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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.

Let $N_m=|\{i:c_i=m\}|$. The label likelihood for the mixture weights is proportional to ${\prod }_{m=1}^3{\pi }_m^{N_m}$. Multiplication by the $\mathrm{Dirichlet}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ density and Dirichlet-multinomial conjugacy givesConditional on the labels, the observations and component means provide no further information about $\pi$.