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.