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At a mundane level, Dirichlet processes are a clustering algorithm that determines the number of clusters. However, they are also a way to do Bayesian inference on a single infinite model rather than ad hoc model selection on a series of finite models and are the gateway to the field of Bayesian non-parametric models. Many introductions to Dirichlet processes take a formal measure-theoretic approach. In contrast, if you can understand the multinomial distribution you will understand this primer.