Talk:de Finetti's theorem

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But what the heck does all that mean? This is pure math -- what's the applied math version of exchangeability? Cyan 06:38 Apr 2, 2003 (UTC)

OK, I've attempted to incoporate a sort of answer into the article. Michael Hardy 01:54 Apr 3, 2003 (UTC)

I've read that this theorem can be made to apply to finite exchangeable sequences (or the subsequences of finite exchangeable sequences...?) by removing the restriction that the the mixing function be a distribution and allowing it to assume negative values. I'm not really sure what this means in practical terms... does that information belong in this article? Cyan 08:11 Apr 3, 2003 (UTC)

I think the business about allowing it to assume negative values is from this article: http://bayes.wustl.edu/etj/articles/applications.pdf Ajb

"In particular, one may well doubt the cogency of the underlying assumptions of [the method of inverse probability], i.e. the construction of a prior probability distribution over unknown `true' probability distributions and the choice that this prior distribution should be uniform, or ask whether all this complies with the clause that `nothing else' is known, which was emphasized in the statement of the problem. All I can say is that these assumptions would be deemed appropriate precisely under this clause by most classical probability theorists. Also, it has been shown by De Finetti that this construction can be replaced by the more appealing and parsimonious assumption of `exchangeability'." Jos Uffink, The constraint rule of the maximum entropy principle,Studies in History and Philosophy of Modern Physics 27 (1996) 47-79. ...Somehow, the concept of exchangeability frees us from having to postulate an underlying "true" probability distribution? Cyan 09:56 Apr 3, 2003 (UTC)

Actually, there are versions of this Theorem for finite sequences, see e.g. the well-known work by Diaconis and Freedman, Annals of Probability 8, 745-764 (1980). Here is an excerpt from the abstract: "Let

X_{1},X_{2},\ldots ,X_{k},X_{k+1},\ldots ,X_{n}

be exchangeable random variables taking values in the set S. The variation distance between the distribution of

X_{1},\ldots ,X_{k}

and the closest mixture of independent, identically distributed random variables is shown to be at most 2ck/n, where c is the cardinality of S. If c is infinite, the bound k(k - l)/n is obtained. These results imply the most general known forms of de Finetti's theorem. Examples are given to show that the rates k/n and k(k - l)/n cannot be improved." Actually, it might make sense to mention this in the main text and add a reference... —Preceding unsigned comment added by 129.194.8.73 (talk) 13:44, 18 April 2008 (UTC)[reply]

Negative values

FWIW, negative coefficients were first proposed by Jaynes, but later analysed more deeply, as in [1]. However, allowing negative coefficients breaks the assumption that the coefficients themselves come from a distribution, and this would possibly run against de Finetti's ideas. -- Zz 15:43, 20 November 2006 (UTC)[reply]

so, where's the actual statement of the theorem? —Preceding unsigned comment added by 141.162.101.50 (talk • contribs)

The theorem as revised by Kerns? - a finite collection of exchangeable random variables is a signed mixture of i.i.d. random variables, the signs coming from the range of [-1,1]. In the original wording by de Finetti, the signs come from [0,1], which is nice because it is a weighted distribution, however it is not true for all finite cases. -- Zz 12:01, 15 May 2007 (UTC)[reply]

Introduction is difficult to read

I can't parse this bit from the introduction: "[...] there are underlying, generally unobservable, quantities which are i.i.d. – exchangeable sequences are (not necessarily i.i.d.) mixtures of i.i.d. sequences." The preceding paragraph is not much better. Would someone who understand this please re-write? — Preceding unsigned comment added by 66.188.89.180 (talk) 17:08, 13 February 2012 (UTC)[reply]