Nicole Wang writes:
I am a PhD student in statistics starting this year. I read the Dawid et al. (1973) marginalization paradoxes paper. I found several approaches claiming their “resolution” to this paradox. I am very curious whether you think there is a complete resolution to this paradox so far. If there is one, I sincerely wonder what it is. If not, which approach(s) do you think it(they) is(are) closer to the resolution?
I replied by pointing to these two posts:
Christian Robert on the Jeffreys-Lindley paradox; more generally, it’s good news when philosophical arguments can be transformed into technical modeling issues
and
My problem with the Lindley paradox.
It’s not a topic I’ve thought about much lately because to me it’s already been resolved.
Paul Gustafson and I wrote a piece on the question “Is a spike and slab prior ever a good idea?” as a rejoinder for a Bayesian Analysis paper of ours. From my perspective, this question is exactly equivalent to asking “Are Bayes factors ever a good idea?” Perhaps this might be of interest. Here is a link to the paper with rejoinder:
“Defining a Credible Interval Is Not Always Possible with “Point-Null” Priors: A Lesser-Known Correlate of the Jeffreys-Lindley Paradox (with Discussion)”
https://projecteuclid.org/journals/bayesian-analysis/advance-publication/Defining-a-Credible-Interval-Is-Not-Always-Possible-with-Point/10.1214/23-BA1397.full
You can always relax a spike and slab prior into a narrow and wide Normal mixture. In Julia you could do for example:
MixtureModel([Normal(0.0,0.001),Normal(0.0,100.0)],[0.5,0.5])
To define a tight peak near 0 with 50% and a broad hump around 0 with 50%. Obviously the appropriate scales and weights are up to your knowledge of the applied problem.
While this leaves everything infinitely differentiable and well defined, It’s pretty hard to sample this kind of thing and it has all sorts of numerical issues often.
> The marginalization or Jeffreys-Lindley paradox
The marginalization paradox in the question (Stone, Dawid 1972; Dawid, Stone, Zidek 1973) doesn’t have anything to do with Jeffreys-Lindley’s paradox. (Or does it?)
It’s true that there is no paradox there either. I would say there is a broad agreement that having two different procedures designed to answer two different questions giving two different answers does not make a paradox.