `, and compute the fra ..."
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Cited by 9 (2 self)
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Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for firstorder sentences. Given firstorder sentences ' and `, we consider the structures with domain f1; : : : ; Ng that satisfy `, and compute the fraction of them in which ' is true. We then consider what happens to this fraction as N gets large. This extends the work on 01 laws that considers the limiting probability of firstorder sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kii [Lio69], if there is a nonunary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kii also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is welldefined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable.
Generating Degrees of Belief from Statistical Information: An Overview
, 1993
"... Consider an agent (or expert system) with a knowledge base KB that includes statistical information (such as "90% of patients with jaundice have hepatitis"), firstorder information ("all patients with hepatitis have jaundice"), and default information ("patients with jau ..."
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Cited by 2 (2 self)
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Consider an agent (or expert system) with a knowledge base KB that includes statistical information (such as "90% of patients with jaundice have hepatitis"), firstorder information ("all patients with hepatitis have jaundice"), and default information ("patients with jaundice typically have a fever"). A doctor with such a KB may want to assign a degree of belief to an assertion ' such as "Eric has hepatitis". Since the actions the doctor takes may depend crucially on this degree of belief, we would like to specify a mechanism by which she can use her knowledge base to assign a degree of belief to ' in a principled manner. We have been investigating a number of techniques for doing so; in this paper we give an overview of one of them. The method, which we call the random worlds method, is a natural one: For any given domain size N , we consider the fraction of models satisfying ' among models of size N satisfying KB . If we do not know the domain size N , but know that it is large, we can approximate the degree of belief in ' given KB by taking the limit of this fraction as N goes to infinity. As we show, this approach has many desirable features. In particular, in many cases that arise in practice, the answers we get using this method provably match heuristic assumptions made in many standard AI systems.
Abstract
"... Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for firstorder sentences. Given firstorder sentences ϕ and θ, we consider the structures with domain {1,..., N} that satisfy θ, and compute the fraction of the ..."
Abstract
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Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for firstorder sentences. Given firstorder sentences ϕ and θ, we consider the structures with domain {1,..., N} that satisfy θ, and compute the fraction of them in which ϕ is true. We then consider what happens to this fraction as N gets large. This extends the work on 01 laws that considers the limiting probability of firstorder sentences, by considering asymptotic conditional probabilities. As shown by Liogon’kiĭ [Lio69], if there is a nonunary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon’kiĭ also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable. ∗ Some of this research was done while all three authors were at IBM Almaden Research Center. During this