Search results
Results From The WOW.Com Content Network
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.
For example, the solution to the Dirichlet problem for the unit disk in R 2 is given by the Poisson integral formula. If f {\displaystyle f} is a continuous function on the boundary ∂ D {\displaystyle \partial D} of the open unit disk D {\displaystyle D} , then the solution to the Dirichlet problem is u ( z ) {\displaystyle u(z)} given by
The Birnbaum–Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. The chi distribution. The noncentral chi distribution; The chi-squared distribution, which is the sum of the squares of n independent Gaussian random variables.
For example, the following would be considered Dirichlet boundary conditions: In mechanical engineering and civil engineering (beam theory), where one end of a beam is held at a fixed position in space. In heat transfer, where a surface is held at a fixed temperature. In electrostatics, where a node of a circuit is held at a fixed voltage.
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.
For any real number x and any positive rational number T, (+) = (). The Dirichlet function is therefore an example of a real periodic function which is not constant but whose set of periods, the set of rational numbers, is a dense subset of R {\displaystyle \mathbb {R} } .
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 ...