Design-indexed dominated inverse experiment

Let $D$ be a finite set of design indices, let $(\mathrm{\Theta },\mathcal{T})$ and $(\mathsf{Y},\mathcal{Y})$ be measurable spaces, let ${\mu }_0$ be a probability measure on $(\mathrm{\Theta },\mathcal{T})$, and let $\nu$ be a sigma-finite measure on $(\mathsf{Y},\mathcal{Y})$. For each $d\in D$, let $Q_d$ be a probability kernel from $\mathrm{\Theta }$ to $\mathsf{Y}$ with a specified nonnegative jointly measurable density $p_d(y\mid \theta )$ relative to $\nu$, so $Q_d(\theta ,B)={\int }_B{p}_d(y\mid \theta )\nu (dy)(\theta \in \mathrm{\Theta },B\in \mathcal{Y})\mathrm{.}$ Define the prior-predictive density $p_d(y)={\int }_{\mathrm{\Theta }}{p}_d(y\mid \theta ){\mu }_0(d\theta )$. For each datum $y$ with $0<p_d(y)<\mathrm{\infty }$, define the posterior probability measure ${\mu }_d^y(A)=\frac{{\int }_A{p}_d(y\mid \theta ){\mu }_0(d\theta )}{p_d(y)}(A\in \mathcal{T})\mathrm{.}$ The dominated-kernel Bayes formula makes $p_{d}d\nu$ the observation law and identifies ${\mu }_d^y$ as a posterior version on these positive finite evidence fibers. No posterior formula is assigned on fibers with zero or infinite evidence; their prior-predictive probability is zero.

A finite set has a finite cardinality.

A measurable space consists of a set and a sigma-algebra.

A probability measure has total mass one.

Sigma-finiteness is witnessed by a countable cover with finite measure.

A probability kernel has measurable evaluations and probability-measure rows.

Joint measurability is with respect to the product sigma-algebra.

A measurable function has measurable preimages.

A specified dominated sampling density yields the predictive density and the posterior kernel on positive finite evidence fibers.

A posterior is a regular conditional law of the parameter given the observations.

Chaloner and Verdinelli, §1.2

MacKay, §2