By J. A. Hartigan (auth.)

ISBN-10: 1461382424

ISBN-13: 9781461382423

ISBN-10: 1461382440

ISBN-13: 9781461382447

This ebook relies on lectures given at Yale in 1971-1981 to scholars ready with a path in measure-theoretic likelihood. 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, specially in developing correspondences among Bayesian and Fisherian concepts. countless possibilities create fascinating issues in defining conditional chance and restrict options. the most effects are theoretical, probabilistic conclusions derived from probabilistic assumptions. an invaluable idea calls for principles for developing and reading chances. percentages are computed from similarities, utilizing a formalization of the concept that the longer term might be just like the prior. percentages 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, in particular with bankruptcy thirteen. Dan Barry learn evidence. vii Contents bankruptcy 1 Theories of chance 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 types five 1. 6. Subjective Theories: De Finetti 6 7 1. 7. Subjective Theories: solid eight 1. eight. all of the possibilities 10 1. nine. endless Axioms eleven 1. 10. likelihood and Similarity 1. eleven. References thirteen bankruptcy 2 Axioms 14 2. zero. Notation 14 2. 1. likelihood Axioms 14 2. 2.

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Z to T, OJ/ where T denotes the positive integers and Y consists offunctions g where L Ig(i) I < CIJ. A conditional probability P; is (P~'X)(j) = I X(i, j). For eachj, (P;X) (j) defines a probability on:T. For each X, P;X is a function in W. The function (P~X)(Y) has value at (i,j): P~XY(i,j) = (P;X)(j) = L;~jX(i,j) defining a conditional probability of fll' given y- l (OJ/). Now let X(i,j) = i, and define fll' and OJ/ to be the space offunctionsfthat take finitely many non-zero values. Then P;,'YW(j) = Li~j W(i,j) where W is any real valued function finitely non-zero.

Freedman, David (1963), On the asymptotic behaviour of Bayes estimates in the discrete case, Ann. of Math. Statist. 34, 1386-1403. Loeve, Michel (1955), Probability Theory. Princeton: Van Nostrand. Schwartz, L. (1965), On Bayes procedures, Z. 4, 10-26. 0. Introduction The essence of Bayes theory is giving probability values to bets. Methods of generating such probabilities are what separate the various theories. If probabilities are personal opinions, then they are determined by asking questions or observing which of a family of bets an individual accepts.

I-X ~ 1, so that X I[Pi, pY] = (1 - X) log 1 _ PX + X log PX' If pX{O} = pX{1} = 1/2, then I(Pi, pY) = In 2 at X = 0,1 and I[Pi, pY] = (1 - X) 10g(1 - X) + X log X -In 2 ~ In 2 for all X. Thus pX is a maximal learning probability, and it is unique. 3. Invariance observations and converges weakly to the Jeffreys distribution with sin - 1 uniform. 3. l' to S, fl'. Let a be a 1 - 1 transformation of S onto itself such thatfaefl' eachfefl'. A probability pX is relatively invariant under a, or a-invariant if pX = kP"X some k, that is Pf(X) = kPf(aX), each f in fl'.

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