a new paradigm for improper priors

Gunnar Taraldsen and co-authors have arXived a short note on using improper priors from a new perspective. Generalising an earlier 2016 paper in JSPI on the same topic. Which both relate to a concept introduced by Rényi (who himself attributes the idea to Kolmogorov). Namely that random variables measures are to be associated with arbitrary measures [not necessarily σ-finite measures, the later defining σ-finite random variables], rather than those with total mass one. Which allows for an alternate notion of conditional probability in the case of σ-finite random variables, with the perk that this conditional probability distribution is itself of mass 1 (a.e.).  Which we know happens when moving from prior to proper posterior.

I remain puzzled by the 2016 paper though as I do not follow the meaning of a random variable associated with an infinite mass probability measure. If the point is limited to construct posterior probability distributions associated with improper priors, there is little value in doing so. The argument in the 2016 paper is however that one can then define a conditional distribution in marginalisation paradoxes à la Stone, Dawid and Zidek (1973) where the marginal does not exist. Solving with this formalism the said marginalisation paradoxes as conditional distributions are only defined for σ-finite random variables. Which gives a fairly different conclusion from either Stone, Dawid and Zidek (1973) [with whom I agree, namely that there is no paradox because there is no “joint” distribution] or Jaynes (1973) [with whom I less agree!, in that the use of an invariant measure to make the discrepancy go away is not a particularly strong argument in favour of this measure]. The 2016 paper also draws an interesting connection with the study by Jim Hobert and George Casella (in Jim’s thesis) of [null recurrent or transient] Gibbs samplers with no joint [proper] distribution. Which in some situations can produce proper subchains, a phenomenon later exhibited by Alan Gelfand and Sujit Sahu (and Xiao-Li Meng as well if I correctly remember!). But I see no advantage in following this formalism, as it does not impact whether the chain is transient or null recurrent, or anything connected with its implementation. Plus a link to the approximation of improper priors by sequences of proper ones by Bioche and Druihlet I discussed a while ago.

5 Responses to “a new paradigm for improper priors”

  1. First of all: Thank You for commenting on the arXived paper and related work.

    Regarding “as I do not follow the meaning of a random variable associated with an infinite mass probability measure” I recommend the books by Renyi. The short explanation: In stead of one probability measure, there is a family of (conditional) probability measures.

    These are interpreted similarly to the case of where only one probability is given: Subjective in the most general case, but in some cases (physics!) also in a ‘natural law frequentist sense’.

    This is my personal interpretation, and other people are free to disagree. It may also happen that I disagree myself if I think about this for 30 more years :-)

  2. X:

    I haven’t read the paper in question, but more and more I’ve been thinking that improper priors are a bad idea, so it might well be that these authors are engaged in a useless endeavor!

  3. Cointegration is basically a special case of the null-recurrent-Markov chain-with-a-positive-recurrent-sub-chain thing, right?

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