Test (1995) Vol. 4, No. 1, pp. 19-38 19 Information Tradeoff L. WASSERMAN* and B. CLARKE** *Deparm~ent of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, U. S. A. **Department of Statistics, University of British Columbia 2021 West Mall,, Vancouver, B. C., Canada. V6T 1Z2 SUMMARY A prior may be noninformative for one parameter at the cost of being in- formative for another parameter. This leads to the idea of tradeoff priors: priors that give u~ noninformativity for some parameters to achieve nonin- formativity for others. We propose a general framework where priors are selected by optimizing a functional with two components. The first com- ponent formalizes the requirement that the optimal prior be noninformative for the parameter of interest. The second component is a penalty term that forces the optimizing prior to be close to some target prior. Optimizing such a functional results in a parameterized family of priors from which a specific prior may be selected as tile tradeoffprior. An important particular example of such Tunctionals is provided by choosing the first term to be the marginal missing information for the parameter of interest (generalizing Bernardo's notion of missing information) and the second term to be the relative en- tropy between the unknown prior and the Jeffreys prior. In this case we find a closed form expression for the tradeoff prior and we make explicit connections wittl the Berger-Bernardo prior. In particular, we show that un- der certain conditions, the Berger-Bernardo prior and the Jeffreys prior are special cases of the tradeoffprior. We consider several examples. Keywords: ASYMPTOTIC INFORMATION; NONINFORMATIVE PRIORS; NUISANCE PARAMETERS. 1. INTRODUCTION The most common method for constructing noninformative priors is due to Jeffreys (1961). Although this method works well in the absence of nuisance parameters, some authors have argued that Jeffreys prior leads Received January 94; Revised February 95.