Dirichlet-process posterior update

For $x=(x_1,\dots ,x_n)$ put ${\beta }_x=\alpha +{\sum }_{i=1}^n{\delta }_{x_i}$. This is a finite nonzero measure because its total mass is $\alpha (\mathsf{X})+n$ by [F8]. The existence theorem [F1] therefore defines $DP({\beta }_x)$ for every $x$.

Fix a finite evaluation cylinder $C=\{G:(G(A_1),\dots ,G(A_s))\in H\}$ with Borel $H$, and form the finite Boolean partition generated by $A_1,\dots ,A_s$. On each cell $P_j$, the updated parameter is ${\beta }_x(P_j)=\alpha (P_j)+{\sum }_i{\mathbf{1}}_{P_j}(x_i)$. Each count is a measurable finite-valued function of $x$ by [F9], so $x\mapsto DP({\beta }_x)(C)$ is measurable: it is constant on each of finitely many count-pattern preimages.

For each $G\in \mathcal{P}(\mathsf{X})$, the map $A\mapsto G(A)$ is a probability kernel from the evaluation-measurable space $\mathcal{P}(\mathsf{X})$ to $(\mathsf{X},\mathcal{A})$. Its finite product $G^{\otimes n}$ is a probability kernel by [F2]; on rectangles $D_1\times \cdots \times D_n$ it has value ${\prod }_{i}G(D_i)$, which is evaluation-measurable. The proper model in [F3] therefore gives a joint probability $J$ for $(G,X_{1:n})$ and its data marginal $M$.

Fix a finite-partition cylinder $C$ and a data rectangle $D=D_1\times \cdots \times D_n$. Refine the partition defining $C$ by all $D_i$ to obtain a finite measurable partition $P_1,\dots ,P_m$ of $\mathsf{X}$. The data rectangle is a disjoint union of ordered cell-pattern blocks $P_{j_1}\times \cdots \times P_{j_n}$. For one such block let $y_j=\mathrm{\#}\{i:j_i=j\}$. If $y_j>0$ for a cell with $\alpha (P_j)=0$, then $G(P_j)=0$ almost surely by [F1], so the block has prior-predictive mass zero.

The finite evaluation cylinders form a generating pi-system. The class of sets $C$ for which $x\mapsto DP({\beta }_x)(C)$ is measurable contains this pi-system, contains the whole space, is closed under complements, and is closed under countable disjoint unions by pointwise limits of partial sums. Dynkin's theorem [F7] gives measurability on the full evaluation sigma-algebra. Each section is a probability measure by [F1], so $K$ is a probability kernel.

Otherwise omit cells with $\alpha (P_j)=0$ (their counts are zero). On the remaining cells the prior vector is Dirichlet with positive parameters $a_j=\alpha (P_j)$. Given that vector, the likelihood of this particular ordered pattern is ${\prod }_{j}G(P_j{)}^{y_j}$, a constant multiple ${\prod }_j{y}_j!\mathrm{/}n!$ of the multinomial count likelihood. By [A2] and [F5], its posterior vector is Dirichlet$(a_1+y_1,\dots ,a_m+y_m)$ with positive finite evidence. Coarsening to the cells defining $C$ gives the same cylinder probability as $DP({\beta }_x)$ by [F6]. This probability is constant over the ordered block because its cell counts are fixed.

Summing step 2.2 over the ordered blocks contained in $D$ gives $J(C\times D)={\int }_{D}K(x,C)M(dx)$ for every finite-partition cylinder $C$ and data rectangle $D$. For fixed such $C$, the two sides define finite measures of $D$; their total masses agree by taking $D={\mathsf{X}}^n$. Finite-measure uniqueness [F7] extends the identity to every $D\in {\mathcal{A}}^{\otimes n}$.

Now fix any $D\in {\mathcal{A}}^{\otimes n}$. The set functions $C\mapsto J(C\times D)$ and $C\mapsto {\int }_{D}K(x,C)M(dx)$ are finite measures on the evaluation sigma-algebra; the second is a measure because $K$ is a probability kernel. They have common total mass $M(D)$ and agree on every finite-partition cylinder by step 3.1. These cylinders form a generating pi-system, so [F7] extends equality to every measurable $C$. By [F4], $K(x;\cdot )$ is the claimed posterior version. Null data blocks contribute zero to both sides, so the proof imposes no pointwise uniqueness there.