invariant conjugate analysis for exponential families

Here is a paper from Bayesian Analysis that I somehow missed and only become aware thanks to a (more) recent paper of the first author: in 2012, Pierre Druilhet and Denis Pommeret published invariant conjugate analysis for exponential families. The authors define a new class of conjugate families, called Jeffreys’ conjugate priors (JCP) by using Jeffreys’ prior as the reference density (rather than the uniform in regular conjugate families). Following from the earlier proposal of Druilhet and Marin (2007, BA). Both families of course coincide in the case of quadratic variance exponential families. The motivation for using those new conjugate priors is that the family is invariant by a change of parametrisation. And to include Jeffreys’ prior as a special case of conjugate prior. In the special case of the inverse Gaussian distribution, this approach leads to the conjugacy of the inverse normal distribution, a feature I noticed in 1991 when working on an astronomy project. There are two obvious drawbacks to those new conjugate families: one is that the priors are not longer always proper. The other one is that the computations associated with those new priors are more involved, which may explain why the authors propose the MAP as their default estimator. Since posterior expectations of the mean (in the natural representation [in x] of the exponential family) are no longer linear in x.

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