Evaluating default priors with a generalization of Eaton's Markov chain

Shea, Brian P.; Jones, Galin L.

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 1069-1091.

Nous considérons l’évaluation de lois a priori impropres dans un cadre Bayésien formel en fonction des conséquences de leur utilisation. Soit $𝛷$ une classe de fonctions sur l’espace des paramètres, que l’on cherche à estimer sous une fonction de perte quadratique. Si l’estimateur Bayésien de toute fonction dans $𝛷$ est admissible, alors la loi a priori est fortement admissible par rapport à $𝛷$. La méthode d’Eaton pour établir l’admissibilité forte est basée sur l’étude des propriétés de stabilité d’une certaine chaîne de Markov associé au cadre inférentiel. Dans des travaux précédents, nous considérions une nouvelle chaîne de Markov qui permet d’unifier et de généraliser les approches existantes tout en élargissant simultanément son champ d’application. Nous utilisons cette théorie générale pour étudier des conditions d’admissibilité forte pour des modéles à paramètre de position, une loi a priori donnée par la mesure de Lebesgue et la loi normale multivariée de dimension $p$ et moyenne $\theta$, et une loi a priori de la forme $\nu (\parallel \theta {\parallel }^2)\mathrm{d}\theta$.

We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let $𝛷$ be a class of functions on the parameter space and consider estimating elements of $𝛷$ under quadratic loss. If the formal Bayes estimator of every function in $𝛷$ is admissible, then the prior is strongly admissible with respect to $𝛷$. Eaton’s method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with the inferential setting. In previous work, this was handled differently depending upon whether $\varphi \in 𝛷$ was bounded or unbounded. We consider a new Markov chain which allows us to unify and generalize existing approaches while simultaneously broadening the scope of their potential applicability. We use our general theory to investigate strong admissibility conditions for location models when the prior is Lebesgue measure and for the $p$-dimensional multivariate Normal distribution with unknown mean vector $\theta$ and a prior of the form $\nu (\parallel \theta {\parallel }^2)\mathrm{d}\theta$.

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