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On Wikipedia a Dirichlet Process $\text{DP}$ is described by the procedure;

  1. Draw a distribution $P$ from $\text{DP}(\alpha, H)$
  2. Draw observations $X_1, X_2, \dots$ independently from $P$

where $\alpha$ is the concentration parameter and $H$ is some 'base distribution'. I understand that a Dirichlet Process can also be described by the Chinese Restaurant Process (CRP) metaphor, which I won't describe here.

My question is, when using the CRP metaphor to explain the Dirichlet Process, what is the base distribution $H$?

Another answer elsewhere on Math Stack Exchange claims this can be any distribution, and gives the Normal distribution as an example;

Let's take a Gaussian distribution $\mathcal{N} \sim (\mu, \sigma^2)$ as an example. Note that a DP also depends on the concentration parameter α, let's consider two extreme cases: 1) when α is very small. 2) when α is very large. In case 1), we will find that the number of customers on most tables is around μ; In case 2), we will find that the distribution of the number of customers on those tables approximately follows the base distribution.

But this can't be true - I could select a base distribution with a negative mean and then the number of customers on tables would tend to be negative. Indeed, any distribution with unbounded support on the reals invalidates this claim.

aaronsnoswell
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1 Answers1

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From the paper that formally defines Dirichlet Process,

Proposition 1. Let $P$ be a Dirichlet process on ( $\mathscr{X}, \mathscr{A}$ ) with parameter $\alpha$, and let $A \in \mathscr{A}$. If $\alpha(A)=0,$ then $P(A)=0$ with probability one. If $\alpha(A)>0,$ then $P(A)>0$ with probability one. Furthermore, $\mathscr{E} P(A)=\alpha(A) / \alpha(\mathscr{X})$

Clearly the base distribution $\alpha$ cannot have support on the negative real line. So my guess for the CRP is that any base method that has support only on the positive real line would do.

honeybadger
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