Dirichlet process

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Dirichlet processes were introduced in (Ferguson 1973). As stated therein, they can be described as follows:

Let $\mathcal{H}$ be a space and $\mathcal{A}$ a σ-field of subsets and let α be a finite non-null measure(mathematics) on $\mathfrak{F}=(\mathcal{H}, \mathcal{A})$ . Then a stochastic process $P$ indexed by elements $A$ of $\mathcal{A}$ is said to be a Dirichlet process on $\mathfrak{F}$ with parameter α if for any measurable partition $(A_1, \ldots, A_k)$ of $\mathcal{H}$ , the random vector $\left(P(A_1), \ldots, P(A_k)\right)$ has Dirichlet distribution with parameter $\left(\alpha(A_1), \ldots, \alpha(A_k)\right)$ , then $P$ may be considered a random probability measure on $\mathfrak{F}$ .

The main theorem therein states that if $P$ is a Dirichlet process on $\mathfrak{F}$ with parameter α and $X_1, \ldots, X_n$ is a sample from $P$ , then the posterior distribution of $P$ given $X_1, \ldots, X_n$ is also a Dirichlet process on $\mathfrak{F}$ with parameter $\alpha + \sum_{i=1}^n \delta_{X_i}$ , where $δ x$ denotes the process giving unit mass to the point $x$ .

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[edit] References

Ferguson, Thomas (1973). "Bayesian analysis of some nonparametric problems". Annals of Statistics 1: 209--230.

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Dirichlet process

From Wikipedia, the free encyclopedia

[edit] References

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