Conjugate-form prior for an exponential family

Let $(\mathrm{\Theta },\mathcal{T},\rho )$ be a sigma-finite measure space. Let $\eta :\mathrm{\Theta }\to \mathrm{\Omega }\subseteq H$ be measurable, where $\mathrm{\Omega }$ has the trace Borel sigma-algebra and is the parameter set in Exponential families and full rank. Let $A$ be its log-partition function. For $\chi \in {\mathbb{R}}^k$ and $\kappa \in \mathbb{R}$, define $Z(\chi ,\kappa )={\int }_{\mathrm{\Theta }}\exp \{\eta (\theta {)}^T\chi -\kappa A(\eta (\theta ))\}\rho (d\theta )\mathrm{.}$ When $0<Z(\chi ,\kappa )<\mathrm{\infty }$, the conjugate-form prior ${\mathrm{\Pi }}_{\chi ,\kappa }$ is the probability measure with density $\frac{\exp \{\eta (\theta {)}^T\chi -\kappa A(\eta (\theta ))\}}{Z(\chi ,\kappa )}$ relative to $\rho$. If this finite-positive condition fails, this notation does not designate a probability prior.

Log-partition measurability. The log-partition function is Borel on its natural-parameter domain $H=\{\eta :0<Z(\eta )<\mathrm{\infty }\}$. On the product measurable space $({\mathbb{R}}^k\times \mathcal{X},\mathcal{B}({\mathbb{R}}^k)\otimes \mathcal{A})$ (The product sigma-algebra and its finite iterates), the coordinate functions of $\xi$ and $T(x)$ are measurable. Finite sums and products preserve measurability (Arithmetic and lattice operations preserve measurability whenever they are defined), and composition with the continuous exponential is measurable (A continuous map has Borel preimages of Borel sets, The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$); hence $(\xi ,x)\mapsto h(x)e^{{\xi }^{T}T(x)}$ is product-measurable. Regard $\mu$ as the constant kernel $K(\xi ,B)=\mu (B)$ from ${\mathbb{R}}^k$ to $\mathcal{X}$ (Measure kernel and probability kernel). Choose a measurable exhaustion $X_m\uparrow \mathcal{X}$ with $\mu (X_m)<\mathrm{\infty }$, supplied by sigma-finiteness; then $K(\xi ,X_m)=\mu (X_m)$ for every $\xi$, so this is a uniformly sigma-finite kernel. The parameterized-integral measurability theorem (Measurability of integration against a kernel) therefore makes $Z(\xi )=\int h(x)e^{{\xi }^{T}T(x)}\mu (dx)$ a Borel map into $[0,\mathrm{\infty }]$. Consequently $H=\{\xi :0<Z(\xi )<\mathrm{\infty }\}$ is Borel, and $A(\xi )=\log Z(\xi )$ is Borel on $H$ because logarithm is differentiable, hence continuous, on $(0,\mathrm{\infty })$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, A function differentiable at $c$ is continuous at $c$, A continuous map has Borel preimages of Borel sets). Thus $A\circ \eta$ and $\theta \mapsto \exp \{\eta (\theta {)}^T\chi -\kappa A(\eta (\theta ))\}$ are measurable. Since $\rho$ is sigma-finite, the integral defining $Z(\chi ,\kappa )$ is a well-defined extended nonnegative integral; under its stated finite-positive condition the displayed density is measurable and normalizes to one.