COS 597 C : Bayesian nonparametrics

In this first lecture, we begin by introducing the Chinese Restaurant Process. After a brief review of finite mixture models, we describe the Chinese Restaurant Process mixture, where the latent variables are distributed according to a Chinese Restaurant Process. We end by noting a connection between Dirchlet processes and the Chinese Restaurant Process mixture. We will define a distribution on the space of partitions of the positive integers, N. This would induce a distribution on the partitions of the first n integers, for every n ∈ N. Imagine a restaurant with countably infinitely many tables, labelled 1, 2,. . .. Customers walk in and sit down at some table. The tables are chosen according to the following random process. 1. The first customer always chooses the first table. 2. The nth customer chooses the first unoccupied table with probability α n−1+α , and an occupied table with probability c n−1+α , where c is the number of people sitting at that table. In the above, α is a scalar parameter of the process. One might check that the above does define a probability distribution. Let us denote by k n the number of different tables occupied after n customers have walked in. Then 1 ≤ k n ≤ n and it follows from the above description that precisely tables 1,. .. , k n are occupied.