Talk:Conditional expectation

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[edit] One of the worst

Um, this has to be one of the worst Wikipedia pages I've ever read. It is written like a probability theory lecture or a textbook, not an encyclopedia article. For example, "In order to handle the general case, we need more powerful mathematical machinery." has no place here. I've studied significant amounts of math and statistics, and it's somewhat hard for even me to read, I'd imagine it is completely inaccessible to a layperson. Compare, for example, to the article on conditional probability. And at the same time, it attempts to explain things like discrete random variable right there. A complete (ok, you can keep the first paragraph) rewrite is needed.Zalle 18:41, 26 March 2007 (UTC)

I agree; this page is extremely arcane, and could use a major rewrite. Cazort (talk) 17:59, 30 March 2008 (UTC)
Definitely it needs improvement, but the early section titled "special cases" shouls be easy to read for anyone who's had "significant amounts of math and statistics". But the topic of the article necessarily requires that at least some of what is said will not be accessible to the "layperson". Michael Hardy 19:21, 26 March 2007 (UTC)
And the first sentence really should be accessible to anyone (except those who don't know what conditional probability distributions are, and it's not appropriate to ask that this be made accessible to those people, since "conditional probability distribution" is quite naturally a prerequisite to this topic), and for some purposes, says most of what need to be said. Michael Hardy 19:24, 26 March 2007 (UTC)
Well, looking at your "revert" I think you've failed to comprehend the difference between a mathematics textbook and an encyclopedia. "It can be shown that" is nothing but silly mathematical jargon and does not "signify that there is more to this argument". As this is an encyclopedia, it is by definition true (at least in an ideal situation) that all the claims contained can be "shown" to be true, and it is obvious without stating it that such a proof exists. I'd argue and edit more, but apparently this is a pet project so I don't think it's really worth the bother. Have fun, good luck.88.112.25.211 21:49, 26 March 2007 (UTC)

You seem to misunderstand. Yes, it is obvious that such a proof exists, but it may not be obvious that there is more to the proof than what is given here. Michael Hardy 22:40, 26 March 2007 (UTC)

[edit] Reworking suggested

This page needs substantial reworking.

  • Motivation involving finite probability spaces
  • Intuitive generalizations
  • CLear statements of abstract framework general theorems and the general probability framework
  • Cond Exp as factorization (important for defining sufficient statististics)
  • References.

If nobody objects, I'll do it in the next few hours CSTAR 15:45, 9 May 2004 (UTC)

Go for it. Charles Matthews 15:58, 9 May 2004 (UTC)


I'm still working on it. But I'd like to get some stuff out so I can ponder more on this. If I've screwed things up, please tell me. CSTAR 21:31, 9 May 2004 (UTC)

This article is terrible. It's fairly well written for an article addressed to mathematicians who know a bit of measure theory and have a bit of intuition for probability. Therefore, it's terrible. Obviously the main ideas can be stated simply in a way that can be understood by someone who knows only as much probability as can be understood without knowing even calculus. Well, it's not as much of an Augean stable as some things on Wikipedia, so maybe I'll do something with it at some point. Michael Hardy 01:55, 8 Oct 2004 (UTC)

I'm perfectly willing to believe the article is terrible...but does your argument really establish that it's terrible? Too abstract yes, not enough intuition yes etc etc. Please be more specific about what you think should be done with it, whether the abstract stuff should be removed etc. I'd be somewhat unhappy if conditioning as projection were to be removed, since without this it is hard to talk about martingales etc., but hey I won't lose any sleep over it. But simply concluding in an abrupt non-sequitur that it's terrible isn't very helpful!CSTAR 02:19, 8 Oct 2004 (UTC)

I would not remove the abstract stuff, but I would attempt to make the article comprehensible to everyone who understands the basic definition of conditional probability, not just to mathematicians who know, e.g., the Radon-Nikodym theorem. Even the definition of E(X | Y) for continuous random variables X and Y can be clearly stated in such terms if you don't require examination of the sort of issues addressed only in measure theory. Mathematical rigor is important in its place, but so is communication. I'll return to this when I've got some time. Michael Hardy 20:29, 8 Oct 2004 (UTC)

I have to say - the 'intuitive' explanation always went right by me. Charles Matthews 21:15, 8 Oct 2004 (UTC)

OK, here's something from another Wikipedia article:

The conditional expected value E( X | Y ) is a random variable in its own right, whose value depends on the value of Y. Notice that the conditional expected value of X given the event Y = y is a function of y (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!). If we write E( X | Y = y) = g(y) then the random variable E( X | Y ) is just g(Y). Similar comments apply to the conditional variance.

Charles, is that the intuitive explanation that went by you? Michael Hardy 00:01, 9 Oct 2004 (UTC)

Is the previous paragraph an example of a clear explanation? Is it too impolitic to say it doesn't seem to me to be hardly an improvement? I'm also curious as to what you would point to as being more of an Augean stable in wikipedia, though I do agree that there are many, many articles which I think fit this bill.CSTAR 00:28, 9 Oct 2004 (UTC)

Guys, I think everyone's ambitions here are compatible, at least. Charles Matthews 08:52, 9 Oct 2004 (UTC)

[edit] 1_{...}

What does the 1 in the E(X 1_{...}) notation stand for?

1_A is the indicator function of A. --CSTAR 17:38, 14 Apr 2005 (UTC)
..., and, in the context of probability theory, 1A can be defined as a random variable that is equal to 1 if the event A occurs and is equal to 0 if the event A fails to occur. Michael Hardy 18:23, 14 Apr 2005 (UTC)
But what does E(X f) mean, f being a function? Does it mean the E of the function composition of X and f? (http://www.stats.uwo.ca/courses/ss357a/handouts/cond-expec.pdf uses a completely different formula)

[edit] Split into sections?

Does anybody else feel as if the table of contents is too down in the article, and some more splitting in sections could be done at the top? I don't know what is a good way of splitting it myself. Oleg Alexandrov 20:07, 14 Apr 2005 (UTC)

Yeah I agree; but don't look at me for changes..CSTAR 20:44, 14 Apr 2005 (UTC)


[edit] Proofs

===Can somebody provide the profes for conditional expectation is contractions? or should the reference for the profes be provided? I can add a good profe for Jensen's Inequality later. —Preceding unsigned comment added by Pondyyuan (talk • contribs) 18:09, 31 December 2006

Please describe what statement you want to prove regarding contractions in more detail. Jmath666 18:32, 28 March 2007 (UTC)

[edit] Pretty good, proposal to make it even better

This article is pretty good, surely better than what I could find in any book I looked and I found it of great help when I needed to clarify this stuff. Compared with the treatment in the classical books, such as Feller, Varadhan, Levy,.. I found it really lucid. I ended up writing notes for myself and few others, which are hopefully even more lucid and some may find more satisfactory. Any comments welcome.

I plan on merging the notes with the article in future. For now, my original lives in LaTeX so any edits will be overwritten next time when I run the translator. Jmath666 22:44, 27 March 2007 (UTC)

[edit] Merge with Conditional distribution

There does not seem to be a need for separate Conditional distribution article, that concept should be defined here anyway. The current Conditional distribution article is elementary and incorrect anyway. This could be done separately or in conjunction with the proposal above. Jmath666 18:27, 28 March 2007 (UTC)

There is now draft of the merged page. The original still lives in LaTeX and is not ready for public editing, that's why it is in user space. Jmath666 15:50, 29 March 2007 (UTC)

Methink they are quite different objects, so they could stay on separate pages. Conditional distribution may be further developed.User:unregistered user

I agree, separate pages would be better, though conditional distribution does need a bit of development.GromXXVII 12:17, 1 November 2007 (UTC)
Aren't they two completely different things??? They are to me. I have some sort of understanding what conditional probability is, i.e. Pr(y|x), but I've almost no idea what a conditional pdf p(y|x) might mean, which is why I'm looking it up. I know I'm pig ignorant, but presumably so are 99.9% of Wikipedia's users -- or they wouldn't be consulting this fount of wisdom. Please remember the ordinary users. --84.9.83.26 (talk) 20:46, 14 December 2007 (UTC)

I agree, pages should not be merged. A conditional mean is just one part of the many aspects related to conditional distributions. A distribution is the starting point for all random variables. Then there are conditional distributions. For conditional distributions there are conditional means, variances etc. But I am not a statistician or mathematician either. 12 February 2008

[edit] Composition

The diagram

  -------- X --------> 
Ω                      R
  --Y--> U --E(X|Y)-->

you draw is not correct, because also E(X|Y): Ω --> R. It should look like:

  -------- X --------> 
Ω                      R
  ---Y---> U ----g--->
  ---E(X|Y) = g(Y) -->

Sorry, wasn't logged in.Nijdam 11:51, 25 April 2007 (UTC)

[edit] properties

properties of conditional expectation should be list in the article. We can find them in any textbook. such as E(X|X)=?, tower rule, independent rule, ... Jackzhp (talk) 20:46, 11 March 2008 (UTC)

[edit] Relation to estimators

I would like to request someone adding material on using conditional expectation to "improve" estimators (like in the Rao-Blackwell theorem) to this page; I would add the material, except that I feel I don't understand it 100%. Cazort (talk) 17:58, 30 March 2008 (UTC)