Editing De Finetti's theorem (section)

== Background ==
A Bayesian statistician often seeks the conditional probability distribution of a random quantity given the data. The concept of [[exchangeability]] was introduced by de Finetti.  De Finetti's theorem explains a mathematical relationship between independence and exchangeability.<ref>See the Oxford lecture notes of Steffen Lauritzen  http://www.stats.ox.ac.uk/~steffen/teaching/grad/definetti.pdf</ref>

An infinite sequence

:<math>X_1, X_2, X_3, \dots </math>

of random variables is said to be exchangeable if for any [[natural number]] ''n'' and any finite sequence  ''i''<sub>1</sub>, ..., ''i''<sub>''n''</sub> and any permutation of the sequence π:{''i''<sub>1</sub>, ..., ''i''<sub>''n''</sub> } → {''i''<sub>1</sub>, ..., ''i''<sub>''n''</sub> },

:<math>(X_{i_1},\dots,X_{i_n}) \text{ and } (X_{\pi(i_1)},\dots,X_{\pi(i_n)}) </math>

both have the same [[joint probability distribution]].

If an identically distributed sequence is [[statistical independence|independent]], then the sequence is exchangeable; however, the converse is false—there exist exchangeable random variables that are not statistically independent, for example the [[Pólya urn model]].