## Jaynes’ marginalisation paradox

**A**fter delivering my one-day lecture on Jaynes’ ** Probability Theory**, I gave as assignment to the students that they wrote their own analysis of Chapter 15 (

*Paradoxes of probability theory*), given its extensive and exciting coverage of the marginalisation paradoxes and my omission of it in the lecture notes… Up to now, only Jean-Bernard Salomon has returned a (good albeit short) synthesis of the chapter, seemingly siding with Jaynes’ analysis that a “good” noninformative prior should avoid the paradox. (In short, my own view of the problem is to side with Dawid, Stone, and Zidek, in that the paradox is only a paradox when interpreting marginals of infinite measures as if they were probability marginals…) This made me wonder if there could be a squared marginalisation paradox: find a statistical model parameterised by

*θ*with a nuisance parameter

*η=η(θ)*such that when the parameter of interest is

*ξ=ξ(θ)*the prior on

*η*solving the marginalisation paradox is not the same as when the parameter of interest is

*ζ=ζ(θ)*…

*[I have not given the problem more than a few seconds thought so this may prove a logical impossibility!]*

January 6, 2014 at 6:48 pm

Just seen this – 2.5 years late! With reference to Xian’s “squared marginalisation paradox”, there were already examples of this in the original Dawid/Stone/Zidek paper.

January 11, 2012 at 12:12 am

[…] valid environment for working with improper priors. For instance, the final section on the marginalisation “paradoxes” is illuminating in this respect as it does not demand using a limit of proper priors. […]

June 20, 2011 at 8:36 pm

Hi Xian!

Regarding the Marginalization Paradox, Dawid et al wrote

an extensive answer to Chapter 15 of Jaynes Book. It is

available here:

http://www.ucl.ac.uk/statistics/research/pdfs/172.zip

All my best,

Paulo

June 21, 2011 at 7:01 am

Paulo: thank you for pointing out the reference. I was aware of it and had in mind to write a separate piece about this very special and fascinating chapter in an indefinite future….