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

Thank you, Nick! While it may be modern day (!), it does not sound comprehensible enough for me to make the link with the uniqueness of Jeffreys’s prior as the invariant prior.

]]>Here’s a modern day version of Cencov

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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