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I am reading about the Dirichlet process and I can understand the construction from Chinese restaurant process or stick-breaking process or Polya urn scheme. Now I am trying to understand why Dirichlet process is a distribution of distribution from its formal/original definition from link of Wikipedia:

Given a measureable set S, a base probability distribution H and a positive real number $\alpha$, the Dirichlet process $DP(H, \alpha)$ is a stochastic process whose sample path is a probability distribution over $S$, such that the following holds:

for any measureable finite partition of $S$, denoted $\{B_i\}_{i=1}^n$, if $X \sim DP(H, \alpha)$, then $$(X(B_1), ..., X(B_n)) \sim Dir(\alpha H(B_1), ..., \alpha H(B_n))$$ where $Dir$ denotes Dirichlet distribution.

I have the basic understanding of the concept of measure theory and I can understand the terms in this definition. However, I fail to get a picture from the above definition by linking with Chinese restaurant processes:

  1. Considering from the Chinese restaurant process, what should this measureable set $S$ and its parition correspond to? Is $S$ all customers and does the partition correspond to a particular way how all customers are partitioned into tables?
  2. What does $X(B_i)$ mean in Chinese restaurant process? Does it mean the probability that a specific group of customers sit at table $i$?
  3. What does the vector $(X(B_1), ..., X(B_n))$ mean? Is it the vector of probabilities for each table to have a specific group of customers? Or is it the multinomial distribution vector for a new customer to sit at any table?
  4. For Chinese restaurant process, what should be an intuitive example of the base distribution $H$? And what is the meaning of $H(B_1)$ here?
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Can't help with the theoretical aspects and this is not really an answer to your question (just couldn't fit this in the comment section):

A good way to think about Dir Process is to start from the Dirichlet distribution itself. Dirichlet distribution is a probability distribution over a K-dimensional simplex. Where a draw from the simplex is a K-dimensional vector. A draw from it is a distribution. Now Dir has two properties that allows you to split (or merge) it as many times (i.e., infinite number of times) as you want and still get Dir distribution back. So, you can always take an infinite-dimensional simplex and imagine a construction process that starts from a Dir over 1-D simplex, and splitting it to two, and so on.

The chinese restaurant process is just a realization of a draw from a Dirichlet. It allows you to generate samples from the Dirichlet without really dealing without dealing with $B_i$ partitions. Otherwise, to generate a Dirichlet draw you would need to decide about the partitioning of the space first (for which there are infinite possibilities). You can understand this better if you watch this (great lecturer).

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