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I have a question regarding sticking-breaking model of Dirichlet process, which is defined as follows:

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There are further statements that

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I am not clear that how to derive equation 1 from that posterior distribution and why does the equation 1 is equal to equation 2. Here $\mathcal{B}$ is Beta distribution.

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  • $\begingroup$ Beta distribution is just a two dimensional Dirichlet. $\endgroup$
    – Mo Chen
    Commented Nov 15, 2014 at 22:50
  • $\begingroup$ Isn't the Beta distribution a one-dimensional distribution of continuous proportions? $\endgroup$
    – gung - Reinstate Monica
    Commented Nov 15, 2014 at 23:10
  • $\begingroup$ Yes. Assume the proportion is p. p follows a Beta, then [p,1-p] follows a 2d Dirichlet. $\endgroup$
    – Mo Chen
    Commented Nov 15, 2014 at 23:36
  • $\begingroup$ Doesn't that mean that the Dirichlet is a 2D Beta? $\endgroup$
    – gung - Reinstate Monica
    Commented Nov 15, 2014 at 23:38
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    $\begingroup$ @gung I don't know if you sorted that yet, but no, the Dirichlet Distribution is not a 2d beta. In general, its support is the simplex $(p_1,p_2,...,p_k)$ with the sum of $p_j$ equals to 1 and $p_j>0$. i.e. its a $k$-dimensional distribution lives on a $k-1$-dimensional surface. E.g.: the 2d Dirichlet only has parameters $p$ and $1-p$. So the 2d Dirichlet distribution is just directly the beta distribution. $\endgroup$
    – jgyou
    Commented Apr 26, 2015 at 1:01

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I would recommend you to read Introduction to the Dirichlet Distribution and Related Processes by Bela A. Frigyik, Amol Kapila, and Maya R. Gupta. It is written in a very accessible manner, and I could imagine, it would be a great introduction to those topics (measure theory, etc)

Hope it helps

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    $\begingroup$ Wow, this is a great tutorial. $\endgroup$
    – jgyou
    Commented Apr 25, 2015 at 23:39

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