Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form

Let $\tilde{l}\ge1$ and $n\ge1$ be natural numbers, 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 $\mathbf{R}$ and their likelihoods are as in those definitions; integrals over $\mathbf{R}$ of nonnegative measurable functions are those of Lebesgue Integral of a Nonnegative Measurable Function, integrable means integrable with respect to $\rho$, $\int_{[0,T]}\cdot\,ds$ is the Lebesgue integral over the compact interval $[0,T]$, $\exp$ is the real exponential function, and $\mathbf{1}_A$ denotes the indicator of a set $A$.

Let $\mu=(\mu^\upsilon)_{\upsilon\in V}$ be a causal intensity on $\mathbf{R}$ with bound $\bar\mu$ for which there is a real number $\underline\mu>0$ with $\mu^\upsilon_s(r)\ge\underline\mu$ for all $s\in[0,T]$, $r\in\mathbf{R}$ and $\upsilon\in V$ (the base intensity), and let $\mu_1,\dots,\mu_n$ be causal intensities on $\mathbf{R}$ with bounds $\bar\mu_1,\dots,\bar\mu_n$ (the perturbed intensities). Write $\ell=\ell_\mu$ and $\ell_q=\ell_{\mu_q}$ for the likelihoods and $L_q=\ell_q/\ell$ for the likelihood ratios ($1\le q\le n$). For $q,q'\in\{1,\dots,n\}$ let $\tilde\mu_{qq'}=(\mu_q^\upsilon\mu_{q'}^\upsilon/\mu^\upsilon)_{\upsilon\in V}$ be the pair intensity and $E_{qq'}:\mathbf{R}\to\mathbb{R}$ the pair exponent, $E_{qq'}(r)=\int_{[0,T]}\sum_{\upsilon\in V}\frac{\bigl(\mu^\upsilon_{q,s}(r)-\mu^\upsilon_s(r)\bigr)\bigl(\mu^\upsilon_{q',s}(r)-\mu^\upsilon_s(r)\bigr)}{\mu^\upsilon_s(r)}\,ds,$ as in claim 3 of Integer Power Products of Causal-Intensity Likelihoods: Pathwise Identity, Integral Bounds, and the Pair Identity (applied to $(\mu,\mu_q,\mu_{q'})$), and define the pair covariance $C_{qq'}=\int_{\mathbf{R}}\ell\,(1-L_q)(1-L_{q'})\,d\rho .$ Finally let $w=(w_1,\dots,w_n)$ be a point of Euclidean space $\mathbb{R}^n$ (the weights) and let $\mathsf{r}_1,\dots,\mathsf{r}_n\ge0$ be real numbers (the ratio factors).

1. (Pair covariances) $\ell>0$ on $\mathbf{R}$; each $L_q$ is $\mathcal{R}$-measurable; $\tilde\mu_{qq'}$ is a causal intensity on $\mathbf{R}$ with $\int_{\mathbf{R}}\ell_{\tilde\mu_{qq'}}\,d\rho=1$; $E_{qq'}$ is $\mathcal{R}$-measurable and bounded; the functions $\ell$, $\ell_q$, $\ell L_qL_{q'}$, $\ell(1-L_q)(1-L_{q'})$ and $\ell_{\tilde\mu_{qq'}}(\exp(E_{qq'})-1)$ are integrable; and $\int_{\mathbf{R}}\ell\,d\rho=\int_{\mathbf{R}}\ell L_q\,d\rho=1,\qquad \int_{\mathbf{R}}\ell L_qL_{q'}\,d\rho=1+C_{qq'},\qquad C_{qq'}=C_{q'q}=\int_{\mathbf{R}}\ell_{\tilde\mu_{qq'}}\bigl(\exp(E_{qq'})-1\bigr)\,d\rho .$ In particular $\mathrm{Var}_q=C_{qq}=\int_{\mathbf{R}}\ell(L_q-1)^{2}\,d\rho$ (the ratio variance) is finite and nonnegative, and $\ell L_q=\ell_q$ is integrable. Here $|E_{qq'}|\le\tilde{l}T(\bar\mu_q\bar\mu_{q'}/\underline\mu+\bar\mu+\bar\mu_q+\bar\mu_{q'})$ on $\mathbf{R}$.

2. (Exact expansion) The function $\ell\bigl(\sum_{q=1}^{n}w_q(1-\mathsf{r}_qL_q)\bigr)^{2}$ is integrable and $\int_{\mathbf{R}}\ell\Bigl(\sum_{q=1}^{n}w_q\bigl(1-\mathsf{r}_qL_q\bigr)\Bigr)^{2}d\rho=\Bigl(\sum_{q=1}^{n}w_q(1-\mathsf{r}_q)\Bigr)^{2}+\sum_{q=1}^{n}\sum_{q'=1}^{n}w_qw_{q'}\,\mathsf{r}_q\mathsf{r}_{q'}\,C_{qq'} .$

3. (First-order form) Fix $q,q'\in\{1,\dots,n\}$, let $\bar{E}\ge0$ be a real number, and let $\mathsf{N}\in\mathcal{R}$ with $\rho(\mathsf{N})=0$ be such that $|E_{qq'}(r)|\le\bar{E}$ for all $r\in\mathbf{R}\setminus\mathsf{N}$. Then $\ell_{\tilde\mu_{qq'}}E_{qq'}$ is integrable, and $|C_{qq'}|\le\exp(\bar{E})-1,\qquad \Bigl|C_{qq'}-\int_{\mathbf{R}}\ell_{\tilde\mu_{qq'}}E_{qq'}\,d\rho\Bigr|\le\tfrac12\,\bar{E}^{2}\exp(\bar{E}) .$