Good-Bad Splitting of the Integrated Symmetrised Score Functional: Chebyshev Bound for the Under-Likelihood Set, Transfer of Mass Between Densities, and the Split Bound

Let $n\ge1$ be a natural number, let $w=(w_1,\dots,w_n)$ be a point of Euclidean space $\mathbb{R}^n$ with $\lVert w\rVert_1=\sum_{q=1}^{n}|w_q|$, and let $\Phi_w:[0,\infty)^{1+n}\to[0,\infty)$ be the symmetrised score functional for the weights $w$. Let $(\mathsf{Z},\mathcal{Z},\nu)$ be a measure space; measurability of real-valued maps on $\mathsf{Z}$ is with respect to $\mathcal{Z}$ and the Borel $\sigma$-algebra, integrals of $[0,\infty]$-valued measurable maps are those of Lebesgue Integral of a Nonnegative Measurable Function, $\int_Ah\,d\nu$ denotes $\int_{\mathsf{Z}}h\mathbf{1}_A\,d\nu$ for $A\in\mathcal{Z}$ with $\mathbf{1}_A$ the indicator of $A$, and $t^{1/2}$ is the nonnegative square root of $t\in[0,\infty)$, with $\infty^{1/2}=\infty$ and the conventions $0\cdot\infty=0$ and $t+\infty=\infty$ in $[0,\infty]$.

Let $\ell:\mathsf{Z}\to(0,\infty)$ and $\ell_1,\dots,\ell_n:\mathsf{Z}\to[0,\infty)$ be measurable with $\int_{\mathsf{Z}}\ell\,d\nu=1\qquad\text{and}\qquad\int_{\mathsf{Z}}\ell_q\,d\nu=1\quad(1\le q\le n),$ and put $L_q=\ell_q/\ell:\mathsf{Z}\to[0,\infty)$ (the likelihood ratios) and $V_q=\int_{\mathsf{Z}}\ell\,(L_q-1)^{2}\,d\nu\ \in[0,\infty]\qquad(1\le q\le n),$ the ratio variances. Let $\mathsf{r}_1,\dots,\mathsf{r}_n\ge0$ be real numbers (the ratio factors) and let $\delta$ be a real number with $0<\delta\le1$. Put $\pi_q=\int_{\mathsf{Z}}\ell\,\mathbf{1}\{\mathsf{r}_qL_q<1-\delta\}\,d\nu\in[0,1]\quad(1\le q\le n),\qquad \pi=\sum_{q=1}^{n}\pi_q,$ where $\mathbf{1}\{\mathsf{r}_qL_q<1-\delta\}$ is the indicator of $\{z\in\mathsf{Z}:\mathsf{r}_qL_q(z)<1-\delta\}$, and let $\mathsf{U}=\{z\in\mathsf{Z}:\ \mathsf{r}_qL_q(z)<1-\delta\text{ for some }q\}$, the under-likelihood set.

1. (Measurability and the second moment of a ratio) Each $L_q$ is measurable, $\mathsf{U}\in\mathcal{Z}$, $\int_{\mathsf{Z}}\ell L_q\,d\nu=1$, and $\int_{\mathsf{Z}}\ell L_q^{2}\,d\nu=1+V_q$ in $[0,\infty]$.

2. (Chebyshev bound for the under-likelihood set) For every $q$, $\pi_q\le\frac{(1-\mathsf{r}_q)^{2}+\mathsf{r}_q^{2}V_q}{\delta^{2}}.$

3. (Transfer of mass) For every $A\in\mathcal{Z}$ and every $q$, $\int_A\ell_q\,d\nu\le\int_A\ell\,d\nu+V_q^{1/2}\Bigl(\int_A\ell\,d\nu\Bigr)^{1/2}.$

4. (Split bound) The maps $z\mapsto\Phi_w\bigl(\ell(z),\mathsf{r}_1\ell_1(z),\dots,\mathsf{r}_n\ell_n(z)\bigr)$ and $z\mapsto\ell(z)\bigl(\sum_{q=1}^{n}w_q(1-\mathsf{r}_qL_q(z))\bigr)^{2}$ are measurable, and in $[0,\infty]$ $\int_{\mathsf{Z}}\Phi_w\bigl(\ell,\mathsf{r}_1\ell_1,\dots,\mathsf{r}_n\ell_n\bigr)\,d\nu\ \le\ (1+\delta)\int_{\mathsf{Z}}\ell\Bigl(\sum_{q=1}^{n}w_q\bigl(1-\mathsf{r}_qL_q\bigr)\Bigr)^{2}d\nu\ +\ 2n\lVert w\rVert_1^{2}\Bigl[\pi+\sum_{q=1}^{n}\mathsf{r}_q\bigl(\pi+V_q^{1/2}\pi^{1/2}\bigr)\Bigr].$ More precisely, the first integral on the right may be restricted to $\mathsf{Z}\setminus\mathsf{U}$, and the bracket is an upper bound for $\int_{\mathsf{U}}\ell\,d\nu+\sum_q\mathsf{r}_q\int_{\mathsf{U}}\ell_q\,d\nu$.