This week, our discussion focused on estimating the hyperparameter for Dirichlet process models. We began by working through a couple of theorems in Antoniak (1974) for mixtures of Dirichlet processes. Importantly, it can be shown, in the language of Chinese restaurant processes, that the number of occupied tables and number of samples are sufficient to find a distribution over the Dirichlet hyperparameter.

Given a gamma prior for the hyperparameter, we worked through the derivation of a posterior distribution for the hyperparamter given the number of occupied tables and number of observations given by Escobar & West (1995). This results in an easily samplable mixture of two gamma distributions which can be added to the Gibbs sampling scheme we reviewed last week.

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