By Christian P. Robert
ISBN-10: 0387715983
ISBN-13: 9780387715988
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Additional info for The Bayesian Choice
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47). , corresponds to N (x, 1). Notice that the constant does not play a role in the posterior distribution, and that the posterior distribution is actually the likelihood function. Therefore, even though improper priors cannot be normalized, it does not matter because the constant is of no interest for the statistical inference (but see Chapter 5 for an important exception). 5 Improper prior distributions 29 According to the Bayesian version of the Likelihood Principle, only posterior distributions are of importance.
I=1 In this case, the stopping rule is obviously incompatible with frequentist modeling since the resulting sample always leads to the rejection of the null hypothesis H0 : θ = 0 at the level 5% (see Chapter 5). On the contrary, a Bayesian approach avoids this difficulty (see Raiffa and Schlaifer (1961) and Berger and Wolpert (1988, p. 81)). 3 Derivation of the Likelihood Principle A justification of the Likelihood Principle has been provided by Birnbaum (1962) who established that it is implied by the Sufficiency Principle, conditional upon the acceptance of a second principle.
As a function of θ, which is unknown, depending on the observed value x. 1). A general description of the inversion of probabilities is given by Bayes’s Theorem: If A and E are events such that P (E) = 0, P (A|E) and P (E|A) are related by P (A|E) 4 5 6 = P (E|A)P (A) P (E|A)P (A) + P (E|Ac )P (Ac ) coherence, from Savage (1954) to Heath and Sudderth (1989), which all lead to the conclusion that a procedure is coherent if, and only if, it is Bayesian. This chapter and Chapter 11 are worth re-reading once the more technical points of the inferential process and the issues at hand have been fully understood.
The Bayesian Choice by Christian P. Robert
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