The differential of probabilistic entailment
Section snippets
Introduction and statement of the main results
Let be elements of a boolean algebra A, , and a -valued map. Interpreting each as a probability, Boole asked to determine the probability of some other event given the assignment . De Finetti's solution of Boole's problem, Theorem 2.7, yields a closed interval such that the set of probabilities of a compatible with the assignment coincides with . The interval is nonempty iff the assignment is consistent in the sense
Preliminaries: Boole's extension problem and de Finetti's solution
Boole [2, Chapter XVI, 4, p. 246] writes:
Let us call this Boole's extension problem. This section is devoted to giving a short proof of de Finetti's solution of the problem. His result and the preliminary material for the proof will find repeated use below.the object of the theory of probabilities might be thus defined. Given the probabilities of any events, of whatever kind, to find the probability of some other event connected with them.
So let us first assume “events” are assigned
Minimal paths in the state space induced by a consistent book
Let A be a finite boolean algebra together with elements and a consistent book . Theorem 2.7 yields the de Finetti interval of consistent probabilities for given β. We may write where is as defined in (3). For fixed β, our aim in this section is to study the dependence of on the probability .
Suppose A has u atoms. Given affinely independent points a convenient geometric representation of the state space is
The dynamics of Boole's extension problem
In this section we consider the totality of consistent books on a fixed but otherwise arbitrary k-tuple of elements of a finite boolean algebra A. Our aim is to investigate the computability of the differential version of Boole's extension problem. We let In Proposition 4.2 we will provide a concrete realization of the set . To this purpose we prepare the following easy lemma, whose proof we have not been able to
Examples
Example 5.1 The faces of a tetrahedral biased dice are denoted and are identified with the atoms of the free two-generator boolean algebra . The outcomes have equal probability 1/3. Problem: How does the probability of rolling depend on the probability of rolling ? Fig. 1 gives a geometric representation of the problem and its solution. The boolean algebra is presented as the powerset of the set , the latter being in one-one correspondence with both and
Related work and final remarks
In [10, §§ 1-5] Hailperin considered the possible probabilities of a boolean formula ϕ given a -valued map defined on the free generating set of the free n-generator boolean algebra . In Hailperin's terminology, “ are arbitrary events of which nothing is known except that their respective probabilities are .” An effective procedure is described, based on Linear Programming, by which, given any formula one can obtain the best possible upper and
Acknowledgements
The author thanks the two Referees for their competent and careful reading of this paper and their valuable comments and suggestions for improvement.
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