Talk:Regular conditional probability

If I didn't feel too rushed right now, I'd immediately start getting rid of all or nearly all the inline TeX in this article in favor of non-TeX notation. Michael Hardy (talk) 06:42, 18 January 2009 (UTC)

Yes, I know, boys

It's a big mess, and it will take a long time to clean up. Deepmath (talk) 02:07, 25 July 2009 (UTC)

Let $(\Omega ,{\mathcal {F}},{\mathfrak {P}})$ be a probability space, and let $T:\Omega \rightarrow E$ be a random variable, defined as a measurable function from $\Omega$ to its state space $(E,{\mathcal {E}}).$

We may also define a regular conditional probability for an event A given a particular value t of the random variable T in the following manner:

${\mathfrak {P}}(A|T=t)=\lim _{U\ni t}{\frac {{\mathfrak {P}}(A\cap U)}{{\mathfrak {P}}(U)}},$

where the limit is taken over the net of open neighborhoods U of t as they become smaller with respect to set inclusion. This limit is defined if and only if the probability space is Radon, and only in the support of T, as described in the article. This is the restriction of the transition probability to the support of T. To describe this limiting process rigorously:

For every $\epsilon >0,$ there exists an open neighborhood U of t, such that for every open V with $t\in V\subset U,$

$\left|{\frac {{\mathfrak {P}}(A\cap V)}{{\mathfrak {P}}(V)}}-L\right|<\epsilon ,$

where $L={\mathfrak {P}}(A|T=t)$ is the limit. Deepmath (talk) 02:17, 29 July 2009 (UTC)

I think it is still quite a mess:

It should be noted, that though such function ν exists by the Radon–Nikodym theorem, it is not unique. It is unique only modulo the pushforward measure ${\mathfrak {P}}(T^{-1}(\cdot ))$. That is, ν can be any function in an equivalence class of ${\mathcal {E}}$-measurable functions which mutually differ on 0 ${\mathfrak {P}}(T^{-1}(\cdot ))$-measure sets. So for $x\in E$ with ${\mathfrak {P}}(T^{-1}(x))={\mathfrak {P}}(T=x)=0$, ν(x) may take more value (as far as ν is ${\mathcal {E}}$-measurable), and thus this concept does not really overcome the difficulties in defining the conditional probability as a specific number for a 0-measure event, only formalize it with a certain terminology. That is, the value of ν is not necessarily determined for all $x\in \mathrm {supp} T$ (though it is surely determined for atoms of ${\mathfrak {P}}(T^{-1}(\cdot ))$). Defining the conditional probability as a specific number in general for all 0-measure events in the Kolmogorov setting seems to be impossible, see Borel-Kolmogorov paradox.

By the way, to speak about support in E, E should be topological space, but it is not required in the Definition.

Also, what does "Borel-measurable function" refer to here? A random variable is defined as a $({\mathcal {F}},{\mathcal {E}})$-measurable function. Does Ω have to be a topological space with Borel-σ-algebra which has to be subset of ${\mathcal {F}}$? Why? 80.98.239.192 (talk) 15:40, 3 November 2013 (UTC)

Standard probability space, is it Radon?

As far as I know, only a topological space may be Radon or non-Radon; if no topology is given on a space, the term "Radon" is not applicable to it. A probability space is a set endowed with a sigma-algebra and a measure, not a topology. Also a standard probability space is not endowed with a topology. This is why one cannot ask whether it is Radon or not. However one can ask whether it can be topologized appropriately; the answer is affirmative. Strangely enough, Deepmath disagrees and attacks the corresponding reservation made by me (and even calls me "rabbi" for this reason; what could it mean?? see my talk page). Anyway, I revert, thus starting an edit war. Sorry. Boris Tsirelson (talk) 06:30, 26 July 2009 (UTC)

Stop trying to "complete" my regular measures. Deepmath (talk) 04:28, 29 July 2009 (UTC)

Now it is more or less OK with me, except for the claim (in the table) that pathological cases exist for a standard probability space. No, they exist only for non-standard probability spaces. Boris Tsirelson (talk) 06:45, 29 July 2009 (UTC)

Non-regular conditional probability

Is there a common example of a non-regular conditional probability (i.e. a conditional probability that isn't regular)? I've looked in a couple of probability books, but have never found one. —3mta3 (talk) 08:29, 29 July 2009 (UTC)

If you want an example of a non-regular conditional probability in a situation where a regular conditional probability exists, look at Section 4.4 of Conditioning (probability). If you prefer a situation where a regular conditional probability does not exist, see Section 3.2 of Standard probability space. There, in Sect. 7.1, you can also find a ref: "see Durrett 1996, Sect. 4.1(c)". Boris Tsirelson (talk) 09:03, 29 July 2009 (UTC)

Ridiculously, the "Alternate definition" in THIS article is an example of a NON-REGULAR conditional probability. Let U be a random variable distributed (say) uniformly on (0,1). Let us calculate conditional probabilities, given (say) U = 0.5, by the limiting procedure proposed there. Then the conditional probability of the event 0.5 – ε < U < 0.5 + ε is equal to 1 for every positive ε, however, the conditional probability of the event U = 0.5 is equal to zero! This is just failure of regularity. It is also ridiculous that I did not note it before. Thanks to your question, I did now. Boris Tsirelson (talk) 18:19, 30 July 2009 (UTC)

Cheers, thanks for your help. I think the "alternative definition" might actually be ill-defined (e.g. look at the example on Borel-Kolmogorov paradox). —3mta3 (talk) 07:53, 31 July 2009 (UTC)

Rather, the "alternative definition" does give a conditional probability (though, open neighborhoods should probably be replaced with intervals), defined almost sure; the only problem is that it is not regular (which already manifests itself in that it is defined only ALMOST sure). Boris Tsirelson (talk) 09:10, 31 July 2009 (UTC)

Ah yes, I think I see what you mean (so U should be a member of the sigma-algebra on which we are conditioning?). Would it be better to use a definition like "a regular conditional probability is a conditional probability $\nu :E\times {\mathcal {F}}\to [0,1]$ such that $\nu (x,\cdot )$ is a probability measure for all $x\in E$"? —3mta3 (talk) 15:27, 31 July 2009 (UTC)

"U a member of the sigma-algebra"? I'd say, "U generates the sigma-algebra". Yes, "your" definition is easier to understand, and it is equivalent to the "official" definition under a very reasonable technical condition: the "big" sigma-algebra must be countably generated. Boris Tsirelson (talk) 17:16, 1 August 2009 (UTC) But probably you mean $\nu :\mathbb {R} \times {\mathcal {F}}\to [0,1],$ or else it is just the "official" definition. Boris Tsirelson (talk) 17:20, 1 August 2009 (UTC)

Comparison of spaces

What does "$\mu (\mathbb {Q} \cap [0,1])$ is undefined" in the table at the end exactly mean? $\mathbb {Q} \cap [0,1]$ is Borel-measurable and therefore can be assigned a value (for the Lebesgue-Borel measure, the value is 0). Patschy (talk) 05:12, 23 August 2012 (UTC)

You are right: the table is a nonsense (as well as most of this article). This table is a strange Point-Of-View of the strange banned User:Deepmath trying to convince everyone that "Standard probability space" is "Extremely complicated and weak" while "Radon space" is "Simple and powerful" (clearly, he was acquainted with Radon space but not measure space). Boris Tsirelson (talk) 05:40, 23 August 2012 (UTC)

Other articles

Presently there are quite a few articles that attempt to deal with conditional probability and also regularity, with varying levels of quality:

• conditioning (probability)
• conditional probability
• conditional probability distribution
• regular conditional probability
• conditional expectation

etc. It seems quite arbitrary which contains what. I think (some of) these should be reorganized, unified, and probably combined and merged into fewer articles. 80.98.239.192 (talk) 15:41, 3 November 2013 (UTC)

I second this, although a decade has passed. I see the following drawbacks:

1. If this article to exist, it must focus on regularity and why it is important.
2. Poor story-telling. For example, instead of giving separate definitions for discrete and a.c. variables, a general definition must be provided and followed by the special expressions.
3. A better explanation is needed for why "[these probability are] called a Markov kernel". It is not clear if there is any difference between the regular conditional probability and the Markov kernel, and if there is none, why the same concept comes under two different names.

AVM2019 (talk) 17:30, 28 October 2022 (UTC)

¿x0 = 2/3 or x0=3/2?

Hey, anyone. I really don´t know much about measure theory, but I got into this page wondering what exactly goes on when moving from the definition of a conditional density in the discrete case, were you are conditioning on an event with probability > 0, to the continuous case, were you are conditioning on an event with probability zero. Now, from the example on the page, I understand that this is allowed since it's some sort of limit behaviour; but in the last couple of lines it reads that it's meaningless to assign a conditional density when x0=3/2... I thought we were considering the case when x0=2/3, as was said in the motivational example? So is it x0=3/2 or x0=2/3? If you can answer, thank you. 201.141.171.20 (talk) 05:09, 3 September 2014 (UTC)

x0=3/2 indeed, since the point is, "this limit fails to exist for x0 outside the support of X"; 2/3 is within the support, 3/2 is outside. Boris Tsirelson (talk) 06:03, 3 September 2014 (UTC)

Which regularity condition is trivial

Quote:

For working with $\kappa _{Y|X}$, it is important that it be regular, that is:
# For almost all x, $A\mapsto \kappa _{Y|X}(x,A)$ is a probability measure
# For all A,  $x\mapsto \kappa _{Y|X}(x,A)$ is a measurable function
In other words $\kappa _{Y|X}$ is a Markov kernel.
The first condition holds trivially, but the proof of the second is more involved. It can be shown that if Y is a random element $\Omega \to S$ in a Radon space S, there exists a $\kappa _{Y|X}$ that satisfies the measurability condition. It is possible to construct more general spaces where a regular conditional probability distribution does not exist.

The statement in bold is false. The contrary is true: $\kappa _{Y|X}$ satisfies the measurability trivially by the definition of the conditional expectation. This should be corrected and supplied with an explanation for why the first condition is not guaranteed to be satisfied. Such an explanation can be found in many texts (in a nutshell, the condition holds for a fixed combination of events A_1, ..., A_k for almost any x; however the sets of exceptional x may accumulate to a non-null set, when one considers all possible combinations of events A). AVM2019 (talk) 13:11, 29 November 2022 (UTC) AVM2019 (talk) 13:11, 29 November 2022 (UTC)

References

Sufficient Conditions for Existence of Regular Conditional Probability

This pages seems to suggest that regular conditional probability distributions exist for random variables valued in a Radon space, but the terminology on Radon spaces can be ambiguous. The best I've been able to prove (as well as find a reference for) is that the precise conditions are that the space be a separable metrizable Radon space (i.e. a separable metrizable space on which every Borel probability measure is tight/inner regular). I think some clarification here would be nice. The same applies for the page on Disintegration.

Source: Ambrosio, L.; Gigli, N.; Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. ETH Zürich, Birkhäuser Verlag, Basel. ISBN 978-3-7643-2428-5. AJ LaMotta (talk) 14:31, 16 May 2024 (UTC)