Reference priors

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James Berger at the at the Mathematical Research Institute of Oberwolfach in 2005

The reference method is a very successful mathematical rule to derive objective priors that was originally proposed by José Miguel Bernardo in [1] and later developed in collaboration with James O. Berger in the series of papers [2],[3],[4].

The logic behind the reference method is to notice that, the more informative a prior is, the small is the amount of information that can be obtained from the data. Conversely, an objective prior should maximize some sort of expected information from the data (what in the reference literature is called the missing information).

Ordering the importance of parameters

For univariate continuous parameters, and under regularity conditions, the reference prior is unique and coincides with Jeffreys prior.

When the model depends on several parameters, the reference prior is derived through a sequential use of the univariate reference rule. As a result, the reference prior does not have to be unique and depends on the order chosen (to be interpreted from more to less important for the purposes of the study). For example, in the case with two parameters the order means that is the parameter of interest and is a nuisance parameter.

References

  1. Bernardo, J.M. (1979), "Reference posterior distributions for Bayesian inference", Journal of the Royal Statistical Society, Series B, 41, 113-128.
  2. Berger, J.O and Bernardo, J.M. (1989), "Estimating a product of means: Bayesian analysis with reference priors", Journal of the American Statistical Association, 84, 200-207.
  3. Berger, J. O. and Bernardo, J. M. (1992), "Ordered group reference priors with application to the multinomial problem", Biometrika 79, 25–37.
  4. Berger, J. O. and Bernardo, J. M. (1992), "Reference priors in a variance components problem", in Bayesian Analysis in Statistics and Econometrics (P. K. Goel and N. S. Iyengar, eds.). Berlin: Springer, pp. 323–340.