Likelihood functions for a dominated parametric family

Let $(S,\mathcal{S})$ be a measurable space, the data space (Measurable spaces and measurable sets), let $\nu$ be a measure on it (Measures on sigma-algebras), let $\mathrm{\Theta }$ be a nonempty set, the parameter set, and let $(p_{\theta }{)}_{\theta \in \mathrm{\Theta }}$ be a family of measurable maps $p_{\theta }:(S,\mathcal{S})⟶([0,+\mathrm{\infty }],{\mathcal{B}}_{+}),$ where $[0,+\mathrm{\infty }]$ is the nonnegative part of the extended real line $\bar{\mathbb{R}}$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined) and ${\mathcal{B}}_{+}:=\{A\cap [0,+\mathrm{\infty }]:A\in \mathcal{B}(\bar{\mathbb{R}})\}$ is the trace of its order-topology Borel sigma-algebra (The Borel sigma-algebra on the extended real line, A measurable function between measurable spaces), such that ${\int }_S{p}_{\theta }d\nu =1\text{for every }\theta \in \mathrm{\Theta }$ (The nonnegative Lebesgue integral). Each $p_{\theta }$ is the density of a probability measure ${\mathbb{P}}_{\theta }$ with respect to $\nu$, ${\mathbb{P}}_{\theta }(A)={\int }_A{p}_{\theta }d\nu ,A\in \mathcal{S}$ (The measure with density $f$ relative to $\mu$); the fixed measure $\nu$ is the dominating measure of the family.

For an observed datum $s\in S$ the likelihood function is $L_s:\mathrm{\Theta }⟶[0,+\mathrm{\infty }],L_s(\theta ):=p_{\theta }(s),$ and the log-likelihood function is ${\mathrm{\ell }}_s:\mathrm{\Theta }⟶[-\mathrm{\infty },+\mathrm{\infty }],{\mathrm{\ell }}_s(\theta ):=\{\begin{array}{ll}\log L_s(\theta ),& 0<L_s(\theta )<+\mathrm{\infty },\\ +\mathrm{\infty },& L_s(\theta )=+\mathrm{\infty },\\ -\mathrm{\infty },& L_s(\theta )=0.\end{array}$ Here $\log$ is the real logarithm on $(0,+\mathrm{\infty })$, with the extended-value convention $\log (+\mathrm{\infty }):=+\mathrm{\infty }$.

Well-definedness and conventions

For fixed $s$ the value $p_{\theta }(s)$ is a well-defined element of $[0,+\mathrm{\infty }]$ for each $\theta$, so $L_s$ and ${\mathrm{\ell }}_s$ are defined as functions on $\mathrm{\Theta }$; no measurability or integrability in $\theta$ is asserted, and no topology on $\mathrm{\Theta }$ is assumed. The densities $p_{\theta }$ and the dominating measure $\nu$ are data of the model: a likelihood claim must name the pair $(\nu ,(p_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ it uses, since the same family of probability measures can be represented by different pairs.

The value $L_s(\theta )$ is the density of the observed datum under the parameter $\theta$; it is not a probability of the parameter value $\theta$, and no distribution on $\mathrm{\Theta }$ is part of this definition.