# Probability distribution on a subset of a simplex

I want to define a probability distribution on a subset of a simplex. for example, on a 3-simplex, we know that $x_1+x_2+x_3+x_4=1$ and $X \sim Dirichlet$. Would it be possible to constraint $X$ more (e.g. $x_1+x_2=0.5$)? If it's possible, how? If it's not, what could be done instead to apply the constraint on the support?

In case you are curios why I need that: I want to concatenate two Dirichlet distributed vectors, then define a probability distribution on the resulting vector and use it in a mixture model.

• A single linear constraint of this nature intersects a hyperplane with the simplex, producing a simplex of lower dimension. Why not just choose a (scaled, translated) Dirichlet distribution on the intersection, then? – whuber May 25 '16 at 17:51
• @whuber, Thats what I'm looking for I think, would you kindly refer me to some literature or url where I can get more info? – Ramin Barati May 25 '16 at 18:00
• Literature on what? I merely made a basic observation about solving systems of equations; you already appear to know the difficult stuff, which is awareness of Dirichlet and related distributions. – whuber May 25 '16 at 18:02

To whuber's point, if you want to generate from such a distribution in a way that's most similar to the Dirichlet distribution, first draw samples from a 2-D Dirichlet distribution with $0.5+x_3+x_4=1$, and then draw $x_1,x_2$ from another 2-D Dirichlet distribution: $u_1+u_2=1$, by setting $x_i=u_i/2$.