Integer Power Products of Causal-Intensity Likelihoods: Pathwise Identity, Integral Bounds, and the Pair Identity

Let $\tilde{l}\ge1$ be a natural number, let $T>0$ be a real number, and let $(\mathbf{R},\mathcal{R},\rho)$ be the observation record space with horizon $T$ and $\tilde{l}$ channels, with channel set $V$. Causal intensities on $\mathbf{R}$, their total intensities and their likelihoods are as in those definitions; for a causal intensity $\mu$ write $\Lambda_T(r)=\int_{[0,T]}\mu^{\mathrm{tot}}_s(r)\,ds$ (the Lebesgue integral over $[0,T]$), which is $\mathcal{R}$-measurable in $r$ by claim 2 of Survival Identity, Measurability, Event-Count Recursion, and Normalization of the Likelihood of a Causal Intensity. Integrals over $\mathbf{R}$ are those of Lebesgue Integral of a Nonnegative Measurable Function, $\exp$ is the real exponential function, and for a real $x>0$ and an integer $\theta$ the power $x^{\theta}$ is the finite product of $\theta$ copies of $x$ if $\theta\ge1$, $1$ if $\theta=0$, and $1/x^{-\theta}$ if $\theta\le-1$ (for $x\ge0$ and $\theta\ge0$ the same reading applies).

Products over a subset of the index range, such as $\prod_{i:\theta_i>0}$, mean the product over $1,\dots,n$ with the omitted factors replaced by $1$; an empty product is $1$.

1. (Power product) Let $n\ge1$ be a natural number, let $\mu_1,\dots,\mu_n$ be causal intensities on $\mathbf{R}$ with bounds $\bar\mu_1,\dots,\bar\mu_n$, written $\mu_i=(\mu^\upsilon_i)_{\upsilon\in V}$ with $\mu^\upsilon_{i,s}(r)=\mu^\upsilon_i(s,r)$, and let $\theta_1,\dots,\theta_n$ be integers. Assume there is a real $\underline\mu>0$ such that $\mu^\upsilon_{i,s}(r)\ge\underline\mu$ for all $s$, $r$, $\upsilon$ and every $i$ with $\theta_i<0$. Define $\tilde\mu=(\tilde\mu^\upsilon)_{\upsilon\in V}$ by $\tilde\mu^\upsilon_s(r)=\prod_{i=1}^{n}\bigl(\mu^\upsilon_{i,s}(r)\bigr)^{\theta_i},$ and $E:\mathbf{R}\to\mathbb{R}$ by $E(r)=\tilde\Lambda_T(r)-\sum_{i=1}^{n}\theta_i\Lambda_{i,T}(r)$, where $\tilde\Lambda_T$ and $\Lambda_{i,T}$ are the integrated total intensities of $\tilde\mu$ and $\mu_i$. Then $\tilde\mu$ is a causal intensity on $\mathbf{R}$ with bound $\bar{\tilde\mu}=\prod_{i:\theta_i>0}\bar\mu_i^{\theta_i}\prod_{i:\theta_i<0}\underline\mu^{\theta_i}$; $E$ is $\mathcal{R}$-measurable with $|E(r)|\le\tilde{l}T\bigl(\bar{\tilde\mu}+\sum_{i=1}^{n}|\theta_i|\bar\mu_i\bigr)$; for every $i$ with $\theta_i<0$ the likelihood $\ell_{\mu_i}$ is strictly positive; and for every $r\in\mathbf{R}$, $\prod_{i=1}^{n}\ell_{\mu_i}(r)^{\theta_i}=\ell_{\tilde\mu}(r)\exp\bigl(E(r)\bigr).$

2. (Integral bounds) In the setting of claim 1, the function $r\mapsto\prod_{i=1}^{n}\ell_{\mu_i}(r)^{\theta_i}$ is $\mathcal{R}$-measurable and nonnegative, and if $\underline E$ and $\bar E$ are real numbers with $\underline E\le E(r)\le\bar E$ for every $r\in\mathbf{R}$, then $\exp(\underline E)\le\int_{\mathbf{R}}\prod_{i=1}^{n}\ell_{\mu_i}^{\theta_i}\,d\rho\le\exp(\bar E).$ In particular, if $E$ is constant with value $E_0$, the integral equals $\exp(E_0)$.

3. (Pair identity) Let $\mu,\mu',\mu''$ be causal intensities on $\mathbf{R}$ with bounds $\bar\mu,\bar\mu',\bar\mu''$, and assume there is a real $\underline\mu>0$ with $\mu^\upsilon_s(r)\ge\underline\mu$ for all $s,r,\upsilon$. Then claim 1 applies to $(\mu,\mu',\mu'')$ with exponents $(-1,1,1)$, giving $\tilde\mu^\upsilon=\mu'^\upsilon\mu''^\upsilon/\mu^\upsilon$ and $E(r)=\int_{[0,T]}\sum_{\upsilon\in V}\frac{\bigl(\mu'^\upsilon_s(r)-\mu^\upsilon_s(r)\bigr)\bigl(\mu''^\upsilon_s(r)-\mu^\upsilon_s(r)\bigr)}{\mu^\upsilon_s(r)}\,ds ;$ moreover $\ell_\mu>0$, the likelihood ratios $L'=\ell_{\mu'}/\ell_\mu$ and $L''=\ell_{\mu''}/\ell_\mu$ are measurable, the functions $\ell_\mu$, $\ell_{\mu'}$, $\ell_{\mu''}$, $\ell_{\mu'}\ell_{\mu''}/\ell_\mu$ and $\ell_\mu(1-L')(1-L'')$ are integrable with respect to $\rho$, and $\int_{\mathbf{R}}\ell_\mu\,(1-L')(1-L'')\,d\rho=\int_{\mathbf{R}}\frac{\ell_{\mu'}\ell_{\mu''}}{\ell_\mu}\,d\rho-1 .$