Likelihood and Bayesian Inference by Leonhard Held & Daniel Sabanés Bové

Author:Leonhard Held & Daniel Sabanés Bové
Language: eng
Format: epub
ISBN: 9783662607923
Publisher: Springer Berlin Heidelberg

is optimal with respect to if and only if, for all and ,

(6.18)

i.e. if is an HPD region.

Proof

To prove Result 6.4, consider some set with expected loss

For fixed , the credible region with smallest size will therefore minimise the expected loss. Now let with property (6.18), and let be some other element in . We need to show that . Let denote the disjoint union of and , i.e. and . Then:

It therefore remains to show that . From (6.18) it follows that

(6.19)

and from it follows that

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