While studying Haar measures I came across the following paper:

George, EI, McCulloch, R (1993). On obtaining invariant prior distributions,

which might relate to the uniqueness of Jeffreys’s prior. I believe that the paper states the following:

1. Ω-invariance: All priors that are proportional to det(D”), where D is a divergence measure and D” its Hessian, are parametrisation invariant.

2. S-invariance: Divergence measures of the form are sample space invariant, if d_{0} is homogenous. Such a divergence measure can be related to a divergence measure such that $latexd_{0}(p, q)= p d(q/p)$ , which is related to Csiszar’s f-divergence measures. As an example they discuss D_{0} being the squared Hellinger distance and D_{1} the KL-divergence and these two divergence measures have the Fisher information as their Hessian.

3. Group-invariance: According to Kass (1980, 1989) Jeffreys prior is the left Haar density whenever the data can be written as being acted upon by the parameter (left-group action).

I’m not sure whether these three statements points towards the uniqueness of Jeffreys’s prior. I’m not sure whether the converse of 1 holds? Do you know whether a parametrisation invariant prior is necessarily proportional to the root of a divergence’s Hessian? Clearly, there’s a relation between Jeffreys prior, Hellinger distance and the KL-divergence, but is this relationship uniquely connected to the Riemannian metric on model space?

Cheers,

Alexander

Here’s a modern day version of Cencov

http://m.pnas.org/content/108/25/10078.full.pdf

Nick ]]>

The uniqueness argument comes from the uniqueness of the Fisher-Rao metric, not the resulting measure itself.

Basically Fisher-Rao is the only metric that is consistent with the usual properties of Frequentist statistics, such as sufficiency and the like. Then, using the fact that a metric induces a unique measure (not guaranteed to be a probability measure, of course), you can argue that the Jeffreys’ prior is unique to a given likelihood function.

If you relax the condition on Fisher-Rao (in particular if you do mathematically suspect things like adding the Hessian of a prior density) then you no longer get a unique metric and hence no more unique Jeffreys’ prior.

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