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Talk:Conditional expectation

Errors
This article has several significant mathematical errors. In the 'Conditioning as factorization' section, X must take values in a field so that the integral makes sense at all. And X and Y do not need to be both U-valued. In fact, U doesn't have to be a field, and so the current definition is broken. I suggest making X real valued, leaving Y U-valued, and patching things from there. —Preceding unsigned comment added by 140.247.149.155 (talk) 14:19, 14 February 2009 (UTC)

 * I have tried to fix this error. --Bdmy (talk) 16:27, 14 February 2009 (UTC)

 * Looks good, thanks. —Preceding unsigned comment added by 140.247.142.164 (talk) 06:05, 16 February 2009 (UTC)

 * The section 'Conditioning as factorization' is still really problematic. Defining E(X|Y) as a mapping over U is wrong, it is a random variable on the probability space (Omega,F,P). This section should simply mention that there exists a function g over U such that E(X|Y) = g(Y), the latter being the composition of g with Y, thus defined over Omega. Note that this is currently redundant with the previous section. 132.169.4.223 (talk) 14:53, 28 June 2017 (UTC)

 * Yes. See "" below. Boris Tsirelson (talk) 18:54, 28 June 2017 (UTC)

I think, at the end of Section 'Conditioning relative to a subalgebra', $$ L^2_{\operatorname{P}}(X;M) \rightarrow L^2_{\operatorname{P}}(X;N)$$ should be $$ L^2_{\operatorname{P}}(\Omega;M) \rightarrow L^2_{\operatorname{P}}(\Omega;N)$$. Right? 80.98.239.192 (talk) 17:10, 2 November 2013 (UTC)
 * Right! Corrected. Thank you. Boris Tsirelson (talk) 20:56, 2 November 2013 (UTC)
 * Thank you. 80.98.239.192 (talk) 22:47, 2 November 2013 (UTC)

Does anybody agree to put the fact that conditional expectation is the best regression in the mean square error sense? --Memming (talk) 15:51, 20 September 2008 (UTC)
 * This has been done since 2008. AVM2019 (talk) 13:58, 23 May 2022 (UTC)

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)

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)

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)

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)

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)

 * Is not contraction an obvious application of Jensen's inequality for $$f(x)=|x|^s$$? (Unconditional) Jensen's inequality has already some proof on Jensen%27s_inequality. 22:59, 2 November 2013 (UTC) — Preceding unsigned comment added by 80.98.239.192 (talk)

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)

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

I would rather not merge the pages too. I think it's clearer to have separate, shorter pages that link to each other appropriately. That's the point of WP: Wikify. Cazort (talk) 05:16, 14 February 2009 (UTC)

 * Presently there are quite a few articles that attempt to deal with conditional probability (and expectation), with varing 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. Yes, conditional probability and conditional expectation are very closely related concepts, e.g. through $$\mathop{P}(A|...)=\mathop{E}(\mathbf{1}_A|...)$$. 80.98.239.192 (talk) 13:01, 3 November 2013 (UTC)

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)

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)

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)

Connecting formal definition back to common usage
Regarding recent edits by Eclecticos: The "formal definition" is now a mixture of the formal definition and some discussion (even before the "discussion" subsection). In addition, the discussion is not always correct. About the "dividing by zero" problem in this context, see also Conditioning (probability). Boris Tsirelson (talk) 11:21, 5 January 2010 (UTC)

 * The common notation P[X|Y∈S] also has to be formally defined. What you characterize as "discussion before the discussion subsection " was an attempt to do this.  However, as my log comments indicated, I realized that this attempt was incorrect and have removed it.  More discussion of the issue is immediately below. Eclecticos (talk) 03:21, 6 January 2010 (UTC)

The article doesn't explain how we get from the formal $$\scriptstyle \operatorname{E}[X|\mathcal{B}]$$ back to the common notation for conditional expectation as used in the introduction.

Last night I tried to fix that, adding this text to Conditional expectation:

 * Note that $$\scriptstyle \operatorname{E}[X|\mathcal{B}]$$ is simply the name of the conditional expectation function. Given this function, we can compute specific conditional expectations such as E[X|Y=y] — in general, we define
 * $$\operatorname{E}[X|Y\in S] = \int_{Y^{-1}(S)} \operatorname{E}[X|\mathcal{B}](\omega)\ \operatorname{d}\omega$$
 * provided that $$\scriptstyle Y^{-1}(S) \in \mathcal{B}$$.

Here I was trying to remove the measure of Y-1(S) from the integration, replacing dP(ω) with just dω. However, I quickly realized that there was something wrong with that, and so have removed the second sentence. The problem is that I don't think we can integrate directly with respect to dω -- we need some kind of measure for integration. And I don't think we yet have a probability measure on the set Y-1(S); we can't just renormalize the P measure over that set if its original measure under P is 0.

So in fact, how do we define E[X|Y∈S]? I am tempted to say that we really should define conditional probability as a ratio, and handle the 0-denominator case by taking a limit. In fact, the Jaynes book quoted in the Borel-Kolmogorov paradox article suggests that we do just that (the relevant pages are on Google Books), with different ways of taking the limit giving different answers. That is also what the conditional probability article says, and it is hinted at by Conditioning (probability) as well.

However, the introduction to the present conditional expectation article motivates the whole conditional expectation trick and formalism as a way of solving the division by 0 problem, presumably without any stinkin' limits. And the Borel-Kolmogorov paradox article also suggests that the resolution to the division by 0 paradox can be found here in this article. So I am confused. Do the introduction and Borel-Komogorov article make false promises?

 * Where does that article suggest this? It does not even link here now. 80.98.239.192 (talk) 21:31, 2 November 2013 (UTC)

If we want a limit-free solution to the paradox, I would really have expected a different one. Specifically, I would have expected that we would start from conditional probability and be required to define not just a single probability measure P over the measurable subsets S of the sample space Ω, but rather an appropriately consistent family of probability measures PS where each PS is a probability measure over the measurable subsets of S. In particular, we would have some freedom to define PS when PΩ(S) = 0.

Notice that that approach (see Rao (1993) citation below for details) starts from conditional probability. The article starts instead from conditional expectation (as Kolmogorov did). Here's my concern. The article gives us freedom in defining $$\scriptstyle \operatorname{E}[X|\mathcal{B}]$$ on sets of measure 0, but the article seems to allow us to exercise that freedom differently for $$\scriptstyle \operatorname{E}[X|\mathcal{B}]$$ and $$\scriptstyle \operatorname{E}[Z|\mathcal{B}]$$, say, even if Z=X/2. Can't that lead to inconsistencies?

Eclecticos (talk) 03:10, 6 January 2010 (UTC)

 * First of all, there is no hope to define P(A|B) in general, when P(B)=0. This sad fact is the point of the Borel-Kolmogorov paradox. And in particular, there is no hope to define $$E(Y|X\in A)$$ when $$P(X\in A)=0.$$ Boris Tsirelson (talk) 06:39, 6 January 2010 (UTC)

 * Of course p(A|B) can be defined by stipulation, subject to certain requirements. The "Formal Definition" section of the article appears to be stating these requirements.  It says that you can define "a conditional expectation" of a random variable X (not necessarily unique!), conditioned on the sets of "a sub-σ-algebra $$\scriptstyle \mathcal B \subseteq \mathcal A$$".  Nothing in the article says that this algebra has to exclude sets of measure 0.  In fact, on the contrary, the conditional probability article points here with a suggestion that this article will provide a way of defining conditional probabilities that condition on sets of measure 0.

 * My question about this approach is given above: if I want to define two conditional expectations $$\scriptstyle \operatorname{E}[X|\mathcal{B}]$$ and $$\scriptstyle \operatorname{E}[Z|\mathcal{B}]$$, shouldn't I be forced to choose both at once in a way that makes them consistent with each other? If Z=X/2 (this is meaningful since Z and X are functions on the probability space), then I am bothered that the article does not force $$\operatorname{E}[Z|B] \neq \operatorname{E}[X|B]/2$$ for all $$B \in \mathcal{B}$$ (in particular, it does not for B of measure 0).  Is this a flaw in Kolmogorov's axiomatization, or an error in the article?

 * (Could someone try to answer the immediately preceding question, which is still bothering me?) Eclecticos (talk) 06:23, 23 August 2010 (UTC)

 * The problem is certainly solvable, as I wrote above ("If we want a limit-free solution...").  Namely, one can replace Kolmogorov's axiomiatization of probability measures with an axiomatization of conditional probability measures.  In this case, all the conditional probabilities (and conditional expectations!) are defined directly and consistently by the conditional measure from the start.  The authoritative book on this approach, I believe, is Rao (1993), "Conditional Measures and Applications," who attributes the basic ideas to Rényi (1955) although he develops them further.

 * To be clear, the approach is well known. For example, the undergraduate textbook Makinson (2008, pp. 165-166) says that "This approach is popular among philosophers." Eclecticos (talk) 04:07, 16 September 2011 (UTC)

 * By the way, Jaynes (2003) -- who founds probability theory on rather a different basis but (as he says in his preface) ends up agreeing that Kolmogorov's axioms are correct if not necessarily complete -- also takes conditional probability to be fundamental, unlike Kolmogorov. He complains in his Appendix A that "The Kolmogorov axioms make no reference to the notion of conditional probability; indeed, KSP finds this an awkward notion, really unwanted ... In contrast, we considered it obvious from the start that all probabilities referring to the real world are necessarily conditional on the information at hand."  (As noted in the Borel-Kolmogorov paradox article, Jaynes does have his own views on how to commonsensically choose a definition of the conditional probability in this case: "Whenever we have a probability density on one space, and we wish to generate from it one on a subspace of measure 0, the only safe procedure is to pass to an explicitly defined limit by a process like (15.55).")  Eclecticos (talk) 05:16, 30 June 2010 (UTC)

 * About the limiting procedure, its merits and demerits are discussed in Conditioning (probability), see especially "The limiting procedure" there. See also Talk:Regular conditional probability. Boris Tsirelson (talk) 06:39, 6 January 2010 (UTC)

 * See also Conditional probability (not quite well-done, though) and Talk:Conditional probability. Boris Tsirelson (talk) 06:44, 6 January 2010 (UTC)

 * I think the Rao (1993) axiomatization should be mentioned as an alternative in the above articles, no? Eclecticos (talk) 05:17, 30 June 2010 (UTC)

 * If you believe that conditional probability makes sense in all cases, then please consider the following two questions.
 * First. Let U be a random variable distributed uniformly on [0,1]. Find the conditional probability of U=0.4 given that U is a rational number.
 * Second. Let $$U_1,U_2,\dots$$ be independent random variables distributed uniformly on [0,1]. Find the conditional probability of $$U_{10} > 0.5$$ given that $$U_n\to1$$ as n tends to infinity.
 * Boris Tsirelson (talk) 10:28, 1 July 2010 (UTC)

 * Sure, these are natural questions about Rao's axiomatization (please look at it before answering, in section 4.2 of his book on Google Books). Let me focus on your first question since it's simpler.  The answer is that this conditional probability might be undefined.  It is defined only if the set of rationals $${\mathbb Q}$$ is an element of the class of conditions $${\mathcal B}$$.  (Not all sets are conditions, just as not all sets are measurable.)  So I have to ask you to make your question more precise: what do you mean by "distributed uniformly" when your distribution is to be specified as a two-place function $$P(X\mid Y)$$ satisfying Rao's conditions?  The class $${\mathcal B}$$ will be specified as part of your definition of the distribution.  Eclecticos (talk) 06:23, 23 August 2010 (UTC)

 * (For any reasonable definition, the answer to your question is indeed "undefined." Why?  There are many ways to extend the classical uniform distribution (along with the conditional distributions derived from it) to a Rao-style family of conditional distributions.  These extensions differ in the conditional probabilities $$P(X\mid Y)$$ that they assign where Y is a set of measure zero.  They could for example be chosen to reflect different limiting procedures, in the sense of the Borel-Kolmogorov paradox.  However, it is easy to see that any extension that allows you to condition on the countably infinite set $${\mathbb Q}$$ cannot possibly treat the elements of $${\mathbb Q}$$ symmetrically -- so neither of us would be very happy using the name "uniform distribution" for any such extension!  If we really want to allow $${\mathbb Q}$$ in the denominator, we could perhaps get away with it by forbidding singleton sets in the numerator (in other words, choose a smaller sigma-algebra of measurable sets: I suspect this can be made to work out for the sigma-algebra generated by non-null intervals).  But then this is no longer an extension of the classical uniform distribution.  Either way -- regardless of whether we ban {0.4} from the numerator or $${\mathbb Q}$$ from the denominator -- the answer to your question would be that $$P(U=0.4 \mid U \in {\mathbb Q})$$ is not defined.) Eclecticos (talk) 06:23, 23 August 2010 (UTC)

 * "please look at it before answering" — no, sorry, I know it is not very polite, but I did not. Your answer convince me that the Rao-style theory is not a really important progress in the theory of conditioning, and so, not worth of my time. It is probably a reformulation of what I know in a different language. Of course, this is just my point of view; different people have different idea of "really important progress". Boris Tsirelson (talk) 11:24, 23 August 2010 (UTC)

 * Yes, the idea is to better formalize what you "know"! Your current (standard) formalization gives rise to the Borel-Kolmogorov paradox; whereas this one doesn't because the conditional probabilities are defined directly with no limiting procedure needed.  I brought up Rao not to claim his notability, but because he appears to solve an apparent problem with the formalization in the current wikipedia article.  I don't know yet whether you agree or disagree that it is a problem, but I gave a concrete example of it.  See above plea "Could someone try to answer ..."?

 * Rao's axiomatization is indeed obvious (and very short) and based on a 1955 proposal of Rényi, whom you surely respect. It permits you to condition on specified sets, including sets of measure zero.  Conditioning on sets of measure zero is necessary if you want to permit counterfactual reasoning, i.e., relativize to a world that you believe doesn't actually exist.  By defining an obvious notion of "conditional measure" and making you work with a single conditional measure, Rao ensures that all the conditional probabilities and conditional expectations within any world (including a world of measure zero) will be consistent with one another (in the sense of satisfying ordinary identities), so that the counterfactual reasoning is consistent (in the sense of logic).

 * cf.: Counterfactual_conditional 80.98.239.192 (talk) 21:31, 2 November 2013 (UTC)

 * My concern stated earlier is that the framework for conditional expectation in the current wikipedia article does allow conditioning on sets of measure zero, but does not require any consistency (not even among conditional expectations with the same condition!). The limiting procedures discussed in other wikipedia articles have a similar problem: defining many conditional probabilities requires choosing many limiting procedures, and if those choices are free and independent, what ensures that the resulting conditional probabilities are consistent?  Eclecticos (talk) 12:50, 25 August 2010 (UTC)

 * Eclecticos: About inconsistency in case of Z=X/2: As I see, the situation is that $$\operatorname{E}[X|\mathcal{B}]$$ is defined as an (equivalence) class of $$\mathcal{B}$$-measureable functions (random variables), where the equivalence relation is that two function agree outside a 0-measeure set. The same for $$ \operatorname{E}[Z|\mathcal{B}]$$. Any representative of $$ \operatorname{E}[X|\mathcal{B}]$$ divided by 2 and any representative of $$\operatorname{E}[Z|\mathcal{B}]$$ will equal outside a 0-measeure set. Moreover, the class $$\operatorname{E}[X|\mathcal{B}]$$ consists of just the functions which are 2 times the functions in the class $$\operatorname{E}[Z|\mathcal{B}]$$. In this sense, $$\operatorname{E}[Z|\mathcal{B}] = \operatorname{E}[X|\mathcal{B}]/2$$. But it is not determined which functions we have to choose from the classes, and that the two chosen such function should equal for every ω. Only almost surely.

 * You write more times, that this article allows conditioning on 0-measure sets, e.g., defining $$\operatorname{E}[X|B]$$. On the contrary! This article does not mention $$\operatorname{E}[X|B]$$ for 0-measure B. (Though other articles, like Conditioning_(probability) does allow it for special distributions with a joint density.) It cannot be done for the general case in the Kolmogorov setting, as you also referred to the Borel-Kolmogorov paradox. (By the way, that article does not suggests that the resolution to the division by 0 paradox can be found in this article, as you write.) This one defines only the class of functions $$\operatorname{E}[X|\mathcal{B}]$$, where $$\mathcal{B}$$ is a σ-algebra.

 * Also I think this consistency has something todo with the regularity in Section 'Definition of conditional probability', which ensures that, e.g., for disjoint A,C ⊆ Ω and for all fixed ω∈Ω, the represatatives are chosen such that the linearity $$\operatorname{E}(\mathbf{1}_A + \mathbf{1}_C|\mathcal{B})(\omega) = \operatorname{E}(\mathbf{1}_A|\mathcal{B})(\omega) + \operatorname{E}(\mathbf{1}_C|\mathcal{B})(\omega)$$ holds.

 * Anyway, this article is supposed to be on the standard Kolmogorov axiomatization. There can be another article on conditional probability under Rényi-Rao axiomatization which is referenced from here as an alternative (as you wrote), but this one probably should not be extended to this not-(so)-standard direction. 80.98.239.192 (talk) 21:31, 2 November 2013 (UTC)

I think, Section 'Conditioning relative to a subalgebra' has basicly almost the same content as Section 'Formal definition', only with different terminology, e.g. M, N and B instead of $$\mathcal{F}, \mathcal{H}$$, and H, and a precise theorem instead of Discussion. So I do not see what "This version is preferred by probabilists" refers to. These two should be merged probably. I prefer having mathcal letter, but also the formal Theorem.

I do not understand why the notation for the basic σ-algebra and its sub-σ-algebra should change throughout the article. 80.98.239.192 (talk) 21:31, 2 November 2013 (UTC)

relation with Conditional probability
Since conditional probability is a special case of conditional expectation, it will be good to treat the conditional probability in this article, rather than an independent article. Jackzhp (talk) 19:56, 12 March 2011 (UTC)

Different Conditioning
We have defined condition expectation on a sigma field and on a (measurable) set, the latter one is very messy. Conditioning on a random variable has not been defined yet.

As for conditioning on a set, let's show P(A|B)=P(AB)/P(B)) satisfies the conditional expectation definition. Here, the sigma field is $$\left\{ \emptyset,\Omega,B,\bar{B}\right\}$$, so we have to show 4 equalities. Can someone please elaborate this? Jackzhp (talk) 16:10, 12 March 2011 (UTC)

Formal definition
Is the formal definition given actually correct? As stated, it says that that $$E[X|\Beta] = X$$ a.e. I'd imagine you should explicitly state what the who probability measures are, and clarify that E is a Radon–Nikodym derivative times the random variable. —Preceding unsigned comment added by 128.176.122.34 (talk) 11:52, 8 May 2011 (UTC)

I would also add perhaps: the formal definition of conditional expectation w.r.t a $$\sigma$$-algebra doesn't mention the $$\sigma$$-algebra defined on the random variable's output. If we assume that the random variable's $$\sigma$$-algebra is the Borel one, then the conditional expectation is unique (right?), and results like $$E(X | H) = X$$ (when X is H-measurable) make sense. However, if we say nothing about the output's $$\sigma$$-algebra, then the conditional expectation can have other values and results like the above aren't necessarily true. So perhaps it would be good to mention this (or mention that texts on stochastic processes and expectation often assume that the random variable is defined with a Borel $$\sigma$$-algebra. See: https://stats.stackexchange.com/questions/495562/how-to-understand-conditional-expectation-w-r-t-sigma-algebra-is-the-conditiona/495667#495667

Error
Regretfully, an incorrect paragraph was introduced by User:3mta3 on 14:55, 14 March 2009, and is unnoticed during almost 6 years. In the end of Section "Calculation" we see:
 * $$ \operatorname{E} (X | Y=y) \operatorname{P}(Y=y) = \sum_{x \in \mathcal{X}} x \ \operatorname{P}(X=x,Y=y), $$
 * and although this is trivial for individual values of y (since both sides are zero), it should hold for any measurable subset B of the domain of Y that:
 * $$ \int_B \operatorname{E} (X | Y=y) \operatorname{P}(Y=y) \ \operatorname{d}y = \int_B \sum_{x \in \mathcal{X}} x \ \operatorname{P}(X=x,Y=y) \ \operatorname{d}y. $$

This is a nonsense. Just "Since both sides are zero" in the former, the integrands are zero in the latter. Thus, we still have 0=0. The author of this paragraph believes naively, that integration is able to gather a continuum of zeros into a positive number. I understand his intuition, but no, it does not work this way. This formulation could be acceptable in Wikipedia of 18-th century, but not now. :-) Boris Tsirelson (talk) 12:13, 15 January 2015 (UTC)

Definition of conditional probability
Should it not be [0,1] instead of (0,1)? After all, A's probability could be zero or one. — Preceding unsigned comment added by Doubaer (talk • contribs) 08:11, 3 February 2015 (UTC)
 * Really, both are a nonsense. The indicator is a random variable. That is, measurable w.r.t. the sigma-algebra given on the relevant probability space. Now fixed. Boris Tsirelson (talk) 12:02, 3 February 2015 (UTC)

Integrability forgotten
Several times, conditional expectation is discussed without assuming integrability of the given random variable (while in fact it is essential). Boris Tsirelson (talk) 16:12, 30 December 2015 (UTC)

Agreed. Conditional expectation is defined on integrable random variables. A reference would be Patrick Billingsley's Probability and Measure. --Han (talk) 01:31, 8 January 2016 (UTC)

section 4.2 error?
In section 4 it is stated
 * $$\operatorname{E}(X\mid Y) = \operatorname{E}(X\mid\mathcal{H}) \circ Y $$,

while $$Y:\Omega \to U$$ and $$\operatorname{E}(X\mid\mathcal{H}) : \Omega \to \mathbb{R}^n$$. It seems to me that it should be
 * $$\operatorname{E}(X\mid Y) = \operatorname{E}(X\mid\mathcal{H}) \circ Y^{-1} $$

instead. — Preceding unsigned comment added by 178.37.84.106 (talk) 15:03, 22 June 2017 (UTC)

 * A mess indeed.
 * "Then the random variable $$g(Y)$$, denoted as $$\operatorname{E}(X\mid Y)$$, is a conditional expectation of X given $$Y$$."
 * So, what is this conditional expectation, is it the function $$g$$ or the random variable $$g(Y)$$? If it is $$g(Y)$$, then it is just $$\operatorname{E}(X\mid\mathcal{H})$$ (and not $$ \operatorname{E}(X\mid\mathcal{H}) \circ Y $$ nor $$\operatorname{E}(X\mid\mathcal{H}) \circ Y^{-1} $$). And then, Sections 4.1 and 4.2 deal with the same object, in slightly different notations. Or alternatively, if that conditional expectation is the function $$g,$$ then it is a (formally) different notion, and $$ \operatorname{E}(X\mid\mathcal{H}) = \operatorname{E}(X\mid Y) \circ Y = g \circ Y = g(Y). $$ And of course, all that should be understood up to equivalence (the equivalence being equality almost everywhere). Boris Tsirelson (talk) 19:00, 22 June 2017 (UTC)
 * A third option: define $$\operatorname{E}(X\mid Y=y)$$ as $$g(y)$$ for $$y\in U,$$ then $$ g = \operatorname{E}(X\mid Y=\cdot) = (y\mapsto\operatorname{E}(X\mid Y=y))$$ and $$ \operatorname{E}(X\mid\mathcal{H}) = \operatorname{E}(X\mid Y=\cdot) \circ Y = \operatorname{E}(X\mid Y=Y); $$ this "Y=Y" looks rather ridiculous, but really, the first Y denotes the random variable "as whole", while the second Y means the function $$Y(\cdot)$$; rather clumsy... Boris Tsirelson (talk) 19:39, 22 June 2017 (UTC)

"The definition of $$\operatorname{E}(X \mid \mathcal{H})$$ may resemble that of $$\operatorname{E}(X \mid H)$$ for an event $$H$$ but these are very different objects. The former is a $$\mathcal{H}$$-measurable function $$\Omega \to \mathbb{R}^n$$, while the latter is an element of $$\mathbb{R}^n$$.  Evaluating the former at $$H$$ yields the latter."

However, $$\operatorname{E}(X \mid \mathcal{H})$$ is a function $$\Omega \to \mathbb{R}^n$$ (which is explicitely stated in this paragraph); how can one evaluate this function at an arbitrary event $$H\in\mathcal{H}$$, when $$\mathcal{H}\neq \Omega$$? — Preceding unsigned comment added by 84.114.30.251 (talk) 21:10, 16 December 2018 (UTC)

 * You are right. If the σ-algebra $$\mathcal{H}$$ is finite (which is elementary case), then it corresponds to a finite partition of $$\Omega$$ into parts of positive measure, and on every part $$H$$ the function $$\operatorname{E}(X \mid \mathcal{H})$$ is constant (almost sure), and this constant is indeed $$\operatorname{E}(X \mid H).$$ But generally that function is not constant on $$H,$$ and $$\operatorname{E}(X \mid H)$$ is rather its mean (average) value over $$H$$ according to the given probability measure, provided that $$H\in\mathcal{H}$$ and $$P(H)\ne0.$$ Boris Tsirelson (talk) 22:28, 16 December 2018 (UTC)

 * This inconsistency has been resolved. "Evaluating the former at $$H$$ yields the latter." means taking an integral of the former over $$H$$. AVM2019 (talk) 13:54, 23 May 2022 (UTC)

title: Talk:Conditional expectation


text:
James Vashon

Admiral James Vashon (9 August 1742 – 20 October 1827) was a British officer of the Royal Navy. He saw service during the Seven Years' War, the American War of Independence and the French Revolutionary and Napoleonic Wars. He was first captain of HMS Dreadnought, between 1801 and 1802. Previously, he had commanded Alert (1781), Europa (1786), and Formidable.

Washington's Vashon Island was named after him by the explorer George Vancouver, who had been Vashon's subordinate in Europa. Vashon rose to the rank of Admiral of the White. He is reported to have been stationed for many years in the West Indies.

Family and early life
James Vashon was born on 9 August 1742 in the village of Eye, near Leominster, Herefordshire. He was the son of the Vicar of Eye, the Reverend James Volant Vashon. The family was descended from Peter Vashon, a Huguenot apothecary c. 1636, who had immigrated from Paris. The family name is of French origins and became English. James Vashon entered the Royal Navy at age thirteen, in 1755, and first served aboard the 28-gun frigate HMS Revenge (1755) under Captain Cornwall. As a midshipman he served on a number of vessels and saw combat service in the Mediterranean, the coast of North America, and in the Caribbean. In 1763 he passed his examination for lieutenant, but continued to serve as a midshipman until 1772 for lack of a lieutenant's posting. Vashon was serving on HMS Quebec (1760) when the Seven Years' War ended and went ashore on half-pay until 1774 when the Navy began re-arming for the approaching American War of Independence.

American War of Independence
He was appointed lieutenant on the frigate HMS Maidstone (1758) and was assigned to the West Indies. He was quickly promoted to first-lieutenant when Lieutenant Peter Rainier (for whom Mount Rainier was named) was promoted and transferred off the Maidstone. He achieved recognition with his action during the capture of the French ship Lion, when he was given command of the prize and over 200 prisoners, and successfully reached the British naval base at Antigua. As a reward, he was returned to England, promoted to commander, and given command of the 14-gun brig HMS Alert (1779). During 1780–81, Alert served on convoy duty in the North Atlantic and in the Caribbean. During the second tour in the Caribbean for the Alert, Joseph Baker (after whom Mount Baker was named) served as cabin boy for Commander Vashon. Because of distinguished action as commander of the Alert during the battle of the Saintes, Vashon was promoted to captain and given command of the 64-gun ship-of-the-line HMS Prince William (1780). He then served as flag captain aboard HMS Formidable (1777), and then the frigate HMS Sibyl (1779). With the end of the American wars, Vashon again went ashore on half pay in 1783. He and his wife Jane Bethell, who married in 1779, had a son in 1784, but Jane died suddenly in 1786. Vashon then married Sarah Rainier, the sister of his earlier shipmate Peter Rainier.

French Revolutionary and Napoleonic Wars
In late 1786 Vashon was recalled to service and took command of the 50-gun HMS Europa (1783), where serving under him were Midshipmen James Baker and Peter Puget, and Lieutenant George Vancouver, who would become significant in future explorations of the area of Washington state. Vashon commanded a series of ships during the period from 1789 to 1804 including the 64-gun ships HMS Ardent (1782), HMS St Albans (1764), HMS Dreadnought (1801), and the 80-gun HMS Pompee. It was during this period that George Vancouver commanded the Pacific Northwest expedition that bears his name and during which Vancouver named a number of prominent features of the Northwest after his colleagues and friends in the Royal Navy. On 28 May 1792 Vancouver named Vashon's Island in honour of Captain Vashon. In 1842 Charles Wilkes of the United States Exploring Expedition named Maury Island as a separate island and the usage began to change to Vashon Island.

In April 1804 Vashon was promoted to Rear-Admiral in command of the Leith Station, Scotland with his flag aboard HMS Roebuck (1774). He instituted a formal convoy system out of Leith to protect this vital shipping route, and was so successful that he was promoted to Vice-Admiral in 1808. When Vashon decided to retire in late 1808, the merchants and leaders of Leith were so pleased that they gave him a public dinner, awarded him two commemorative plates, and honoured him as a Freeman of the City of Edinburgh.

After the Navy
Vashon retired to Ludlow where he lived at Number 54 Broad Street (which is marked with a Ludlow Civic Society blue plaque). He was promoted to Admiral in 1814, and died on 27 October 1827 at the age of 85. He was buried at St Laurence Church in Ludlow.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: James Vashon


text:
Flag of the Isle of Wight

The flag of the Isle of Wight was adopted and registered in January 2009. It shows a diamond shape (the island) hovering over ocean waves. The indentation of the top corner of the diamond represents the River Medina, which is the largest river on the island.

History
There is anecdotal evidence that the Isle of Wight county colours have been used as a flag. These consisting of two thinner green vertical stripes surrounding a central thick white stripe and these flags are still available from various sources. Other than this the only established flag of modern times prior to 2007 was the three anchor and castle design used by the council. Those who still fly the green and white colours now often refer to them as 'The rebel green'.

In 2007 an 'Isle of Wight Flag Committee' was founded to create a flag for the Isle of Wight. It ran a public competition, in association with the County Press newspaper, to design the flag, which received over 350 entries. Four designs were shortlisted and put to the public in a vote.

2008 competition finalists

The winning design was by John Graney and was registered by the Flag Institute on 9 January 2009. Although the island does not meet the Flag Institute's preferred criterion of being a historic county, the flag has nevertheless been described as a "county flag" to reflect the island's distinct identity.

In April 2009 the new flag was officially launched and replaced the older design flown from County Hall.

Isle of Wight Council flag
Before this competition, the Council flag was the only popularly known flag for the island. It is taken from the Isle of Wight Arms granted in 1938 and features a representation of Carisbrooke Castle, which was the historic seat of the Governors. The blue surrounding field and three gold anchors represent the island's status and maritime history.

The Council flag can only be used by the Council on its local government buildings. Therefore, until the new island flag was registered, the island's team at events such as the International Island Games used only the Union Flag, or on occasion, St George's Cross.

title: Flag of the Isle of Wight


text:
Home (Dixie Chicks album)

Home is the sixth studio album by American country music band Dixie Chicks, released on August 27, 2002, through Monument and Columbia Records. It is notable for its acoustic bluegrass sound, which stands in contrast with their previous two country pop albums.

The group was promoting the album when lead singer Natalie Maines made controversial comments about U.S. President George W. Bush criticizing his role in the 2003 invasion of Iraq. The album's third single, "Travelin' Soldier", was #1 on the Billboard Country Chart the week that Maines' comments hit the press. The following week, as many stations started a still-standing boycott of the Chicks' music, the song collapsed. None of their following singles gained traction with country radio. Despite these events, the album was certified 6× Multi-platinum status by the RIAA and has sold 5,979,000 copies in the United States up to November 2008. The album also featured a cover of Fleetwood Mac's "Landslide", which was their biggest pop crossover hit until 2007, when "Not Ready to Make Nice" peaked at #4 on the Billboard Hot 100. The album was also successful in Australia, in its 175th week in the country charts it was certified Triple Platinum for shipments of 210,000 copies.

The album was nominated at the 45th Grammy Awards for six awards, including their second attempt for Album of the Year. The group went home with four in 2003, including Best Country Album, Best Recording Package, Best Country Instrumental Performance for "Lil' Jack Slade", and Best Country Performance by a Duo or Group with Vocal for "Long Time Gone". Additionally, they were nominated for Best Engineered Album, Non-Classical and Darrell Scott was nominated for Best Country Song for Long Time Gone. Two years later, they were nominated and won Best Country Performance by a Duo or Group with Vocal, this time for "Top of the World". It debuted at #1 on the Billboard Top 200 Albums chart and stayed there for four non-consecutive weeks. It also debuted at #1 on the Billboard Top Country Albums, and stayed there for 12 non-consecutive weeks.

Songs
"Landslide" was originally recorded by Fleetwood Mac in 1975. "Travelin' Soldier" was originally recorded by its writer, Bruce Robison, in 1996, and then in rewritten form, in 1999; Ty England also recorded the song in 1999. "Godspeed" was originally recorded by Radney Foster in 1999; his wife suggested sending it to the Dixie Chicks because Natalie Maines had just had a baby. "Long Time Gone" and "More Love" were originally recorded by Darrell Scott in 2000. "Truth No. 2" and "Top of the World" were originally recorded by Patty Griffin for a 2000 album that went unreleased. "I Believe in Love" made its debut on the 2001 telethon America: A Tribute to Heroes.

Singles
"Long Time Gone" was released on May 23, 2002 as the lead single from the record. It received positive reviews, with many critics noting the unique bluegrass and acoustic sound that was atypical of mainstream country music at the time. The song debuted at number 37 on the Hot Country Songs (then titled the "Hot Country Singles & Tracks") chart on June 8, 2002 and later rose up to its peak position of number two on August 24. The song also ended up peaking at number seven on the all-genre Billboard Hot 100, becoming the group's first top ten hit on the chart.

A cover of the Fleetwood Mac song, "Landslide" was released on August 26, 2002, as the second single from the album and the only one to be released to adult contemporary radio. The song was also the "Hot Shot Debut" of the week of September 7 (at country radio), debuting at number 32 and peaking at number two on the country charts on November 9. It also topped the Adult Contemporary chart, making it the group's only number one hit on that chart. It went to become the second top ten hit from the album on the Hot 100, peaking at number seven. The song also peaked within the top ten in Canada and Australia.

"Travelin' Soldier" was released on December 9, 2002 as the third single from Home. It was performed a year before its release at the 2001 CMA Awards, footage of which was later used as the song's music video. The song initially debuted on the country charts based solely on its live performance on February 16, 2002, at number 58, peaking at number 57. In its official release as a single, it re-entered the chart on December 14, 2002, at number 57. The song would go on to top the Hot Country Songs chart the week of March 22, 2003.

"Godspeed (Sweet Dreams)" was released on May 22, 2003, as the fourth single from the album. It debuted on the country charts at number 59 on June 7, 2003.

The album's fifth and final single, "Top of the World" was released in September 2003. The song was originally written by Patty Griffin, who had tried to record the song, but a dispute with her label led her to get dropped and the album project shelved.

Critical
The album received positive reviews. Metacritic gave the album an aggregated score of 75 (green label), indicating "generally favorable reviews".

Rhapsody ranked the album #1 on its list of "Country's Best Albums of the Decade". Another music blog, Country Universe, named it as the album of the decade. CMT's Craig Shelburne included it on his "A Dozen Favorite Country Albums of the Decade" list. Engine 145 country music blog lists it No. 4 on the "Top Country Albums of the Decade" list. Entertainment Weekly put the album on its end-of-the-decade, "best-of" list, saying: "Even George W. Bush fans have to respect the Chicks' authentic bluegrass sound on 'Long Time Gone' and 'Landslide'. Okay, maybe they don't. But they should." Allmusic said "They've delivered not just their best album, but what's arguably the best country album yet released in the 2000s. Needless to say, an instant classic."

Accolades

 * Rhapsody – #1 on its "Country's Best Albums of the Decade" list
 * Country Universe – #1 Country Album of the Decade
 * Country Universe - #2 on its "100 Greatest Contemporary Country Albums" List
 * CMT – "A Dozen Favorite Country Albums of the Decade" list
 * Engine 145 – #4 on its "Top Country Albums of the Decade" list
 * Entertainment Weekly – #6 Best Album of the Decade
 * Entertainment Weekly - #2 on its "25 Essential Country Albums" list
 * Entertainment Weekly = #85 on its "100 Best Albums from 1983-2008"
 * Texas Music Magazine – #1 Album of the Decade
 * BSC – #67 on its "Best Albums of the Decade" List
 * CMT – #15 on its Top 40 Greatest Country Albums
 * Taste of Country - #27 on its "100 Greatest Country Albums Ever"
 * 9513 - #4 on its "Top Country Albums of the Decade" List

Personnel

 * Martie Maguire - fiddle, mandolin, viola, vocals
 * Natalie Maines - vocals
 * Emily Robison - banjo, Dobro, accordion, vocals, Papoose

Additional personnel
 * Glenn Fukunaga - upright bass
 * Emmylou Harris - vocals
 * Byron House - upright bass
 * Lloyd Maines - acoustic guitar, Weissenborn slide guitar, Papoose
 * John Mock - percussion, bodhrán, uilleann pipes, tin whistle
 * Sara Nelson - cello
 * Paul Pearcy - percussion
 * Adam Steffey - mandolin
 * Bryan Sutton - acoustic guitar, baritone guitar, Papoose
 * Chris Thile - mandolin, soloist

Production
 * Producers: Dixie Chicks, Lloyd Maines
 * Engineer: Gary Paczosa
 * Assistant engineers: Thomas Johnson, Adam Odor, Fred Remmert
 * Mixing: Gary Paczosa
 * Mixing assistant: Eric Bickel
 * Mastering: Robert Hadley, Doug Sax
 * Production coordination: Mindi Pelletier
 * String arrangements: John Mock
 * Art direction: Kevin Reagan
 * Design: Bret Healey, Kevin Reagan
 * Photography: James Minchin
 * Make-up: Candy Burton

Awards
Grammy Awards

title: Home (Dixie Chicks album)


text: Template:US currency and coinage

title: Template:US currency and coinage


text:
Talk:Sentence

Shortest sentences

 * The shortest sentences in the English language are: "I am" and "I do"

Wrong. "I." can be a perfectly valid sentence. Fredrik 16:01, 9 May 2004 (UTC)

 * "I." is a sentence fragment because it has no verb. Tribunar 16:08, 9 May 2004 (UTC)

 * Okay, that's right. But "Go" is a valid sentence, whereas "I am" is not . So the article should say that the shortest sentence is "Go", not "I am" or "I do". Fredrik 16:13, 9 May 2004 (UTC)

 * "I" is a valid and complete sentence if uttered in response to a question such as "Who's in charge here?", although many native speakers would say "I am" or "Me" instead. The verb is understood. Andrewa 21:21, 9 May 2004 (UTC)

 * A complete sentence needs to be wholly understood as a single unit. If it requires context from another sentence to be understood, than it is not complete. —Preceding unsigned comment added by 70.50.119.175 (talk)

 * "I" cannot be a valid sentence because it does not have a verb. A verb is needed to make a complete sentence. "I" is just a subject, so it isn't a complete sentence, while "Go" could have an understood you, and so the sentence is technically "(You) go." 2601:8C3:4001:F5EE:F9EB:7150:DD0C:3678 (talk) 03:06, 5 October 2018 (UTC)Stella

This is a disambiguation page
User:2409:4062:2e8f:c9b1::1e4b:ab10 is repeatedly vandalizing this page by trying to turn it into a regular article. Two problems with this:

 * 1) You seem to have no knowledge of Wikipedia markup language - which is a serious matter.  You're putting bizarre heading into the article when you should be using things like "== This is a heading ==".
 * 2) You don't understand that this page is **NOT** an article about the linguistic concept of a "sentence" - it is a "disambiguation" page - which is the way that Wikipedia directs people to the various different meanings of the word "Sentence" (for example - a prison sentence).   By trying to turn this into an article about linguistics, you are quite literally vandalizing Wikipedia.

Please don't do that.

SteveBaker (talk) 00:54, 16 February 2021 (UTC)

title: Talk:Sentence


text:
Justus Georg Westphal

Justus Georg Westphal (18 March 1824 – 9 November 1859) was a German astronomer and mathematician.

Westphal is not to be confused with Johann Heinrich Westphal (1794 – 1831). Within the NASA Astrophysics Data System (ADS) he also appears to be confused with a third Westphal, "Alfred Friedrich Julius Westphal" ("Dr. A. Westphal"), a geophysicist who lived from 1 August 1850 to 1924, and who first published in 1881. J. G. Westphal is only mentioned a few times in the ADS, between 1853 and 1859.

Life
Westphal was born on 18 March 1824 in Colborn near Lüchow. He received his PhD at the University of Göttingen in 1852, his thesis being published as Evolutio readicum aequationum algebraicarum e ternis terminis constantium in series infinitas.

He was a student of the famous mathematician Carl Friedrich Gauss, who was also Director of Göttingen Observatory. During this period Westphal assisted at the observatory, where he held the title of Observator. In 1854, Westphal was appointed as a lecturer, but by the time of Gauss's death in 1855, he had resigned from his astronomical duties and had been replaced by his colleague Ernst Friedrich Wilhelm Klinkerfues.

Westphal is chiefly known for discovering the periodic comet 20D/Westphal (a.k.a. 20D/1852 O1, 1852 IV, D/Westphal 1) on 24 July 1852. C. H. F. Peters independently discovered the same comet from Constantinople on 3 August. Although it returned in 1913 (20D/1913 S1, 1913 VI, 1913d), the comet failed to return in 1976 and is now considered lost. There is substantial evidence that the comet completely disintegrated before reaching perihelion during its 1913 apparition. Westphal did not discover any other comets or any asteroids, though he made many observations of asteroids and calculated the orbital elements of several.

Westphal, who had suffered from poor health for much of his life, died on 9 November 1859 in Lüneburg.

title: Justus Georg Westphal


text:
Davis Square

Davis Square is a major intersection in the northwestern section of Somerville, Massachusetts, United States, where several streets meet: Holland Street, Dover Street, Day Street, Elm Street, Highland Avenue, and College Avenue. The name is often used to refer to the West Somerville neighborhood surrounding the square as well.

Davis Square is located approximately four miles from downtown Boston, and two miles from mid-Cambridge. The square is served by the Davis Square Station, one of the stops on the Massachusetts Bay Transportation Authority (MBTA)'s Red Line. Davis abuts the Powder House Square, Ball Square and Spring Hill neighborhoods, as well as Tufts University.

Davis Square is today a vibrant commercial, retail, nightlife and dining district. Businesses in Davis range from stores to restaurants to even a martial arts school. Davis fell into decline after World War II. After the Red Line was extended to Davis in 1984, however, the area began to experience a prolonged renaissance.

History

Davis Square was officially designated as a square by the City of Somerville in 1883. It was named for Person Davis (1819-1894), a grain dealer who moved to the area in 1850 and built his estate near the intersection of Elm, Grove and Morrison Streets. Davis was both a Somerville selectman and alderman. During the 1860s and 1870s, roads were improved and rail connections strengthened, which allowed West Somerville to urbanize.

After the Civil War, between 1870 and 1910, people increasingly chose to live in West Somerville where subdivided land was plentiful for new homes, both modest and substantial, and the commute to Boston for employment was convenient thanks to rail connections. By 1857 horse car railway lines were extended along Massachusetts Avenue from Harvard Square to Arlington, and in 1863 the Somerville Horse Railroad Company connected Union Square with West Somerville via extended tracks along Somerville Avenue. Passenger rail arrived in 1870, when the Lexington and Arlington branch of the Boston and Lowell Railroad (later succeeded by the Boston and Maine Railroad) routed steam rail service through Davis Square. These public improvements stimulated substantial development in the 1870s and 1880s as Davis Square quickly grew into an active commercial center. A boom in residential construction followed in the 1890s.

Brick paving was introduced in Davis Square in 1900 and the area continued to be a vibrant commercial and transportation center until post-World War II, with the exodus to the suburbs and the decline of urban centers throughout the nation. In 1914, the Somerville Theater opened, hosting silent films as well as burlesque shows. In 1927, the trains that had brought many people to Davis Square were re-routed. This also contributed to Davis Square's decline in the middle part of the 20th century. Factories shut, businesses failed and residents began to move out to the suburbs. As a result, Davis Square lost many of its tenants and became home to "burn-out storefronts" and "bars where motorcycle gangs hung out." According to a planning study completed in 1980, Davis Square suffered from "a lack of competitiveness among merchants, traffic congestion, inadequate parking and an increasingly deteriorated physical environment."

The situation began to turn around in the late 1970s thanks largely to two entities: the Somerville Office of Planning and Community Development, and the Davis Square Task Force. Their efforts culminated in the Davis Square Action Plan, which was adopted in 1982.

In the meantime, local officials and citizen groups petitioned the Massachusetts Bay Transportation Authority (MBTA) to create a Red Line subway stop in Somerville at Davis Square. The MBTA agreed and in 1977 business owners and other local residents and officials formed the Davis Square Task Force to act as a citizen advisory committee on community revitalization plans. By the early 1980s, grassroots ideas about new civic spaces, new job and housing opportunities and safe walking connections to surrounding neighborhoods became part of the discussion around the Davis Square train station.

With input from the Davis Square Task Force, the city made the Red Line station the cornerstone for downtown redevelopment. The goal was to strengthen commercial interests while preserving the residential character of the neighborhood. The 1982 Davis Square Action plan set forth the framework that led to the Somerville Community Path, the Harvard Vanguard office building, Buena Vista parking garage and Ciampa Manor senior housing building.

At the heart of Davis Square is a complex, busy six-point intersection, which was radically configured as part of the plan. Until the reconfiguration, two major, pedestrian-unfriendly streets bisected the square; crosswalks, sidewalks, curb extensions and refuge islands were added to enhance pedestrian capacity, circulation and safety. Additionally, several freight trains had run right through the square each day on the Boston and Maine Railroad, forcing traffic to back up for long periods of time.

The MBTA developed a central plaza linking the two station entrance buildings, built on the old railroad right-of-way. This plaza replaced a poorly defined open area containing at-grade parking spaces and debris. The plaza was designed to serve as the center of Davis Square, a gathering place and center for activities, including outdoor entertainment. The plaza and the station were both eligible for state percent-for-art moneys through the Arts on the Line program. One percent of the cost of constructing the new station entrances was used to commission several figurative sculptures including the Davis Square statues.

The city of Somerville and the Task Force initiated many other projects to accompany the Red Line extension and Davis Square improvements. Property redevelopment activities included a storefront and facade improvement grant program, financing for building renovations, and designation of a portion of Davis Square as an urban renewal district. Property acquisition, clearance, infrastructure upgrades, and development took place within the boundaries of this district.

The district was later developed as a 100,000 square foot office and retail complex, including public open space and a parking garage that serves patrons and employees of local businesses.

In 1997, Davis Square was listed by the Utne Reader as one of the fifteen "hippest places to live" in the United States. This reputation has continued; in 2016, real estate company Cushman & Wakefield rated Davis Square as one of "Top 100 Cool Streets" in North America, giving it a score of "Prime Hipness".

In 2005, The Boston Globe reported the first million dollar condo sale in Davis Square, which marked a major shift for a neighborhood once known as affordable and working-class. It now contains some of the priciest homes in Somerville and is significantly more expensive than the average for eastern Massachusetts.

Transportation
Davis Square is served by the Red Line, extended from Harvard to Alewife via Porter and Davis in the 1980s, with the Davis station opening on December 8, 1984. Additionally, Davis Square connects to several MBTA bus lines leading to nearby towns.

As part of the Davis Square Plan, the old Boston and Maine Railroad right-of-way was converted into a mixed-use path known as the Somerville Community Path. Most of the remaining railroad right-of-way between Davis Square and the Red Line's northern terminus at Alewife was redeveloped and landscaped as a linear park or bicycle/pedestrian pathway. A public park was constructed directly behind the Holland Street MBTA head house as part of a later project. The Alewife Linear Park connects Alewife MBTA station with the Minuteman Bikeway, a 10-mile bicycle path that leads to Bedford.

Arts and culture
As Somerville continues to evolve, Davis Square was clearly the vanguard of revitalization and change. In the past three decades, Davis has become a vibrant regional center for retail, nightlife, dining and more. Davis Square is a prime business location with over 200 businesses including healthcare and high-tech corporations, national non-profits, and media offices. Boston Magazine notes half a dozen food and entertainment establishments dating from before the Red Line still extant in 2012, including the Somerville Theatre, The Rosebud, McKinnon's Meat Market, and a candlepin bowling alley.

The Somerville Theatre shows movies, live performances, and has a satellite gallery of the Museum of Bad Art. The Public Radio International show, Living on Earth is recorded in its studios in Davis Square. For five years, the Jimmy Tingle Off-Broadway Theater boasted a variety of nationally and regionally known acts, both comedic and musical, including Jimmy Tingle himself, but closed at the end of October 2007.

Davis Square is host to several popular festivals and events throughout the year. The Somerville Arts Council's popular ArtBeat festival takes place here every year on the third weekend of July, while the HONK! Festival of activist brass bands occurs here every October. During the summer months there are free public folk dances.

title: Davis Square


text:
Johann Heinrich Westphal

Johann Heinrich Westphal (January 31, 1794 – 1831) was a German astronomer. Not to be confused with J. G. Westphal, who discovered the lost periodic comet 20D/Westphal in 1852.

He was born in Schwerin. His professional career was spent mostly in Italy. He translated Giuseppe Piazzi's Lezioni elementari di astronomia ad use del real osservatorio di Palermo into German. He died in Sicily.

title: Johann Heinrich Westphal