Committee for many useful discussions. This work was supported in part by the U.S. Department of Energy under grant Nos. DE-FG02-91ER40651, DE-FG02-04ER41305, and DE-FG02-95ER40896. [1] C. P. Robert, The Bayesian Choice: from Decision-Theoretic Foundations to Computational Implementation (Springer, New York, 2007), 2nd ed.; E. T. Jaynes, Probability Theory: The Logic of Science , edited by G. L. Bretthorst (Cambridge University Press, Cambridge, 2003); A. O’Hagan, Kendall’s Advanced Theory of Statistics, Volume 2B: Bayesian Inference (Ed- ward Arnold, London, 1994); H. Jeffreys, Theory of Probability (Oxford University Press, Oxford, 1961), 3rd ed. [2] See for example A. O’Hagan, op. cit. [3] For a discussion of different types of informative priors, see e.g. D. R. Cox, in Proceedings of the PHYSTAT LHC workshop on statistical issues for LHC Physics , edited by H. B. Prosper, L. Lyons, and A. de Roeck, CERN Yellow Report CERN-2008-001 (2008), pp. 3-7. [4] R. E. Kass and L. Wasserman, J. Amer. Statist. Assoc. 91 , 1343 (1996). [5] I. Bertram, G. Landsberg, J. Linnemann, R. Partridge, M. Paterno, and H. B. Prosper, Fermilab preprint FERMILAB-TM-2104 (2000). [6] L. Demortier, in Proceedings of the Conference on Advanced Statistical Techniques in Particle Physics , edited by M. R. Whalley and L. Lyons, Durham preprint IPPP/02/39 (2002), pp. 145- 151. [7] There is no agreement on what to call such priors. In the literature one finds many names: objective, non-subjective, non-informative, minimally-informative, default, neutral, conven- tional, reference. However, we prefer the term formal prior because it reflects the manner in which they are constructed. [8] J. M. Bernardo, J. R. Statist. Soc. B 41 , 113 (1979). [9] J. O. Berger and J. M. Bernardo, J. Amer. Statist. Assoc. 84 , 200 (1989). [10] J. O. Berger and J. M. Bernardo, Biometrika 79 , 25 (1992). [11] J. O. Berger and J. M. Bernardo, in Bayesian Statistics 4 , edited by J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith (Oxford University Press, Oxford, 1992), pp. 35-60, http://www.uv.es/ ~ bernardo/1992Valencia4Ref.pdf . 26