For a Dirichlet process, there are two parameter $\alpha$ and $H$, and the Dirichlet process $X$ is defined as $$(X(B_1),\cdots,X(B_n))\sim Dir(\alpha H(B_1),\cdots,\alpha H(B_n))$$ where$\{B_i\}_{i=1}^n$is a partition of the measurable space.

While the Dirichlet distribution is like that $$f(x_1,x_2, \cdots ,x_{K - 1};\alpha _1,\alpha _2, \cdots ,\alpha _{K - 1},\alpha _K) = \frac{1}{B(\vec \alpha )}\prod\limits_{i = 1}^K x_i^{\alpha_i - 1} $$

My question is:

If I want to view the Dirichlet distribution as a special case of the Dirichlet process, then how should I set the parameters $\alpha$ and $H$ in the definition of Dirichlet process?


Here $\alpha$ is a number, and $H$ is a measure.

I believe that you should take a finite space, say, $\{1,...,K\}$. Then for any partition of $\{1,...,K\}$, your first formula will hold. $H$ measures the asymmetry of your Dirichlet process, as the parameters $\alpha_1,...,\alpha_K$ do in the second formula.

  • $\begingroup$ if I partition {1,...,K} into {1},{2}...,{K}, should $\alpha$ equals to $\sum _{i=1}^K \alpha_i$ and $H(i)$ equals to $\alpha_i/\alpha$ ? $\endgroup$ – henrysupercool Sep 18 '14 at 0:59

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