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By Christian P. Robert

ISBN-10: 0387715983

ISBN-13: 9780387715988

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26). The derivation of objective posterior distributions actually calls for a more advanced theory of noninformative distributions (see Chapter 3), which shows that the likelihood function cannot always be considered the most natural posterior distribution. Many approaches have been suggested to implement the Likelihood Principle like, for instance, penalized likelihood theory (Akaike (1978, 1983)) or stochastic complexity theory (Rissanen (1983, 1990)). See also Bjørnstad (1990) for a survey of non-Bayesian methods derived from the Likelihood Principle in the prediction area.

4) is the median of the distribution of ξ. Give the quantity δ that minimizes the average squared error IEξ [(ξ − δ)2 ]. 5) and a flat prior π(ξ) = 1 on ξ. 12 Show that, for a sample from a normal N (θ, σ 2 ) distribution, there does not exist an unbiased estimator of σ but only of powers of σ 2 . 13 Consider x ∼ P (λ). Show that δ(x) = II0 (x) is an unbiased estimator of e−λ which is null with probability 1 − e−λ .

A wrong interpretation of the sparse prior information). (4) Such distributions are generally more acceptable to non-Bayesians, partly for reasons (2) and (3), but also because they may have frequentist justifications, such as: 28 Introduction 1 (i) minimaxity, which is related to the usually improper “least favorable distributions”, defined in Chapter 2); (ii) admissibility, as proper and some improper distributions lead to admissible estimators, while admissible estimators sometimes only correspond to Bayes estimators (see Chapter 8); and (iii) invariance, as the best equivariant estimator is a Bayes estimator for the generally improper Haar measure associated with the transformation group (see Chapter 9).

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The Bayesian Choice by Christian P. Robert

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