By Dukk V.

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**Example text**

Let us stress again that the main justiﬁcation for using improper prior distributions is to provide a completion of the Bayesian inferential ﬁeld for subjective, axiomatic (in relation with complete class results, see Chapter 8), and practical reasons. This extension does not modify the complexity of the inference, however, because the posterior distribution is truly a probability distribution. 6 The Bayesian choice To close this introduction, let us impress upon the reader that there is such a thing as a Bayesian choice.

Xn . This implies the following principle. Stopping Rule Principle If a sequence of experiments, E1 , E2 , . , is directed by a stopping rule, τ , which indicates when the experiments should stop, inference about θ must depend on τ only through the resulting sample. 4 illustrates the case where two diﬀerent stopping rules lead to the same sample: either the sample size is ﬁxed to be 12, or the experiment is stopped when 9 viewers have been interviewed. 96/ n. 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).

31). Some numerical procedures, such as the EM algorithm of Dempster et al. (1977) for missing data models or the algorithm of Robertson et al. (1988) for order-restricted parameter spaces, have been tailored to this approach, but unsolved diﬃculties remain. Second, a maximization technique is bound to give estimators that lack smoothness, as opposed to integration for instance. This is particularly true when the parameter space is restricted. For example, Saxena and Alam (1982) show that, if x ∼ χ2p (λ), that is, a noncentral chi-squared distribution with p degrees of freedom13 , the maximum likelihood estimator of λ is equal to 0 for x < p.

### A hierarchical Bayesian approach to modeling embryo implantation following in vitro fertilization (2 by Dukk V.

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