Bayesians conditioning on sets of measure zero
Although I have already discussed this point repeatedly on this ‘Og, I found myself replying to [yet] another question on X validated about the apparent paradox of conditioning on a set of measure zero, as for instance when computing
P(X=.5 | |X|=.5)
which actually has nothing to do with Bayesian inference or Bayes’ Theorem, but is simply wondering about the definition of conditional probability distributions. The OP was correct in stating that
P(X=x | |X|=x)
was defined up to a set of measure zero. And even that
P(X=.5 | |X|=.5)
could be defined arbitrarily, prior to the observation of |X|. But once |X| is observed, say to take the value 0.5, there is a zero probability that this value belongs to the set of measure zero where one defined
P(X=x | |X|=x)
arbitrarily. A point that always proves delicate to explain in class…!
September 25, 2018 at 3:36 pm
Interesting to see your new post about this, Xi’an. Recently we considered exactly the same issue for Bayesian modeling under certain constraint, that gives a conditioning on zero measure issue.
https://arxiv.org/abs/1801.01525
We show that this paradox can be reconciled by considering a Hausdorff measure in lower dimensional space; or alternatively, by an approximation using standard Lebesgue measure, which gives a much broader spectrum of priors and likelihoods to choose from for general constrained problems.
September 25, 2018 at 10:16 pm
Thanks Leo! Actually, for reasons you [and not the general public] can appreciate, I cannot currently comment on this paper of yours!
September 25, 2018 at 8:14 am
The Disintegration Theorem to deal with conditioning on sets of Measure zero is actually exploited in the formal definition of Bayesian Probabilistic Numerical Methods – to appear in SIAM Reviews – https://arxiv.org/pdf/1702.03673.pdf –
September 25, 2018 at 10:00 am
Thanks, Mark! I have to confess I had never heard of a disintegration theorem before, but will definitely add it to my measure toolbox.