The Annals of Statistics

Deriving Posterior Distributions for a Location Parameter: A Decision Theoretic Approach

Constantine A. Gatsonis

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In this paper we develop a decision theoretic formulation for the problem of deriving posterior distributions for a parameter $\theta$, when the prior information is vague. Let $\pi(d\theta)$ be the true but unknown prior, $Q_\pi(d\theta\mid X)$ the corresponding posterior and $\delta(d\theta\mid X)$ an estimate of the posterior based on an observation $X$. The loss function is specified as a measure of distance between $Q_\pi(\cdot\mid X)$ and $\delta(\cdot\mid X)$, and the risk is the expected value of the loss with respect to the marginal distribution of $X$. When $\theta$ is a location parameter, the best invariant procedure (under translations in $R^n$) specifies the posterior which is obtained from the uniform prior on $\theta$. We show that this procedure is admissible in dimension 1 or 2 but it is inadmissible in all higher dimensions. The results reported here concern a broad class of location families, which includes the normal.

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Ann. Statist., Volume 12, Number 3 (1984), 958-970.

First available in Project Euclid: 12 April 2007

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Primary: 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 62C15: Admissibility 62A99: None of the above, but in this section 62F15: Bayesian inference 62H12: Estimation

Location parameter noninformative priors best invariant procedures admissibility Stein phenomenon normal mean


Gatsonis, Constantine A. Deriving Posterior Distributions for a Location Parameter: A Decision Theoretic Approach. Ann. Statist. 12 (1984), no. 3, 958--970. doi:10.1214/aos/1176346714.

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