Noninformative priors for one parameter of many

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).