## Bayesians conditioning on sets of measure zero

**A**lthough 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.