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Ferguson's 1973 paper on the Dirichlet process
If you have a question about this talk, please contact Richard Samworth.
In 1973, Ferguson proposed to perform nonparametric estimation in a Bayesian framework by defining a prior distribution on an infinite-dimensional parameter space (the set of probability measures over a given domain). When applied to a finite set of observations, only a finite number out of the infinitely many degrees of freedom is used to explain the data, which accounts for the term “nonparametric”. The prior model is constructed as a stochastic process, with Dirichlet marginals and pathes in the set of probability measures over a separable metric space, that Ferguson called a “Dirichlet process”. His estimation model has a conjugate form with a closed-form solution for the posterior parameters, which mimics the conjugate posteriors of the Dirichlet marginals under a multinomial sampling model. Measures drawn at random from the Dirichlet process are a.s. discrete.
I will review Ferguson’s construction and his application of the model to sample observations. I also intend to briefly discuss the two major lines of research which developed from Ferguson’s paper: One that attempts to overcome the model’s discreteness in order to construct “universal” priors, and one that exploits discreteness to generalize the notion of finite mixtures and related models.
TS Ferguson, “A Bayesian analysis of some nonparametric problems” Ann. Statist. 1 (1973), 209—230.
This talk is part of the Statistics Reading Group series.
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