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I have read that the parameters of Dirichlet distribution must be strictly positive.


The Dirichlet distribution of order $K \geq 2$ with parameters $\alpha_{1}, \ldots, \alpha_{K} \color{blue}{> 0}$ has a probability density function with respect to Lebesgue measure on the Euclidean space $\mathbf{R}^{K-1}$ given by $$ f\left(x_{1}, \ldots, x_{K} ; \alpha_{1}, \ldots, \alpha_{K}\right)=\frac{1}{\mathrm{~B}(\boldsymbol{\alpha})} \prod_{i=1}^{K} x_{i}^{\alpha_{i}-1} $$ where $\left\{x_{k}\right\}_{k=1}^{k=K}$ belong to the standard $(K-1)$-simplex.


Then the Dirichlet distribution is then used in the definition of Dirichlet process.


Given a measurable set $S$, a base probability distribution $H$ and a positive real number $\alpha$, the Dirichlet process $\mathrm{DP}(H, \alpha)$ is a stochastic process whose sample path (or realization, i.e. an infinite sequence of random variates drawn from the process) is a probability distribution over $S$, such that the following holds. For any measurable finite partition of $S$, denoted $\left\{B_{i}\right\}_{i=1}^{n}$, if $X \sim \operatorname{DP}(H, \alpha)$ then $$ \left(X\left(B_{1}\right), \ldots, X\left(B_{n}\right)\right) \sim \operatorname{Dir}\left(\alpha H\left(B_{1}\right), \ldots, \alpha H\left(B_{n}\right)\right), $$ where Dir denotes the Dirichlet distribution and the notation $X \sim D$ means that the random variable $X$ has the distribution $D$.


If $H(B_1) = 0$, then $\alpha H(B_1)=0$. How is $\operatorname{Dir}\left(\alpha H\left(B_{1}\right), \ldots, \alpha H\left(B_{n}\right)\right)$ well-defined in this case?

Thank you for your explanation!

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  • $\begingroup$ Because every realization of this process is a probability distribution on $S,$ its values on sets of measure zero don't matter. Ergo, you may define the process any way you like when $H(B_1)=0.$ Perhaps the visualizations I posted at stats.stackexchange.com/a/421266/919 might help with the intuition. $\endgroup$
    – whuber
    Jun 1, 2022 at 13:55
  • $\begingroup$ @whuber Let $X$ be the Dirichlet process in our consideration. When $H(B_1)=0$, $B_1$ is a null set w.r.t. the base measure $H$. Clearly, the probability distribution of $X$ is not $H$. Could you elaborate more? $\endgroup$
    – Akira
    Jun 1, 2022 at 14:24
  • $\begingroup$ I do not follow the logic of your "clearly" or even the intended meaning of the words that follow. $X$ is a process while $H$ is a base measure: they cannot be equated or even compared. $\endgroup$
    – whuber
    Jun 1, 2022 at 14:40

1 Answer 1

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The definition of the Dirichlet process, as given on the Wikipedia page, is correct only if we use the definition of the Dirichlet distribution as given in Ferguson, 1973. Ferguson's definition is slightly more general than the one on Wikipedia. Under Ferguson's definition, only one of the $\alpha_j$'s need be positive. With this definition, there is no problem with sets of $H$-measure 0.

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