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Random Worlds and Maximum Entropy
 In Proc. 7th IEEE Symp. on Logic in Computer Science
, 1994
"... Given a knowledge base KB containing firstorder and statistical facts, we consider a principled method, called the randomworlds method, for computing a degree of belief that some formula ' holds given KB . If we are reasoning about a world or system consisting of N individuals, then we can co ..."
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Given a knowledge base KB containing firstorder and statistical facts, we consider a principled method, called the randomworlds method, for computing a degree of belief that some formula ' holds given KB . If we are reasoning about a world or system consisting of N individuals, then we can consider all possible worlds, or firstorder models, with domain f1; : : : ; Ng that satisfy KB , and compute the fraction of them in which ' is true. We define the degree of belief to be the asymptotic value of this fraction as N grows large. We show that when the vocabulary underlying ' and KB uses constants and unary predicates only, we can naturally associate an entropy with each world. As N grows larger, there are many more worlds with higher entropy. Therefore, we can use a maximumentropy computation to compute the degree of belief. This result is in a similar spirit to previous work in physics and artificial intelligence, but is far more general. Of equal interest to the result itself are...
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 48 (13 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...
Logics of Probabilistic Reasoning and Imperfect Agents
"... This paper was originally intended to be a review of current state of affairs in firstorder logics for probabilistic reasoning, giving an outline of major works on logics for probabilistic reasoning. However, the concept of the "imperfect agent" was discovered while searching for computati ..."
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This paper was originally intended to be a review of current state of affairs in firstorder logics for probabilistic reasoning, giving an outline of major works on logics for probabilistic reasoning. However, the concept of the "imperfect agent" was discovered while searching for computational models applicable to intelligent systems based on FOPL (firstorder logics of probability). The paper has now taken a broader, more philosophical tone to it. It is still inspired by our study of past and present work on FOPL and is particularly indebted to the essay "Logical foundations of Probability" by J. Lukasiewicz [Luk13].
Asymptotic Conditional Probability in Modal Logic: A Probabilistic Reconstruction of Nonmonotonic Logic
"... We analyze the asymptotic conditional validity of modal formulas, i.e., the probability that a formula ψ is valid in the finite Kripke structures in which a given modal formula ϕ is valid, when the size of these Kripke structures grows to infinity. We characterize the formulas ψ that are almost sure ..."
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We analyze the asymptotic conditional validity of modal formulas, i.e., the probability that a formula ψ is valid in the finite Kripke structures in which a given modal formula ϕ is valid, when the size of these Kripke structures grows to infinity. We characterize the formulas ψ that are almost surely valid (i.e., with probability 1) in case ϕ is a flat, S5consistent formula, and show that these formulas ψ are exactly those which follow from ϕ according to the nonmonotonic modal logic S5G. Our results provide – for the first time – a probabilistic semantics to a wellknown nonmonotonic modal logic, establishing a new bridge between nonmonotonic and probabilistic reasoning, and give a computational account of the asymptotic conditional validity problem in Kripke structures.
From Statistical Knowledge Bases to Degrees of Belief: An Overview
, 2006
"... An intelligent agent will often be uncertain about various properties of its environment, and when acting in that environment it will frequently need to quantify its uncertainty. For example, if the agent wishes to employ the expected utility paradigm of decision theory to guide its actions, she wil ..."
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An intelligent agent will often be uncertain about various properties of its environment, and when acting in that environment it will frequently need to quantify its uncertainty. For example, if the agent wishes to employ the expected utility paradigm of decision theory to guide its actions, she will need to assign degrees of belief (subjective probabilities) to various assertions. Of course, these degrees of belief should not be arbitrary, but rather should be based on the information available to the agent. This paper provides a brief overview of one approach for inducing degrees of belief from very rich knowledge bases that can include information about particular individuals, statistical correlations, physical laws, and default rules. The approach is called the randomworlds method. The method is based on the principle of indifference: it treats all of the worlds the agent considers possible as being equally likely. It is able to integrate qualitative default reasoning with quantitative probabilistic reasoning by providing a language in which both types of information can be easily expressed. A number of desiderata that arise in direct inference (reasoning from statistical information to conclusions about individuals) and default reasoning follow directly from the semantics of random worlds. For example, random worlds captures important patterns of reasoning such as specificity, inheritance, indifference to irrelevant information, and default assumptions of independence. Furthermore, the expressive power of the language used and the intuitive semantics of random worlds allow the
Helsinki 2009
"... Constructive (intuitionist, antirealist) semantics has thus far been lacking an adequate concept of truth in innity concerning factual (i.e., empirical, nonmathematical) sentences. One consequence of this problem is the difculty of incorporating inductive reasoning in constructive semantics. It is ..."
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Constructive (intuitionist, antirealist) semantics has thus far been lacking an adequate concept of truth in innity concerning factual (i.e., empirical, nonmathematical) sentences. One consequence of this problem is the difculty of incorporating inductive reasoning in constructive semantics. It is not possible to formulate a notion for probable truth in innity if there is no adequate notion of what truth in innity is. One needs a notion of a constructive possible world based on sensory experience. Moreover, a constructive probability measure must be dened over these constructively possible empirical worlds. This study denes a particular kind of approach to the concept of truth in innity for Rudolf Carnap's inductive logic. The new approach is based on truth in the consecutive nite domains of individuals. This concept will be given a constructive interpretation. What can be veriably said about an empirical statement with respect to this concept of truth, will be explained, for which purpose a constructive
The Temporal Calculus
"... We consider the problem of defining conditional objects (a I b), which would allow one to regard the conditional probability Pr(alb ) as a probability of a welldefined event rather than as a shorthand for Pr(ab)/Pt(b). The next issue is to define boolean combinations of con ditional objects, a ..."
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We consider the problem of defining conditional objects (a I b), which would allow one to regard the conditional probability Pr(alb ) as a probability of a welldefined event rather than as a shorthand for Pr(ab)/Pt(b). The next issue is to define boolean combinations of con ditional objects, and possibly also the operator of further conditioning.
The Temporal Calculus of Conditional Objects and Conditional Events
, 2008
"... We consider the problem of defining conditional objects (ab), which would allow one to regard the conditional probability Pr(ab) as a probability of a welldefined event rather than as a shorthand for Pr(ab) / Pr(b). The next issue is to define boolean combinations of conditional objects, and poss ..."
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We consider the problem of defining conditional objects (ab), which would allow one to regard the conditional probability Pr(ab) as a probability of a welldefined event rather than as a shorthand for Pr(ab) / Pr(b). The next issue is to define boolean combinations of conditional objects, and possibly also the operator of further conditioning. These questions have been investigated at least since the times of George Boole, leading to a number of formalisms proposed for conditional objects, mostly of syntactical, prooftheoretic vein. We propose a unifying, semantical approach, in which conditional events are (projections of) Markov chains, definable in the threevalued extension (TLTL) of the past tense fragment of propositional linear time logic (TL), or, equivalently, by threevalued counterfree Moore machines. Thus our conditional objects are indeed stochastic processes, one of the central notions of modern probability theory.