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A third approach to the Dirichlet process is the so-called stick-breaking process view. Conceptually, this involves repeatedly breaking off and discarding a random fraction (sampled from a Beta distribution) of a "stick" that is initially of length 1. Remember that draws from a Dirichlet process are distributions over a set . As noted ...
Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution. The infinite-dimensional generalization of the Dirichlet distribution is the Dirichlet process.
In statistics and machine learning, the hierarchical Dirichlet process (HDP) is a nonparametric Bayesian approach to clustering grouped data. [1] [2] It uses a Dirichlet process for each group of data, with the Dirichlet processes for all groups sharing a base distribution which is itself drawn from a Dirichlet process. This method allows ...
Dirichlet also lectured on probability theory and least squares, introducing some original methods and results, in particular for limit theorems and an improvement of Laplace's method of approximation related to the central limit theorem. [21] The Dirichlet distribution and the Dirichlet process, based on the Dirichlet integral, are named after ...
For categorical variables, i.e., when has a finite number of elements, it is known that the Dirichlet process reduces to a Dirichlet distribution. In this case, the Imprecise Dirichlet Process reduces to the Imprecise Dirichlet model proposed by Walley [7] as a model for prior (near)-ignorance for chances.
The dependent Dirichlet process (DDP) originally formulated by MacEachern led to the development of the DDP mixture model (DDPMM) which generalizes DPMM by including birth, death and transition processes for the clusters in the model. In addition, a low-variance approximations to DDPMM have been derived leading to a dynamic clustering algorithm ...
Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class of semi-elliptic second-order partial differential equations the associated Dirichlet boundary value problem can be solved using an Itō process that solves ...
When d = 0, it becomes the Dirichlet process. The discount parameter gives the Pitman–Yor process more flexibility over tail behavior than the Dirichlet process, which has exponential tails. This makes Pitman–Yor process useful for modeling data with power-law tails (e.g., word frequencies in natural language).