SUMMARY We consider the problem of constructing a prior that is 'noninformative' for a single parameter in the presence of nuisance parameters. Our approach is to require that the resulting marginal posterior intervals have accurate frequentist coverage. Stein (1985) derived nonrigorously a sufficient condition for such a prior. Through the use of orthogonal parameters, we give a general form for the class of priors satisfying Stein's condition. The priors are proportional to the square root of the information element for the parameter of interest times an arbitrary function of the nuisance parameters. This is in contrast to Jeffreys (1946) invariant prior for the overall parameter, which is proportional to the square root of the determinant of the information matrix. Several examples are given and comparisons are made to the reference priors of Bernardo (1979).
[1]
H. Jeffreys.
An invariant form for the prior probability in estimation problems
,
1946,
Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.
[2]
M. A. Creasy.
Limits for the Ratio of Means
,
1954
.
[3]
B. L. Welch,et al.
On Formulae for Confidence Points Based on Integrals of Weighted Likelihoods
,
1963
.
[4]
Charles Stein,et al.
On the coverage probability of confidence sets based on a prior distribution
,
1985
.
[5]
B. Efron.
Why Isn't Everyone a Bayesian?
,
1986
.
[6]
L. Tierney,et al.
Accurate Approximations for Posterior Moments and Marginal Densities
,
1986
.
[7]
James O. Berger,et al.
Estimating a Product of Means: Bayesian Analysis with Reference Priors
,
1989
.