I was wondering how conditional probability and the disintegration theorem are related?
How is the conditional probability given by the disintegration theorem? I don't quite understand what Wikipedia says:
The disintegration theorem can be applied to give a rigorous treatment
of conditioning probability distributions in statistics, while
avoiding purely abstract formulations of conditional probability.
The linked paper Chang, J.T.; Pollard, D. (1997). "Conditioning as disintegration". STATISTICA NEERLANDICA 51 (3) has 30 pages and is somehow overwhelming to me now.
Following is a scenario I am considering (but it may be over-complicated and misleading, I am not sure):
Suppose $(\Omega, \mathcal{M},P)$ is a probability space. $(S, \mathcal{S})$ is a measurable space. and $Y:\Omega \to S$ is a measurable mapping.
The conditional probability $P(\cdot | Y)$ is a mapping $\mathcal{M} \times S \to [0,1]$.
I guess it is only when the conditional probability is regular, i.e. $\forall s \in S$, $P(\cdot | Y)(s)$ is a probability measure, that the conditional probability can be given by the disintegration theorem?
Then, how is the conditional probability given by the disintegration theorem?
Note: I think I have got the idea of the disintegration theorem. My source is also Wikipedia.
I also appreciate it if you could let me know about some accessible relevant texts.
Thanks and regards!
Best Answer
Wikipedia probably isn't referring to the conditional probabilities you mention when mentioning the "purely abstract formulations of conditional probability" but to Kolmogorov's approach to conditional expectation by the Radon-Nikodym-theorem. Since the probability of an event is simply the expectation of its indicator function, the Kolmogorov approach gives an abstract approach to conditional probability that doesn't require any regularity conditions.
Now the disintegration theorem tells you essentially that you get a conditional probability when conditioning on a random variable under some conditions. This is already "concrete"!
The general topic of conditioning gets complicated very fast, but for practical purposes it is usually enough to work with conditioning on $\mathbb{R}$ with the Borel sets (most spaces one encounters in practice are isomorphic to $\mathbb{R}$ as measurable spaces). The clearest exposition of conditioning in this case I have seen is in the book Testing Statistical Hypotheses by Lehmann and Romano. A close second is Probability and Measure by Billingsley.