Information Bound on the Synthetic Copy with Mean-Field Data: Prior Energy of the Profile, Observation Information Along the Profile Response, and the Non-Close Record Mass

Adopt the setting, hypotheses (OC), (X), (W), (G), (P) and notation of Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass (and hence 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, 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 expectation of integrable and of nonnegative random variables), the observation record space $(\mathbf{R},\mathcal{R},\rho)$ with horizon $T$ and $\tilde{l}$ channels, the cells indexed by $\mathsf{L}$ ($d$ elements) with lengths $\mu_q$, the cell-count vector $\mathsf{K}$, the move size $\mathsf{m}$, 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 $\ell^{\sharp,\omega}$, the intensities $\lambda^{\sharp,\omega}$ and the effective removed intensities $\lambda^{-q,\omega}$, the pair exponents $E^{\omega}_{qq'}$ and pair covariances $C^{\omega}_{qq'}$, the data $N$, $l$, $\tilde{l}$, $\tilde{B}$, $\tilde{K}$, $\underline{b}$, $T$, the constants $\Gamma$, $A_0$, $\bar{E}_N$, $\varepsilon_0=\Gamma A_0/(N\underline{b})$ and $\mathsf{c}_N=\exp(\bar{E}_N)-1$, the clock-good event $G_{L,D}$, the tracked records $\mathsf{T}_\omega$, the regularised paths $\bar\Sigma^{\sharp,r}(\omega)$, the event $G$ with $\mathsf{g}=P(\Omega\setminus G)$, the real number $\delta$, the quantities $\mathsf{j}_q$ ($q\in\mathsf{L}$), $\bar\Pi$, the record part $\mathcal{C}$ and the bad part $\mathsf{B}$, and the indicator $\mathbf{1}_A$ of an event or of a subset of $\mathbf{R}$. Adopt also, for the weights $w=(w_q)_{q\in\mathsf{L}}$ of that theorem, the setting, hypotheses and notation of Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data (and hence of Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound and Injection Weights from a Bounded Profile Along Mean-Field Clocks: Label-Rate Form of the Fluctuation Covariance, Weight Bounds, the Prior Quadratic Form, and the Step-Function Injection): the comparison pair $(S,\mathsf{A})$ with the mean-field label rates $\phi_c(t)=\psi_c(S_t,\mathsf{A}_t)$, the profile $\varpi$ with bound $\Lambda$, the profile energy $\mathcal{P}=\int_{[0,T]}\varpi_s\cdot(\Theta(S_s,\mathsf{A}_s)\varpi_s)\,ds$ (claim 1 of that lemma), the profile response $\bar\psi$ with bound $\mathsf{M}$, the observation information matrix $\tilde{D}(x)$, the cell lengths' minimum $\mu_{\min}=\min_q\mu_q$, the event $G^{\mathsf{m}}$, the pair-exponent quadratic form $\mathcal{Q}^{\omega}(r)$, the control discrepancy $\mathsf{D}^{r}_{\mathrm{ctl}}$, and, for the real numbers $\varepsilon_S\ge0$ and $\varepsilon_{\mathrm{ctl}}\ge0$ fixed in (CL) below, the constants $\mathsf{e}_F$, $\epsilon_\psi$ and $\kappa$ of claims 2, 3 and 4 of that lemma (formed from $\varepsilon_S$, $\varepsilon_{\mathrm{ctl}}$ and the adopted data); the weights $w$ are thus the injection weights of Injection Weights from a Bounded Profile Along Mean-Field Clocks: Label-Rate Form of the Fluctuation Covariance, Weight Bounds, the Prior Quadratic Form, and the Step-Function Injection formed from $\phi_c$ and $\varpi$, with $\lVert w\rVert_1=\sum_q|w_q|$. As in Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data, the letter $\Theta$ denotes here the aggregate fluctuation covariance and never the parameter coordinate map of the synthetic copy, $\mathsf{D}^{r}_{\mathrm{ctl}}$ and $\tilde{D}$ are unrelated to the record coordinate map $\mathsf{D}$ of the copy, the real number $\delta$ of (P) is unrelated to the basis vectors $\delta_1,\dots,\delta_l$ and to coordinate indices, the null set $\mathsf{N}$ of claim 3 is unrelated to the population size $N$, the constant $\kappa_0$ defined below is unrelated to the constant $\kappa$ of claim 4 of Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data, and the letter $\eta$ (the smoothing parameter of the copy in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record) is not used below. Write $\exp$ for the real exponential function, $t^{1/2}$ for the nonnegative square root of a real $t\ge0$, $x\cdot y$ for the dot product, $\int_{[0,T]}\cdot\,dt$ for the Lebesgue integral over the compact interval $[0,T]$, and, on $\mathbf{R}$, integrals of nonnegative measurable functions for those of Lebesgue Integral of a Nonnegative Measurable Function and integrable for integrable with respect to $\rho$. Assume in addition:

(G$'$) $G\subseteq G^{\mathsf{m}}$;

(CL) $\varepsilon_S\ge0$ and $\varepsilon_{\mathrm{ctl}}\ge0$ are real numbers and $(\mathsf{R}^{\mathrm{cl}}_\omega)_{\omega\in\Omega}$ is a family of sets $\mathsf{R}^{\mathrm{cl}}_\omega\in\mathcal{R}$ with $\mathsf{R}^{\mathrm{cl}}_\omega\subseteq\mathsf{T}_\omega$ (the close records) such that, for every $\omega\in G$ and every $r\in\mathsf{R}^{\mathrm{cl}}_\omega$, $|\bar\Sigma^{\sharp,r}_t(\omega)-S_t|\le\varepsilon_S$ for every $t\in[0,T]$ and $\mathsf{D}^{r}_{\mathrm{ctl}}\le\varepsilon_{\mathrm{ctl}}$, and such that the non-close mass $\pi^{\mathrm{nc}}_\omega=\int_{\mathbf{R}}\ell^{\sharp,\omega}\,\mathbf{1}_{\mathbf{R}\setminus\mathsf{R}^{\mathrm{cl}}_\omega}\,d\rho\in[0,1]$ is an $\mathcal{F}$-measurable function of $\omega$; put $\bar\pi^{\mathrm{nc}}=\mathbb{E}[\mathbf{1}_G\,\pi^{\mathrm{nc}}]$.

Fix a real number $\zeta>0$ and put $\mathsf{E}^{\star}_N=\tilde{l}\,T\,N\tilde{B}\,\varepsilon_0^{2},\qquad \mathsf{e}^{\star}_N=\tfrac12(\mathsf{E}^{\star}_N)^{2}\exp(\mathsf{E}^{\star}_N)+\mathsf{E}^{\star}_N\bigl(\exp(9\mathsf{E}^{\star}_N)-1\bigr)^{1/2}$ (the constants $\mathsf{E}_\eta$ and $\mathsf{e}_\eta$ of Relative Perturbations of a Causal Intensity: Pair-Exponent Bound, the Pair Intensity as a Relative Perturbation, Replacement of the Pair Likelihood by the Base Likelihood, and the Weighted Pair-Covariance Sum formed with the intensity bound $N\tilde{B}$ and with its perturbation parameter taken to be $\varepsilon_0$; the square root is defined as noted there), $\kappa_0=1+\frac{\mathsf{m}^{2}}{2\mu_{\min}}\exp\Bigl(\frac{\mathsf{m}^{2}}{\mu_{\min}}\Bigr),\qquad \bar{\mathsf{j}}=\max_{q\in\mathsf{L}}\mathsf{j}_q,\qquad \mathsf{j}^{\star}=\bigl((1+\bar{\mathsf{j}})^{1/2}+1\bigr)\,\bar{\mathsf{j}}^{1/2},$ and the close-record information bound $\mathsf{Q}^{\mathrm{cl}}=(1+\zeta)\,N\Bigl(\int_{[0,T]}\bar\psi_t\cdot\bigl(\tilde{D}(S_t)\bar\psi_t\bigr)\,dt+\tilde{l}\,T\,\kappa\Bigr)+\Bigl(1+\frac1\zeta\Bigr)\frac{9\,\tilde{l}\,l^{2}\tilde{K}^{2}T\lVert w\rVert_1^{2}A_0^{4}}{4N^{3}\underline{b}}$ (its integrand being nonnegative, bounded and measurable by claim 1). Finally, for $\omega\in\Omega$ let $\mathcal{C}^{\omega}=\sum_{q\in\mathsf{L}}\sum_{q'\in\mathsf{L}}w_qw_{q'}\,C^{\omega}_{qq'}$ be the pathwise record part.

1. (Well-posedness and prior part) The map $t\mapsto\bar\psi_t\cdot(\tilde{D}(S_t)\bar\psi_t)$ is nonnegative, bounded and measurable on $[0,T]$ (with respect to the trace Borel $\sigma$-algebra), so $\mathsf{Q}^{\mathrm{cl}}$ is a well-defined real number with $\mathsf{Q}^{\mathrm{cl}}\ge0$; moreover $0<\varepsilon_0\le1$, $\mathsf{E}^{\star}_N\ge0$, $\mathsf{e}^{\star}_N$ is defined and nonnegative, and $\kappa_0\ge1$. For every $q\in\mathsf{L}$, $\mu_q\,\mathsf{j}_q\le\kappa_0\,\mathsf{m}^{2}$; hence $\bar{\mathsf{j}}\le\kappa_0\mathsf{m}^{2}/\mu_{\min}$ and $\sum_{q\in\mathsf{L}}w_q^{2}\,\mathsf{j}_q\le\kappa_0\,N\,\mathcal{P}.$

2. (Removal of the ratio factors) For every $q\in\mathsf{L}$, $\mathbb{E}[(\varrho_q(\mathsf{K}_q)-1)^{2}]=\mathsf{j}_q$; and for all $q,q'\in\mathsf{L}$, $\mathbb{E}\bigl|\varrho_q(\mathsf{K}_q)\varrho_{q'}(\mathsf{K}_{q'})-1\bigr|\le(1+\mathsf{j}_q)^{1/2}\mathsf{j}_{q'}^{1/2}+\mathsf{j}_q^{1/2}\le\mathsf{j}^{\star}.$ The map $\omega\mapsto\mathcal{C}^{\omega}$ is $\mathcal{F}$-measurable with $|\mathcal{C}^{\omega}|\le\lVert w\rVert_1^{2}\mathsf{c}_N$ for $\omega\in G$, so that $\mathbf{1}_G\mathcal{C}^{\omega}$ is integrable, and $\mathcal{C}\le\mathbb{E}\bigl[\mathbf{1}_G\,\mathcal{C}^{\omega}\bigr]+\mathsf{c}_N\,\lVert w\rVert_1^{2}\,\mathsf{j}^{\star}.$

3. (Record part through the pair-exponent quadratic form) Let $\omega\in G$. Then the base intensity $\lambda^{\sharp,\omega}$ (with bound $N\tilde{B}$ and lower bound $N\underline{b}$), the null set $\mathsf{N}=\mathbf{R}\setminus\mathsf{T}_\omega$, the perturbation parameter $\varepsilon_0$ and the perturbed intensities $(\lambda^{-q,\omega})_{q\in\mathsf{L}}$ (each a relative $\varepsilon_0$-perturbation of $\lambda^{\sharp,\omega}$ off $\mathsf{N}$) satisfy the hypotheses of claim 4 of Relative Perturbations of a Causal Intensity: Pair-Exponent Bound, the Pair Intensity as a Relative Perturbation, Replacement of the Pair Likelihood by the Base Likelihood, and the Weighted Pair-Covariance Sum, whose pair exponents, pair covariances and quadratic form are $E^{\omega}_{qq'}$, $C^{\omega}_{qq'}$ and $\mathcal{Q}^{\omega}$; consequently $r\mapsto\ell^{\sharp,\omega}(r)\mathcal{Q}^{\omega}(r)$ is integrable with respect to $\rho$ and $\mathcal{C}^{\omega}\le\int_{\mathbf{R}}\ell^{\sharp,\omega}\,\mathcal{Q}^{\omega}\,d\rho+\lVert w\rVert_1^{2}\,\mathsf{e}^{\star}_N,\qquad \int_{\mathbf{R}}\ell^{\sharp,\omega}\,\mathcal{Q}^{\omega}\,d\rho\le\mathsf{Q}^{\mathrm{cl}}+\lVert w\rVert_1^{2}\,\bar{E}_N\,\pi^{\mathrm{nc}}_\omega .$

4. (Information bound with mean-field data) $\mathsf{J}^{\mathrm{sym}}\ \le\ (1+\delta)\Bigl[\kappa_0\,N\,\mathcal{P}+\mathsf{Q}^{\mathrm{cl}}+\lVert w\rVert_1^{2}\bigl(\mathsf{e}^{\star}_N+\bar{E}_N\,\bar\pi^{\mathrm{nc}}+\mathsf{c}_N\,\mathsf{j}^{\star}\bigr)\Bigr]+2d\,\lVert w\rVert_1^{2}\,\mathsf{B}.$