## statistics with improper posteriors [or not]

**L**ast December, Gunnar Taraldsen, Jarle Tufto, and Bo H. Lindqvist arXived a paper on using priors that lead to improper posteriors and [trying to] getting away with it! The central concept in their approach is Rényi’s generalisation of Kolmogorov’s version to define conditional probability distributions from infinite mass measures by conditioning on finite mass measurable sets. A position adopted by Dennis Lindley in his 1964 book .And already discussed in a few ‘Og’s posts. While the theory thus developed indeed allows for the manipulation of improper posteriors, I have difficulties with the inferential aspects of the construct, since one cannot condition on an arbitrary finite measurable set without prior information. Things get a wee bit more outwardly when considering “data” with infinite mass, in Section 4.2, since they cannot be properly normalised (although I find the example of the degenerate multivariate Gaussian distribution puzzling as it is not a matter of improperness, since the degenerate Gaussian has a well-defined density against the right dominating measure). The paper also discusses marginalisation paradoxes, by acknowledging that marginalisation is no longer feasible with improper quantities. And the Jeffreys-Lindley paradox, with a resolution that uses the sum of the Dirac mass at the null, δ⁰, and of the Lebesgue measure on the real line, λ, as the dominating measure. This indeed solves the issue of the arbitrary constant in the Bayes factor, since it is “the same” on the null hypothesis and elsewhere, but I do not buy the argument, as I see no reason to favour δ⁰+λ over 3.141516 δ⁰+λ or δ⁰+1.61718 λ… (This section 4.5 also illustrates that the choice of the sequence of conditioning sets has an impact on the limiting measure, in the Rényi sense.) In conclusion, after reading the paper, I remain uncertain as to how to exploit this generalisation from an inferential (Bayesian?) viewpoint, since improper posteriors do not clearly lead to well-defined inferential procedures…

March 6, 2019 at 4:02 am

Everything makes sense as long as you are allowed to do the conditioning that generally you are not allowed to do! You can have fun trying to make sense of likelihood functions without a prior (or at least without a general prior) in a similar way.