Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass

Adopt the setting, hypotheses (OC), (X), (W) and notation of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances (and hence of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound and The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record): the probability space $(\Omega,\mathcal{F},P)$ with expectation $\mathbb{E}$, the cells indexed by $\mathsf{L}$ ($d$ elements) with lengths $\mu_q$ ($q\in\mathsf{L}$), the cell-count vector $\mathsf{K}=(\mathsf{K}_q)_{q\in\mathsf{L}}$, the move size $\mathsf{m}$, the weights $w=(w_q)_{q\in\mathsf{L}}$ with $\lVert w\rVert_1=\sum_q|w_q|$, the removal ratios $\varrho_q$ and the random variables $\varrho_q(\mathsf{K}_q)$, the symmetrised kernel-weighted move information $\mathsf{J}^{\mathrm{sym}}$, the likelihoods $\ell^{\sharp,\omega}$ and $\ell^{-q,\omega}$, the pathwise score map $\Psi$, the constants $\Gamma$, $A_0$ and $\bar{E}_N$, the clock-good event $G_{L,D}$, the tracked records $\mathsf{T}_\omega$, the effective removed intensities $\lambda^{-q,\omega}$ with likelihoods $\ell_{\lambda^{-q,\omega}}$ and likelihood ratios $L^{\omega}_q=\ell_{\lambda^{-q,\omega}}/\ell^{\sharp,\omega}$, and the pair covariances $C^{\omega}_{qq'}$ of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form for the base $\lambda^{\sharp,\omega}$ and the perturbed intensities $(\lambda^{-q,\omega})_q$. Let $\varpi_\mu$ be the Chernoff exponent of Series Formula, Exponential Moments, and Chernoff Tail Bounds for the Poisson Distribution, $\exp$ the real exponential function, $t^{1/2}$ the nonnegative square root, and $\mathbf{1}_A$ the indicator of an event $A$. Put $\varepsilon_0=\frac{\Gamma A_0}{N\underline{b}},\qquad \mathsf{c}_N=\exp(\bar{E}_N)-1,\qquad \mathsf{j}_q=\mathbb{E}\bigl[\varrho_q(\mathsf{K}_q)^{2}\bigr]-1=\sum_{i=1}^{\mathsf{m}}\binom{\mathsf{m}}{i}^{2}\,i!\,\mu_q^{-i}\quad(q\in\mathsf{L}),$ the last identity by claim 2 of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound. Assume moreover:

(G) $G\in\mathcal{F}$ is an event with $G\subseteq G_{L,D}$ and $\rho(\mathbf{R}\setminus\mathsf{T}_\omega)=0$ for every $\omega\in G$; put $\mathsf{g}=P(\Omega\setminus G)$;

(P) $\delta$ is a real number with $0<\delta\le1$; $\varepsilon_0\le\tfrac12$; $\theta\ge1$ is an integer with $2\theta\varepsilon_0\le1$; and for every $q\in\mathsf{L}$, $x_q>0$ is a real number with $\mathsf{m}(x_q+\mathsf{m})\le\delta\mu_q/2$. Put $\mathsf{E}^{\mathrm{ch}}_\theta=8\,\tilde{l}\,T\,N\tilde{B}\,\theta^{2}\varepsilon_0^{2},\qquad \bar\Pi=\sum_{q\in\mathsf{L}}\Bigl(\exp\bigl(-\varpi_{\mu_q}(x_q)\bigr)+\exp\bigl(-\theta\delta/2+\mathsf{E}^{\mathrm{ch}}_\theta\bigr)\Bigr).$

For $\omega\in\Omega$ and $q\in\mathsf{L}$ define the under-likelihood mass $\pi^{\omega}_q=\int_{\mathbf{R}}\ell^{\sharp,\omega}\,\mathbf{1}\{\varrho_q(\mathsf{K}_q(\omega))L^{\omega}_q<1-\delta\}\,d\rho\in[0,1]$ and $\pi^{\omega}=\sum_q\pi^{\omega}_q$, and the weighted square $Q^{\omega}=\int_{\mathbf{R}}\ell^{\sharp,\omega}\Bigl(\sum_{q\in\mathsf{L}}w_q\bigl(1-\varrho_q(\mathsf{K}_q(\omega))L^{\omega}_q\bigr)\Bigr)^{2}d\rho=\Bigl(\sum_qw_q(1-\varrho_q(\mathsf{K}_q(\omega)))\Bigr)^{2}+\sum_{q,q'\in\mathsf{L}}w_qw_{q'}\varrho_q(\mathsf{K}_q(\omega))\varrho_{q'}(\mathsf{K}_{q'}(\omega))\,C^{\omega}_{qq'}$ (the identity by claim 2 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form).

1. (Prior part) The family $(\mathsf{K}_q)_{q\in\mathsf{L}}$ is independent, hence so is $(\varrho_q(\mathsf{K}_q))_{q\in\mathsf{L}}$, and $\mathbb{E}\Bigl[\Bigl(\sum_{q\in\mathsf{L}}w_q\bigl(1-\varrho_q(\mathsf{K}_q)\bigr)\Bigr)^{2}\Bigr]=\sum_{q\in\mathsf{L}}w_q^{2}\,\mathsf{j}_q .$

2. (Under-likelihood mass on the good event) The maps $\omega\mapsto\pi^{\omega}_q$ (values in $[0,1]$), $\omega\mapsto Q^{\omega}$ (finite values), $\omega\mapsto C^{\omega}_{qq'}$ (finite values) and $\omega\mapsto\int_{\mathbf{R}}\Psi(r,\omega)\rho(dr)$ (values in $[0,\infty]$) are $\mathcal{F}$-measurable. (Throughout, $q\in\mathsf{L}$ is identified with its image in $\{1,\dots,d\}$ under the fixed bijection when the cited lemmas index by $\{1,\dots,d\}$.) For every $\omega\in G$ and $q\in\mathsf{L}$, $\pi^{\omega}_q\le\mathbf{1}\{\mathsf{K}_q(\omega)<\mu_q-x_q\}+\exp\bigl(-\theta\delta/2+\mathsf{E}^{\mathrm{ch}}_\theta\bigr),\qquad\text{hence}\qquad \mathbb{E}\bigl[\mathbf{1}_G\,\pi^{\omega}\bigr]\le\bar\Pi .$

3. (Assembly) The record part $\mathcal{C}=\mathbb{E}\bigl[\mathbf{1}_G\bigl(Q^{\omega}-(\sum_qw_q(1-\varrho_q(\mathsf{K}_q)))^{2}\bigr)\bigr]$ is a well-defined real number, equal to $\sum_{q,q'}w_qw_{q'}\,\mathbb{E}[\mathbf{1}_G\varrho_q(\mathsf{K}_q)\varrho_{q'}(\mathsf{K}_{q'})C^{\omega}_{qq'}]$, with $|C^{\omega}_{qq'}|\le\mathsf{c}_N$ on $G$, and $\mathsf{J}^{\mathrm{sym}}\ \le\ (1+\delta)\Bigl[\sum_{q\in\mathsf{L}}w_q^{2}\,\mathsf{j}_q+\mathcal{C}\Bigr]+2d\,\lVert w\rVert_1^{2}\,\mathsf{B},$ where the bad part is $\mathsf{B}=\bar\Pi+\sum_{q\in\mathsf{L}}(1+\mathsf{j}_q)^{1/2}\Bigl((d\,\bar\Pi)^{1/2}+(\mathsf{c}_N\bar\Pi)^{1/2}\Bigr)+\mathsf{g}+\mathsf{g}^{1/2}\sum_{q\in\mathsf{L}}(1+\mathsf{j}_q)^{1/2}.$