The Chinese Restaurant Process
As shown above, a simple distribution, the so-called Chinese restaurant process, results from considering the conditional distribution of one component assignment given all previous ones in a Dirichlet distribution mixture model with components, and then taking the limit as goes to infinity. It can be shown, using the above formal definition of the Dirichlet process and considering the process-centered view of the process, that the conditional distribution of the component assignment of one sample from the process given all previous samples follows a Chinese restaurant process.
Suppose that samples, have already been obtained. According to the Chinese Restaurant Process, the sample should be drawn from
where is an atomic distribution centered on . Interpreting this, two properties are clear:
- Even if is a countable set, there is a finite probability that two samples will have exactly the same value. Samples from a Dirichlet process are therefore discrete.
- The Dirichlet process exhibits a self-reinforcing property; the more often a given value has been sampled in the past, the more likely it is to be sampled again.
The name "Chinese restaurant process" is derived from the following analogy: imagine an infinitely large restaurant containing an infinite number of tables, and able to serve an infinite number of dishes. The restaurant in question operates a somewhat unusual seating policy whereby new diners are seated either at a currently occupied table with probability proportional to the number of guests already seated there, or at an empty table with probability proportional to a constant. Guests who sit at an occupied table must order the same dish as those currently seated, whereas guests allocated a new table are served a dish at random according to the chef's taste. The distribution of dishes after guests are served is a sample drawn as described above. The Chinese Restaurant Process is related to the Polya Urn sampling scheme for finite Dirichlet distributions.
Read more about this topic: Dirichlet Process
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