Evidence normalizer and fiberwise posterior update

Let $(\mathrm{\Theta },\mathcal{T})$ be a measurable parameter space, let $\mathrm{\Pi }$ be a sigma-finite measure on it, and fix a data value $y$. Let $L_y:\mathrm{\Theta }\to [0,+\mathrm{\infty }]$ be a $\mathcal{T}$-measurable function. Define its evidence normalizer by $Z_{\mathrm{\Pi }}(y):={\int }_{\mathrm{\Theta }}{L}_y(\theta )\mathrm{\Pi }(d\theta )\in [0,+\mathrm{\infty }]\mathrm{.}$ Only when $0<Z_{\mathrm{\Pi }}(y)<+\mathrm{\infty }$ define the fiberwise posterior update by ${\mathrm{\Pi }}_y(A):=\frac{{\int }_A{L}_y(\theta )\mathrm{\Pi }(d\theta )}{Z_{\mathrm{\Pi }}(y)}(A\in \mathcal{T})\mathrm{.}$ The definition makes no quotient at zero or infinite evidence. A prior is proper when $\mathrm{\Pi }(\mathrm{\Theta })=1$, as in Bayesian model as a prior and sampling kernel. We call a sigma-finite prior improper when $\mathrm{\Pi }(\mathrm{\Theta })=+\mathrm{\infty }$; the formula also applies to other sigma-finite measures. For an improper prior, $Z_{\mathrm{\Pi }}(y)$ is a fiberwise normalizer only; it does not by itself define a prior-predictive probability or a conditional law.

When the data are described by a dominated experiment, the dominating measure and its sigma-finiteness are as in Dominated statistical experiments. A family of pointwise likelihood sections is not assumed to be jointly measurable in parameter and data; that additional condition must be supplied separately whenever Tonelli is used.