Talk:Law of total expectation

Untitled

Perhaps it would be nice to include some text on why this conclusion is important, where it fits in the bigger picture. I don't know about that so there's no way I can do it, but I felt that was missing when I first read the page. Thanks to whoever does this.

Helder Ribeiro 20:26, 2 January 2007 (UTC)

That might take the form of examples of its use. Michael Hardy (talk) 17:50, 24 February 2008 (UTC)

If X and Y do NOT need to be independent then this should be explicitly stated. Taking Back Sunday (talk) 11:54, 24 February 2008 (UTC)

If they were independent, then the whole thing would be trivial and pointless. The fact that they're not assumed independent is needed in order for the article to have any substance. Michael Hardy (talk) 17:50, 24 February 2008 (UTC)

Limitation of Law of Total Expectation

Something is wrong in this section. I have nothing against its limitation but the example is wrong I think. What does E(T|Y=y) mean here? The expectation of T given the first sample is y or the expectation of T given all samples are y. If it's the first case E(T|Y=y) does not equal 2*2^y, if it's the second case P(Y=y) is not simply 1/(2*2^y). Enisbayramoglu (talk) 08:54, 15 December 2008 (UTC)

You're quite right. I'm going to delete the section. Interchanging the order of summation is not problematic when all terms are non-negative, regardless of whether the sum converges. See Tonelli's theorem. Michael Hardy (talk) 17:54, 15 December 2008 (UTC)

Example

This article needs one or more examples, please. New Thought (talk) 09:49, 22 November 2009 (UTC)

Which expectation?

I think it should be mentioned more clearly that the two expectations taken are of different variables. You first take the expectation of the non-conditioned variable and then take the expectation of the variable you are conditioning on. Without this being stated more clearly, the article is hard to follow.Eiad77 (talk) 20:57, 23 July 2010 (UTC)

Agreed: I think it should be written out in integral form as well to make sure it doesn't lead to confusion. Also, there should be a measure theoretic statement of the law. Superpronker (talk) 03:11, 1 June 2011 (UTC)

There is something seriously wrong in the section titled "Notational Shortcut". In the first equation you have a number on the left equal to a random variable on the right. I also have no idea what ostensive ambiguity the original author is trying to intimate; presumably both X and Y are defined on the same probability space with probability measure P.

I am also shocked to find that there is no mention of sigma fields on this page - the representation given here lacks far too much generality. I will fix this shortly if no one else does. — Preceding unsigned comment added by Untendo (talk • contribs) 03:26, 26 February 2014 (UTC)

I would point out that using the subscript notation ($\operatorname {E} _{Y}\left(\operatorname {E} _{X\mid Y}(X\mid Y)\right)$) for expectation is a bad idea until someone points the reader to a definition of this notation that actually works (not a definition of what it should mean). In fact, the section "Notation without indices" is conceptually wrong. It should read "the notation with subscripts is cumbersome, informal and used for didactic purposes. Subscripts are often omitted in favor of proper notation for conditional expectations." Omitting the indices is not a convention, it is the proper way of handling conditional expectations (see, e.g., Shorack's "Probability for Statisticians" for a formal definition of conditional expectation).

Discrete Proof

I would propose to use the exact factorization of an conditonal expectation, since its not obvious for everyone why P(y) gets cancelled and the cond. expectation becomes an unconditional one. Thanks — Preceding unsigned comment added by 132.199.65.210 (talk) 10:41, 12 September 2015 (UTC)

Example 2

I believe I can provide a better real life example and also connect it to other topics in finance and philosophy. I will write up something. Tormine (talk) 09:40, 23 February 2012 (UTC)

Iterated expectations with nested conditioning sets

I've never contributed to Wikipedia, so not sure if I am following the correct process here. I think the example in the "Iterated expectations with nested conditioning sets" is wrong, as it is not generally valid. The quoted reference in the Lindquist and Sargent book is about Markov processes, in which future states only depend on the current states, not on past states. This is of course not always true. It would be good if someone with more knowledge than me could confirm this and correct the article accordingly. — Preceding unsigned comment added by Jstlondon (talk • contribs) 18:55, 3 December 2012 (UTC)

Iterated expectations with nested conditioning sets

Hey, This section seems to be wrong, at least in the beginning. After: $\operatorname {E} (X\mid I_{1})=\operatorname {E} (\operatorname {E} (X\mid I_{2})\mid I_{1})$, it says that $I_{1}$ is determined by $I_{2}$. It should actually be the other way around, $I_{2}$ determines $I_{1}$, since the sigma algebra $\sigma (I_{1})$ must be within sigma algebra $\sigma (I_{2})$. As I'm not familiar with the economics example given here, I don't want to delete this section right away. If nobody suggests a better alternative (or clarify to me my lack of understanding), I will delete this section. — Preceding unsigned comment added by YurigCan (talk • contribs) 18:04, 26 August 2015 (UTC)

Iterated expectations with nested conditioning sets: I'm going to remove this section

1. I can't see how the source justifies the material in this section, and the author did not clarify for several years now.

2. The validity of this section had been called into question before. Two authors questioned the content of this section, but no response from the author ever arrived.

Notation without indices: this section is non-sensical and needs to be removed

The term $E_{X|Y}(X|Y)$ is non-sensical (the shortcut is either $E_{X|Y}$ or $E(X|Y)$; or, even more explicitly, $E_{X|\sigma (Y)}(\omega )$ or, if you prefer, $E[X|\sigma (Y)](\omega )$), and since this term is key to this section, the entire section is non-sensical too. StrokeOfMidnight (talk) 15:15, 4 August 2017 (UTC)

Notation for defined expectation

The proofs make the assumption that the expectation values be defined: $\min(\operatorname {E} [X_{+}],\operatorname {E} [X_{-}])<\infty$. But it isn't clear to me what $\operatorname {E} [X_{+}]$ and $\operatorname {E} [X_{-}]$ are. A link could be beneficial here. 129.240.43.144 (talk) 14:32, 6 September 2019 (UTC)

$X_{+}=\max(X,0)$ and $X_{-}=-\min(X,0)$.

This notation is universal in probability/statistics and analysis. See, for example, this article on Lebesgue integration. This article may still need to define $X_{+}$ and $X_{-}$ for clarity. StrokeOfMidnight (talk) 19:46, 7 September 2019 (UTC)

Naming of Adam's Law

AFAIK this naming comes the book Introduction to Probability by Blitzstein and Hwang and the name is used nowhere else. I'm inclined to believe the term should be removed from the page because it is not a term used anywhere else in statistics. Same for "Eve's Law" for Law of total variance. But maybe it should be kept for inclusivity. Wqwt (talk) 03:57, 15 September 2020 (UTC)

I totally agree. Both names are basically only used by people who took a very specific class at Harvard, taught by Blitzstein, where he used this name coined by a colleague of his, Morris. Altogether, it's basically a private joke/notation, very insular, which should not appear on Wikipedia. Clément Canonne (talk) 10:47, 23 February 2022 (UTC)

I removed the name with this justification. My edit was then reverted (without any justification) by an anonymous user (I have just... re-reverted it?). I would like to know the rationale, if any, for trying to keep this name (Adam's law) on this page. Clément Canonne (talk) 10:37, 1 March 2022 (UTC)

There currently is a back-and-forth of edits to add/remove this name from the page, where the "adding back" edits provide no justification as to whether this name is used anywhere outside this class, and in publications by the professor who coined the term as a joke. Unless such rationale is given (or convincing references to back up the common use of this term -- Googling does not provide any such evidence), I see no reaosn to include it, and would argue it'd be a very bad precedent for Wikipedia. (Anybody giving a nickname of their choosing to various mathematical concepts, including them in their lecture notes, and adding them to Wikipedia.) Clément Canonne (talk) 09:04, 3 March 2022 (UTC)

Confusing inclusion

In the section Proof in the finite and countable cases, the special case with events $A_{i}$ is inserted as the second sentence, but this special case is not proven here but rather, it is proven two sections later in the section Proof of partition formula.

On the other hand, in the section Proof of partition formula, there is no statement of what we are going to prove. This is very confusing.

83.173.212.145 (talk) 11:49, 19 June 2023 (UTC)