Dirichlet process

In probability theory, Dirichlet processes (after Peter Gustav Lejeune Dirichlet) are a family of whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular distribution.

The Dirichlet process is specified by a base distribution ${\displaystyle H}$ and a positive real number ${\displaystyle \alpha }$ called the concentration parameter (also known as scaling parameter). The base distribution is the expected value of the process, i.e., the Dirichlet process draws distributions "around" the base distribution the way a normal distribution draws real numbers around its mean. However, even if the base distribution is continuous, the distributions drawn from the Dirichlet process are almost surely discrete. The scaling parameter specifies how strong this discretization is: in the limit of ${\displaystyle \alpha \rightarrow 0}$, the realizations are all concentrated at a single value, while in the limit of ${\displaystyle \alpha \rightarrow \infty }$ the realizations become continuous. Between the two extremes the realizations are discrete distributions with less and less concentration as ${\displaystyle \alpha }$ increases.

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