Talk:Bayes' theorem/Archive 2

message: Hello everyone. There was a recent substantial revision of Bayes' theorem . I'm afraid it doesn't look like an improvement to me. Here are some points to consider. 
# In the introduction, Bayes' theorem is described in terms of random variables. This isn't necessary to clarify Bayes' theorem, and introduces a whole raft of heavy baggage (to mix metaphors) that is going to be well-nigh incomprehensible to the general readership.
# The two-line derivation of Bayes' theorem is put off for several paragraphs by a lengthy digression which introduces some unnecessary notation and includes a verbal statement of Bayes' theorem which is much less clear than the algebraic statement which it displaces. 
# The example is somewhat problematic. It is formally correct, but it's not very compelling, as it doesn't make use of any relevant prior information; the medical test example and even the cookies example, which were moved to Bayesian inference a while ago, were superior in that respect. Perhaps if an example is needed, we can restore the medical test or cookies (I vote for the medical test fwiw). The example is also misplaced (coming before the algebraic statement) although that's easy to remedy.
Given the difficulties of the recent revision, I'm tempted to revert. Perhaps someone wants to talk me out of it. Regards, Wile E. Heresiarch 22:39, 11 Jul 2004 (UTC)
replies: text: Perhaps you are right that the really simple material should come first.  However, that's not a reason to throw away the example on political opinion polling.  That example is in many respects typical of the simplest applications in Bayesian statistical inference.  I for one find it compelling for that reason.  To say that the simple statement that is followed by the words "... which is Bayes' theorem" is more than just a simple special case is misleading. Michael Hardy 23:41, 11 Jul 2004 (UTC)
replies: text: Also, the "verbal" version is very useful; in some ways it makes a simple and memorable idea appear that is less-than-clearly expressed by the formula expressed in mathematical notation.  The role of the likelihood and the role of the prior are extremely important ideas. Michael Hardy 23:44, 11 Jul 2004 (UTC)
replies: text:  I've moved the example farther down in the article, as it interrupts the exposition. I've also reverted the section "Statement of Bayes' theorem" to its previous form; the newer version did not introduce any new material, and was less clear. I put a paraphrase using the words posterior, prior, likelihood, & normalizing constant into the "Statement" section. -- I'm still not entirely happy with "random variable" in the introduction, but I haven't found a suitable replacement. I'd favor "proposition" but that it is likely not familiar to general readers. Fwiw & happy editing, Wile E. Heresiarch 14:49, 20 Jul 2004 (UTC)
thread_title: Substantial Revision

message: It seems to me that the current version of the Bayes' theorem article contains a little too much Bayesian inference.  This is not to deny from the importance of Bayesian inference as the premier application of Bayes' theorem, but as far as I can see:
# The section explaining terms such as posterior, likelihood, etc. is more appropriate to the Bayesian inference article.  None of it is taught with Bayes' theorem in courses on elementary probability (unless, I assume, Bayesian inference is also taught).
# The example is one of Bayesian inference, not simply Bayes' theorem.  Somewhat ironically, the Bayesian inference article contains some simple examples of Bayes' Theorem that are not Bayesian in nature, and that were moved there from an older version of the Bayes' theorem article!
Some of these things are noted in other posts to this talk page and the talk page of the Bayesian inference article, but I can't see that the current version of either article is a satisfactory outcome of the discussions.  The current versions of the articles appear to muddy the distinction between Bayes' theorem and Bayesian inference/probability.
Hence, I propose to change these articles by
# swapping the cookie jar and false positive examples from the Bayesian inference article for the example from the Bayes' theorem article;
# deleting the section on conventional names of terms in the theorem from the Bayes' theorem article (but noting that there are such conventions as detailed in the Bayesian inference article);
# revising the description of the theorem to refer to probabilities of events, since this is the most elementary way of expressing Bayes' theorem, and is consistent with identities given in (for instance) the conditional probability article.
Since this has been a topic of some discussion on the talk pages of both articles, I would like to invite further comment from others before I just go ahead and make these changes.  In the absence of such discussion, I'll make the proposed changes in a few days.
Cheers, Ben Cairns 07:55, 23 Jan 2005 (UTC)
replies: text: Well, I agree the present state of affairs isn't entirely satisfactory. About (1), if you want to move the medical test to Bayes' theorem in exchange for the voters example, I'm OK with that. I'd rather not clutter up Bayes' theorem with the cookies; it's no less complicated than the medical test, and a lot less interesting. (2) I'm OK with cutting the conventional terms from Bayes' theorem . (3) I guess I'm not entirely happy with stating Bayes' theorem as a theorem about events, since "events" has some baggage. I'd be happiest to say something like P(B|A) = P(A|B) P(B)/P(A) whenever A and B are objects for which P(A), P(B), etc, make sense and that might be OK for mathematically-minded readers but maybe not as friendly to the general readership. Any other thoughts about that? Anyway, thanks for reopening the discussion. Now that we've all had several months to think about, I'm sure we'll make quick progress. 8^) Regards & happy editing, Wile E. Heresiarch 21:56, 23 Jan 2005 (UTC)

text: Well, we seem to have reached an impasse. I'm quite aware that "event" has a prescribed meaning; that's why I want to omit it from article. Technical difficulties with strange sets never arise in practical problems and for this reason are at most a curiosity -- this is the pov of Jaynes the uber-Bayesian. From what I can tell, Bayesians are in fact happy to assign probability  to "just anything" and this is pretty much the defining characteristic of their school. Let me see if I can find some textbook statements from Bayesians to see what is permitted for A and B. Wile E. Heresiarch 16:02, 24 Jan 2005 (UTC)

text: The main problem I have with the event business is that it's not necessary, and not helpful, in this context. Being told that A and B are elements of a sigma-algebra simply won't advance the understanding of the vast majority of readers -- this is the "not helpful" part. One can make a lot of progress in probability without introducing sigma-algebras until much later in the game -- this is the "not necessary" part. I'd prefer to say A and B are variables -- this avoids unnecessary assumptions. "A and B are simply things to which we may assign probabilities" is OK by me too. For what it's worth, Wile E. Heresiarch 16:24, 25 Jan 2005 (UTC)

text: OK, so you saw the "not helpful" part. Can you address the "not necessary" part? Btw I don't have any desire or intent to change the event article. Wile E. Heresiarch 00:31, 27 Jan 2005 (UTC)
thread_title:  Bayes' theorem vs Bayesian inference 

message: The Statement of Bayes' Theorem section is correct but confusing. I had to re-read this section several times before I remembered from graduate school that "likelihood" has a counter-intuitive technical definition. To the average math-oriented reader, you can't just pop P(A|B) = L(B|A) and not explain that likelihood is an unfortunate technical phrase. Most non-statisticians would not equate "Probability of A | B" with "Likelihood of B | A". If someone doesn't already know Bayes' theorem (reason they're reading the article), they probably don't know what statisticians mean when they say "likelihood function" either. I'd suggest eliminating everything about likelihood functions entirely from this section and just stick with probabilities-oriented terms.--Toms2866 13:06, 28 March 2006 (UTC)
thread_title:  The Statement of Bayes' Theorem