Editing Dirichlet distribution (section)

===Neutrality===
{{main|Neutral vector}}

If <math>X = (X_1, \ldots, X_K)\sim\operatorname{Dir}(\boldsymbol\alpha)</math>, then the vector&nbsp;{{mvar|X}} is said to be ''neutral''<ref>{{cite journal | journal=Journal of the American Statistical Association | volume=64 | issue=325 | pages=194–206 | year=1969  | author=Connor, Robert J. | title=Concepts of Independence for Proportions with a Generalization of the Dirichlet Distribution | doi = 10.2307/2283728 | jstor=2283728 | author2=Mosimann, James E | publisher=American Statistical Association }}</ref> in the sense that ''X{{sub|K}}'' is independent of <math>X^{(-K)}</math><ref name=FKG/> where

<math display=block>X^{(-K)}=\left(\frac{X_1}{1-X_K},\frac{X_2}{1-X_K},\ldots,\frac{X_{K-1}}{1-X_K} \right),</math>

and similarly for removing any of <math>X_2,\ldots,X_{K-1}</math>. Observe that any permutation of {{mvar|X}} is also neutral (a property not possessed by samples drawn from a [[generalized Dirichlet distribution]]).<ref>See Kotz, Balakrishnan & Johnson (2000), Section 8.5, "Connor and Mosimann's Generalization", pp. 519–521.</ref>

Combining this with the property of aggregation it follows that {{math|''X''{{sub|''j''}} + ... + ''X''{{sub|''K''}}}} is independent of <math>\left(\frac{X_1}{X_1+\cdots +X_{j-1}},\frac{X_2}{X_1+\cdots +X_{j-1}},\ldots,\frac{X_{j-1}}{X_1+\cdots +X_{j-1}} \right)</math>. In fact it is true, further, for the Dirichlet distribution, that for <math>3\le j\le K-1</math>, the pair <math>\left(X_1+\cdots +X_{j-1}, X_j+\cdots +X_K\right)</math>, and the two vectors <math>\left(\frac{X_1}{X_1+\cdots +X_{j-1}},\frac{X_2}{X_1+\cdots +X_{j-1}},\ldots,\frac{X_{j-1}}{X_1+\cdots +X_{j-1}} \right)</math> and <math>\left(\frac{X_j}{X_j+\cdots +X_K},\frac{X_{j+1}}{X_j+\cdots +X_K},\ldots,\frac{X_K}{X_j+\cdots +X_K} \right)</math>, viewed as triple of normalised random vectors, are [[Independence (probability theory)#More than two random variables|mutually independent]]. The analogous result is true for partition of the indices {{math|{{mset|1, 2, ..., ''K''}}}} into any other pair of non-singleton subsets.