The Stick-breaking Process
A third approach to the Dirichlet process is provided by the so-called stick-breaking process, which can be used to provide a constructive algorithm (the stick-breaking construction) for generating a Dirichlet process. Let be a set of random variables such that
where is the normalisation constant for the measure, so that . Define according to
and let be a set of samples from . The distribution given by the density (where is the Dirac delta function), is then a sample from the corresponding Dirichlet process. This method provides an explicit construction of the non-parametric sample, and makes clear the fact that the samples are discrete.
The name 'stick-breaking' comes from the interpretation of as the length of the piece of a unit-length stick assigned to the kth value. After the first k − 1 values have their portions assigned, the length of the remainder of the stick, is broken according to a sample from a beta distribution. In this analogy, indicates the portion of the remainder to be assigned to the k-th value. The smaller is, the less of the stick will be left for subsequent values (on average).
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