Editing Dirichlet distribution (section)

===Intuitive interpretations of the parameters===

====The concentration parameter====
Dirichlet distributions are very often used as [[prior distribution]]s in [[Bayesian inference]].  The simplest and perhaps most common type of Dirichlet prior is the symmetric Dirichlet distribution, where all parameters are equal.  This corresponds to the case where you have no prior information to favor one component over any other.  As described above, the single value {{mvar|α}} to which all parameters are set is called the [[concentration parameter]].  If the sample space of the Dirichlet distribution is interpreted as a [[discrete probability distribution]], then intuitively the concentration parameter can be thought of as determining how "concentrated" the probability mass of the Dirichlet distribution to its center, leading to samples with mass dispersed almost equally among all components, i.e., with a value much less than 1, the mass will be highly concentrated in a few components, and all the rest will have almost no mass, and with a value much greater than 1, the mass will be dispersed almost equally among all the components.  See the article on the [[concentration parameter]] for further discussion.

====String cutting====
One example use of the Dirichlet distribution is if one wanted to cut strings (each of initial length 1.0) into {{mvar|K}} pieces with different lengths, where each piece had a designated average length, but allowing some variation in the relative sizes of the pieces. Recall that <math>\alpha_0 = \sum_{i=1}^K \alpha_i.</math> The <math>\alpha_i/\alpha_0</math> values specify the mean lengths of the cut pieces of string resulting from the distribution.  The variance around this mean varies inversely with <math>\alpha_0</math>.

[[Image:Dirichlet example.png|center|Example of Dirichlet(1/2,1/3,1/6) distribution]]

====[[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.