A standard-Borel posterior kernel for countable data

The discrete metric on ${\mathbb{N}}_0$ is complete because every Cauchy sequence is eventually constant; ${\mathbb{N}}_0$ itself is a countable dense subset. Its topology is discrete, so its Borel sigma-algebra is the power set. The finite discrete parameter space has the same properties. Thus both spaces are standard Borel by [F5]–[F6]. By [F4], ${\sum }_n{f}_0(n)=1$ and ${\sum }_n{f}_1(n)=1$, so they define probability sampling laws and the prior in [F1] gives a Bayesian model.

Since both spaces are standard Borel and AC holds, [F3] supplies a data-indexed posterior kernel. For every $n$, both $f_0(n)$ and $f_1(n)$ are positive and finite, so $m(n)>0$ and the displayed values are in $[0,1]$ and sum to one. Every function on the discrete data space is measurable, hence $n\mapsto {\mathrm{\Pi }}_n$ is a probability kernel.

For $i\in \{0,1\}$ and every $B\subseteq {\mathbb{N}}_0$, ${\sum }_{n\in B}{\mathrm{\Pi }}_n(i)m(n)=\frac{1}{2}{\sum }_{n\in B}{f}_i(n)=J(\{i\}\times B)$. Adding the identities for $i=0,1$ proves the rectangle identity for every $A\subseteq \mathrm{\Theta }$ and every data event $B$. Thus the explicit kernel is a data-indexed posterior version; all its data values have positive prior-predictive mass.