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I am trying to understand the Dirichlet distribution, and more specifically, its relationship to the Beta distribution. The way I understand it, it is defined as follows:

Dirichlet distribution

where $S_k$ is the probability simplex defined like so:

Probability simplex for Dirichlet distribution

First of all, what is the rational behind the $B(\alpha)$ term.

Secondly, in MATLAB image we were given below, how do we label the axes? What is each one for? The $\alpha$ values for the MATLAB plot are $\alpha = (0.1, 0.1, 0.1)$

enter image description here

And lastly, how do you calculate probability from this distribution? Is it volume under the curve?

Thanks in advance

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Dirichlet distribution is a multivariate generalization of the beta distribution.

  1. $B(\alpha)$ is the normalization constant to satisfy the axiom of probability $P(S)=1$.

  2. Due to the constraint $S_k$, $x_i = 1 - \sum_{k=1, k\neq i} x_k$. So, the plot seems to me to be of the density function with two of the axis on the lower plane indicating two of $x_i$'s out of three (where the value of third can be derived from those two) and the vertical axis indicate the value of probability density function.

  3. For the particular case you showed, the cumulative probability can be calculated by computing the volume under the curve for desired region covered by $x_1$ and $x_2$.

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  • $\begingroup$ Ok, the B(alpha) normalization part makes sense now, but could you explain a bit better about x3? You said "where the value of third can be derived from those two", but why would they not show x3, and what is its purpose? Is it only because we cannot see a 4-dimensional graph? Thanks $\endgroup$ – Josh Jul 12 '17 at 19:03
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    $\begingroup$ Note that when you plot pdf of beta distribution you only plot it against $x$ and not include $1-x$ in the plot. I think the same reason applies here too. You plot the pdf value against say $x_1, x_2, \ldots, x_{k-1}$ and do not include $x_k$ in the plot because it is $1-\sum_{i=1}^{k-1}x_i$. Your argument of $4d$ graph is also reasonable. $\endgroup$ – Dhruv Kohli - expiTTp1z0 Jul 12 '17 at 19:09

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