Jaynes’ marginalisation paradox

After 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!]

4 Responses to “Jaynes’ marginalisation paradox”

  1. 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.

  2. […] 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. […]

  3. 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

    • 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….

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