By J. A. Hartigan (auth.)

This publication relies on lectures given at Yale in 1971-1981 to scholars ready with a path in measure-theoretic chance. It comprises one technical innovation-probability distributions within which the complete chance is limitless. Such incorrect distributions come up embarras singly often in Bayes concept, in particular in setting up correspondences among Bayesian and Fisherian recommendations. countless percentages create fascinating problems in defining conditional chance and restrict innovations. the most effects are theoretical, probabilistic conclusions derived from probabilistic assumptions. an invaluable conception calls for principles for developing and examining percentages. chances are computed from similarities, utilizing a formalization of the concept the longer term will likely be just like the prior. possibilities are objectively derived from similarities, yet similarities are sUbjective judgments of people. after all the theorems stay precise in any interpretation of likelihood that satisfies the formal axioms. My colleague David Potlard helped much, particularly with bankruptcy thirteen. Dan Barry learn evidence. vii Contents bankruptcy 1 Theories of likelihood 1. zero. advent 1 1. 1. Logical Theories: Laplace 1 1. 2. Logical Theories: Keynes and Jeffreys 2 1. three. Empirical Theories: Von Mises three 1. four. Empirical Theories: Kolmogorov five 1. five. Empirical Theories: Falsifiable versions five 1. 6. Subjective Theories: De Finetti 6 7 1. 7. Subjective Theories: solid eight 1. eight. all of the chances 10 1. nine. limitless Axioms eleven 1. 10. likelihood and Similarity 1. eleven. References thirteen bankruptcy 2 Axioms 14 2. zero. Notation 14 2. 1. chance Axioms 14 2. 2.

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This e-book is predicated on lectures given at Yale in 1971-1981 to scholars ready with a direction in measure-theoretic chance. It comprises one technical innovation-probability distributions within which the full likelihood is endless. Such unsuitable distributions come up embarras singly often in Bayes thought, in particular in constructing correspondences among Bayesian and Fisherian innovations.

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Note that pX and may not be unitary, but under the conditions of the theorem is unitary. Renyi (1970) takes unitary conditional probabilities as the basic concept, expressing non-unitary probabilities such as lebesgue measure by families of such conditional probabilities. It seems simpler to go the other way, to define unitary conditional probabilities from nonunitary probabilities; indeed we allow non-unitary conditional probabilities in general, though our Bayes theorem produces only unitary conditional probabilities.

G log f ~ g log g + f - g with equality only if f = g. If a density g satisfies the constraints, Q(gX j ) = a p PROOF. Q(g logf) = Q[gLAjX)] = Q[fLAjX j)] = Q(flogf). Thus Q(flogf)=Q(glogf)~Q(glogg) with equality only if f=g as Q. Therefore P with f = dP/dQ is of minimal information among all P satisfying the constraints, and no other minimal P exists. Note: Since I(P, Q) is convex as a function of P, it may be shown that the minimal information P is unique if it exists. Note: It may happen that no minimal information P exists, but that there exists Po = Q ·fo such that for each A with 0 < P o(A) < 00, P A = Q(foA)/ P o(A) is the minimal probability P under the additional constraints P A = 1, P(X) = 0 if XA = O.

N}, then PnX PROOF. -+ X as P. 3. Almost Sure Convergence of Conditional Probabilities a martingale when P is unitary; in this case the theorem is well known, Doob (1953). Lemma. P{ sup IXil ~ s} ;£ plxnl/s. 1 ~i~n PROOF. Then Let Ai = {IX 11 < e, Ix 21 < e, ... , IXil ~ e}, i = 1,2, '" , n. n IAi={ sup IXil~e}. AiX; = Ai(X; + 2(Xn - X)Xi + (Xn - P[Ai(X n - X)XJ = PPJAiX/X Ii - X/) XJ] = P[AiXiPi(X Ii - X,)] = 0 P(AiX;) ~ e2P Ai since IXil ~ s when Ai =1= 0 PX; ~ e2PIA i = e2p{ sup IXil ~ e} as required.