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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.

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    $\begingroup$ 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? $\endgroup$ – whuber May 25 '16 at 17:51
  • $\begingroup$ @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? $\endgroup$ – Ramin Barati May 25 '16 at 18:00
  • $\begingroup$ 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. $\endgroup$ – whuber May 25 '16 at 18:02
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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 $x_i=u_i/2$.

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  • $\begingroup$ So if I'm understanding you and whuber correctly, if I have a dataset of two concatenated Dirichlet vectors and I want to fit a mixture model to it, first I should map the concatenated vector to a lower dimentional simplex and then fit a Dirichlet mixture model to it, right? $\endgroup$ – Ramin Barati May 25 '16 at 18:28

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