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

Brian P. Shea; Galin L. Jones

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 3, page 1069-1091
  • ISSN: 0246-0203

Abstract

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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 ϕ ∈ 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 θ and a prior of the form ν ( ∥ θ ∥ 2 ) d θ .

How to cite

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Shea, Brian P., and Jones, Galin L.. "Evaluating default priors with a generalization of Eaton’s Markov chain." Annales de l'I.H.P. Probabilités et statistiques 50.3 (2014): 1069-1091. <http://eudml.org/doc/272028>.

@article{Shea2014,
abstract = {We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let $\varPhi $ be a class of functions on the parameter space and consider estimating elements of $\varPhi $ under quadratic loss. If the formal Bayes estimator of every function in $\varPhi $ is admissible, then the prior is strongly admissible with respect to $\varPhi $. 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 \varPhi $ 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 (\Vert \theta \Vert ^\{2\})\,\mathrm \{d\}\theta $.},
author = {Shea, Brian P., Jones, Galin L.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {admissibility; improper prior distribution; symmetric Markov chain; recurrence; Dirichlet form; formal Bayes rule},
language = {eng},
number = {3},
pages = {1069-1091},
publisher = {Gauthier-Villars},
title = {Evaluating default priors with a generalization of Eaton’s Markov chain},
url = {http://eudml.org/doc/272028},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Shea, Brian P.
AU - Jones, Galin L.
TI - Evaluating default priors with a generalization of Eaton’s Markov chain
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 3
SP - 1069
EP - 1091
AB - We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let $\varPhi $ be a class of functions on the parameter space and consider estimating elements of $\varPhi $ under quadratic loss. If the formal Bayes estimator of every function in $\varPhi $ is admissible, then the prior is strongly admissible with respect to $\varPhi $. 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 \varPhi $ 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 (\Vert \theta \Vert ^{2})\,\mathrm {d}\theta $.
LA - eng
KW - admissibility; improper prior distribution; symmetric Markov chain; recurrence; Dirichlet form; formal Bayes rule
UR - http://eudml.org/doc/272028
ER -

References

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