On the relation between plausibility logic and the maximum-entropy principle: a numerical study

What is the relationship between plausibility logic and the principle of maximum entropy? When does the principle give unreasonable or wrong results? When is it appropriate to use the rule ‘expectation = average’? Can plausibility logic give the same answers as the principle, and better answers if those of the principle are unreasonable? To try to answer these questions, this study offers a numerical collection of plausibility distributions given by the maximum-entropy principle and by plausibility logic for a set of fifteen simple problems: throwing dice. PACS numbers: 02.50.Cw,02.50.Tt,01.70.+w MSC numbers: 03B48,60G09,60A05 Dedicato a mia madre per il suo trentesimo compleanno 1 When and how should the maximum-entropy principle be applied? For the student of plausibility logic1, the theory of the principles governing plausible inference, the application of the theory in any given problem is crystal clear in principle: (1) The problem is analysed and reduced to a set of propositions {Ai} and background knowledge I. (2) Some plausibilities P[Ai1 . . . Ai2 | (Ai3 . . . Ai4)∧ I] ∈ [0, 1] are assigned, consistently with the laws below, according to our actual or hypothetical knowledge of the situation and to convenience; the ‘Ai j . . .Aik ’ represent collections of Ais joined by various logical connectives (‘¬’, ‘∧’, ‘∨’, ‘⇒’). (3) Finally, using the basic laws P(¬Ai| I) = 1 − P(Ai| I), (1a) P(Ai ∧ A j| I) = P(Ai| A j ∧ I) P(A j| I), (1b) P(Ai ∨ A j| I) = P(Ai| I) + P(A j| I) − P(Ai ∧ A j| I), (1c) P(Ai ⇒ A j| I) = P(¬Ai| I) + P(A j| Ai ∧ I) P(Ai| I), (1d) Email: [email protected] (remove the z) 1I call ‘plausibility logic’ what many other authors call ‘(Bayesian) probability theory’. ‘Logic’, because it is a generalization of the truth-logical calculus. ‘Plausibility’, because ‘degree of belief’ is unfortunately too unwieldy and many authors still contend that ‘probability’ = ‘frequency’ or, perhaps worse, ‘probability’ = ‘(Lebesgue) measure’.