Editing Dirichlet distribution (section)

====[[Pólya urn model|Pólya's urn]]====
Consider an urn containing balls of {{mvar|K}} different colors. Initially, the urn contains {{math|''α''{{sub|1}}}} balls of color 1, {{math|''α''{{sub|2}}}} balls of color 2, and so on. Now perform {{mvar|N}} draws from the urn, where after each draw, the ball is placed back into the urn with an additional ball of the same color. In the limit as {{mvar|N}} approaches infinity, the proportions of different colored balls in the urn will be distributed as {{math|Dir(''α''{{sub|1}}, ..., ''α{{sub|K}}'')}}.<ref>{{cite journal | journal=Ann. Stat. | volume=1 | issue=2 | pages=353–355 | year=1973  | author=Blackwell, David | title=Ferguson distributions via Polya urn schemes | doi = 10.1214/aos/1176342372 | last2=MacQueen | first2=James B. | doi-access=free }}</ref>

For a formal proof, note that the proportions of the different colored balls form a bounded {{math|[0,1]{{isup|''K''}}}}-valued [[martingale (probability theory)|martingale]], hence by the [[martingale convergence theorem]], these proportions converge [[almost sure convergence|almost surely]] and [[convergence in mean|in mean]] to a limiting random vector. To see that this limiting vector has the above Dirichlet distribution, check that all mixed [[moment (mathematics)|moments]] agree.

Each draw from the urn modifies the probability of drawing a ball of any one color from the urn in the future. This modification diminishes with the number of draws, since the relative effect of adding a new ball to the urn diminishes as the urn accumulates increasing numbers of balls.