A natural prior probability distribution derived from the propositional calculus

Paris, Jeffrey; Vencovská, Alena; Wilmers, George

doi:10.1016/s0168-0072(94)90011-6

Cited by 22 publications

(21 citation statements)

References 6 publications

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“…Throughout we shall avoid going too deeply into the technical mathematical details. The interested reader may find, up to straightforward generalizations, proofs of all the results we state in Paris et al 6 ' 10 , Watton 13 .…”

Section: I=lmentioning

confidence: 84%

“…One partial explanation for this might be that linguistic categories that divide the population roughly in half are descriptively more efficient, although it is hard to see that that can really account for the two examples cited above! [Another possible explanation for such clustering is given in Paris et al 6 . ]…”

Section: Univariate Natural Probability Functionsmentioning

confidence: 89%

“…A second criticism of assuming a uniform distribution (as already remarked in Paris et al 11 ), at least in the sort of contexts where knowledge-based systems would normally be seen as being applicable, is that for probability functions w on SL n representative of such situations we would expect the w(qi) to be rather variable and the distinct <&,qj to be, at least somewhat, dependent i.e. (w(qi Aqj)-w(qi)w(qj)) 2 should be relatively large.…”

Section: I=lmentioning

confidence: 99%

“…As such then the uniform distribution seems to provide an inappropriate likelihood of encountering natural probability functions, at least as far as the objective testing of knowledgebased systems is concerned. Instead of simply opting for the uniform distribution therefore we proceeded in Paris et al 6 ' 10 ' 12 ' 14 , Watton 13 , to develop other candidates for V based on modeling natural probability functions themselves. The plan of the rest of this paper is as follows.…”

Section: I=lmentioning

confidence: 99%

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On the Structure of Probability Functions in the Natural World

Paris

Watton

Wilmers

2000

Int. J. Unc. Fuzz. Knowl. Based Syst.

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The purpose of this paper is to describe the underlying insights and results obtained by the authors, and others, in a series of papers aimed at modeling the distribution of 'natural' probability functions, more precisely the probability functions on {0, l} n which we encounter naturally in the real world as subjects for statistical inference, by identifying such functions with large, random, sentences of the propositional calculus. We explain how this approach produces a robust parameterized family of priors, J n , with several of the properties we might have hoped for in the context, for example marginalisation, invariance under (weak) renaming, and an emphasis on multivariate probability functions exhibiting high interdependence between features.

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Section: I=lmentioning

confidence: 84%

Section: Univariate Natural Probability Functionsmentioning

confidence: 89%

Section: I=lmentioning

confidence: 99%

Section: I=lmentioning

confidence: 99%

See 2 more Smart Citations

On the Structure of Probability Functions in the Natural World

Paris

Watton

Wilmers

2000

Int. J. Unc. Fuzz. Knowl. Based Syst.

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“…The starting point is generally the description of formulae as trees of a suitable shape and suitably labelled. The first efforts in this direction were by Paris et al [5] on And/Or trees (i.e. expressions built on the two connectors ∧ and ∨); the underlying model was that of binary Catalan trees, suitably labelled.…”

Section: Introductionmentioning

confidence: 99%

Complexity and Limiting Ratio of Boolean Functions over Implication

Fournier

Gardy

Genitrini

et al.

Lecture Notes in Computer Science

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Abstract. We consider the logical system of boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of boolean functions expressible in this system. We then show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f . We also prove that most expressions computing any given function in this system are "simple", in a sense that we make precise.

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The fraction of large random trees representing a given Boolean function in implicational logic

Fournier¹,

Gardy²,

Genitrini³

et al. 2011

Random Struct Algorithms

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We consider the logical system of Boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of Boolean functions expressible in this system. Then we show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f . We also prove that most expressions computing any given function in this system are "simple", in a sense that we make precise. The probability of all read-once functions of a given complexity is also evaluated in this model. At last, using the same techniques, the relation between the probability of a function and its complexity is also obtained when random expressions are drawn according to a critical branching process.

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A natural prior probability distribution derived from the propositional calculus

Cited by 22 publications

References 6 publications

On the Structure of Probability Functions in the Natural World

On the Structure of Probability Functions in the Natural World

Complexity and Limiting Ratio of Boolean Functions over Implication

The fraction of large random trees representing a given Boolean function in implicational logic

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