Dirichlet Process - Applications of The Dirichlet Process

Applications of The Dirichlet Process

Dirichlet processes are frequently used in Bayesian nonparametric statistics. "Nonparametric" here does not mean a parameter-less model, rather a model in which representations grow as more data are observed. Bayesian nonparametric models have gained considerable popularity in the field of machine learning because of the above-mentioned flexibility, especially in unsupervised learning. In a Bayesian nonparametric model, the prior and posterior distributions are not parametric distributions, but stochastic processes. The fact that the Dirichlet distribution is a probability distribution on the simplex of non-negative numbers that sum to one makes it a good candidate to model distributions of distributions or distributions of functions. Additionally, the non-parametric nature of this model makes it an ideal candidate for clustering problems where the distinct number of clusters is unknown beforehand.

As draws from a Dirichlet process are discrete, an important use is as a prior probability in infinite mixture models. In this case, is the parametric set of component distributions. The generative process is therefore that a sample is drawn from a Dirichlet process, and for each data point in turn a value is drawn from this sample distribution and used as the component distribution for that data point. The fact that there is no limit to the number of distinct components which may be generated makes this kind of model appropriate for the case when the number of mixture components is not well-defined in advance. For example, the infinite mixture of Gaussians model.

The infinite nature of these models also lends them to natural language processing applications, where it is often desirable to treat the vocabulary as an infinite, discrete set.