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A Dirichlet process is a probability distribution over probability measures. For concentration $\alpha >0$ and base probability measure $G_0$, write $G\sim \mathrm{DP}(\alpha ,G_0)$. Every finite measurable partition $A_1,\dots ,A_k$ satisfieswith zero-base-mass cells assigned zero mass. Consequently $\mathbb{E}G(A)=G_0(A)$ and $\mathrm{Var}(G(A))=G_0(A)(1-G_0(A))/(\alpha +1)$: larger concentration keeps random masses closer to the base measure.

For conditionally independent and identically distributed observations $X_i\mid G\sim G$, the posterior isThis follows by Dirichlet-multinomial conjugacy on each partition. It supplies a tractable prior distribution for an unknown distribution without choosing a fixed-dimensional parameter family.^[ref]

Draws are almost surely discrete on the usual standard Borel sample spaces, even when $G_0$ is continuous. A Dirichlet process prior on a mixing measure can still yield a continuous mixture density.

Thomas S. Ferguson works on statistics, probability theory and game theory. His UCLA page documents research on statistical decisions, stopping problems and games.^[ref]

His 1973 paper introduced a Bayesian framework built on the Dirichlet process, a probability distribution over probability measures. If the prior parameter measure is $\alpha$, observing $x_1,\dots ,x_n$ changes it to $\alpha +{\sum }_i{\delta }_{x_i}$. This conjugate update extends the familiar idea of adding category counts to inference about an entire unknown distribution, providing a foundation for Bayesian nonparametric statistics.^[ref]