Proof of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound

Throughout, measurable for a real-valued map means measurable with respect to the Borel $\sigma$-algebra on the target; expectations and integrals of $[0,\infty]$-valued measurable maps are those of Lebesgue Integral of a Nonnegative Measurable Function. We use freely the following two facts. (Q1) If $Z_n:X\to[0,\infty]$ ($n\in\mathbb{N}$) are measurable on a measure space $(X,\mathcal{X},\nu)$, then $\sum_nZ_n$ is measurable and $\int\sum_nZ_n\,d\nu=\sum_n\int Z_n\,d\nu$ (partial sums increase to the sum; Linearity and Monotonicity of the Lebesgue Integral and Monotone Convergence Theorem). (Q2) $\mathbb{N}_0^{\mathsf{L}}$ is countable (it is the set of values of a sequence, listing for $n=0,1,2,\dots$ the finitely many $y$ with $\sum_{(c,j)}y_{c,j}=n$), so sums over $y\in\mathbb{N}_0^{\mathsf{L}}$ of nonnegative terms, in the sense of Sum of a Nonnegative Function over an Arbitrary Set, are sequential sums along this enumeration, and they are invariant under the bijective reindexing $x=y/\sqrt{N}$ between $\mathbb{N}_0^{\mathsf{L}}$ and $\mathsf{S}$ (both being least upper bounds of finite partial sums, as in the preamble of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions). Finally, for $y\in\mathbb{N}_0^{\mathsf{L}}$ the indicator $\mathbf{1}\{\mathsf{K}=y\}$ is $\mathcal{V}$-measurable and $P(\mathsf{K}=y)=\prod_{(c',j')\in\mathsf{L}}\mathrm{poi}_{\mu_{c',j'}}(y_{c',j'})>0$, by claim 1 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, and $\mathbb{E}[\mathbf{1}\{\mathsf{K}=y\}Z]=P(\mathsf{K}=y)\mathbb{E}[Z]$ for every $\mathcal{U}$-measurable $Z:\Omega\to[0,\infty]$ (the factorisation of that claim).

Step 1 (claim 1: removed clocks). Fix $(c,j)$ and $\omega$ with $\mathsf{K}_{c,j}(\omega)\ge\mathsf{m}$ and put $y=\mathsf{K}(\omega)-\mathsf{m}e_{c,j}$, a point of $\mathbb{N}_0^{\mathsf{L}}$ with $y+\mathsf{m}e_{c,j}=\mathsf{K}(\omega)$ and $y_{c,j}=\mathsf{K}_{c,j}(\omega)-\mathsf{m}$. By the definition of the copy clocks, $\mathsf{P}^{\sharp}(\omega)=\mathsf{P}^{(\mathsf{K}(\omega))}(\omega)=\mathsf{P}^{(y+\mathsf{m}e_{c,j})}(\omega)$, and claim 2 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record (insertion, with $(c_0,j_0)=(c,j)$) gives exactly the two displayed identities, the inserted points being $U^{c,j}_i(\omega)$ for $y_{c,j}<i\le y_{c,j}+\mathsf{m}$, together with their location and distinctness on $\Omega^{U}_0$. The identification of $\ell^{-(c,j),\omega}=\ell^{(y),\omega}$ as the likelihood for the clocks $\mathsf{P}^{(y)}(\omega)$ is claim 3 of that lemma.

Step 2 (claim 2: measurability and moments). The removed-clock likelihood. For every $(r,\omega)$, $\ell^{-(c,j),\omega}(r)=\sum_{y\in\mathbb{N}_0^{\mathsf{L}}:\ y_{c,j}\ge\mathsf{m}}\mathbf{1}\{\mathsf{K}(\omega)=y\}\,\ell^{(y-\mathsf{m}e_{c,j}),\omega}(r),$ since at most one term is nonzero, namely the one with $y=\mathsf{K}(\omega)$, present exactly when $\mathsf{K}_{c,j}(\omega)\ge\mathsf{m}$. Each term is the product of the $\mathcal{R}\otimes\mathcal{F}$-measurable maps $(r,\omega)\mapsto\mathbf{1}\{\mathsf{K}(\omega)=y\}$ (indicator of the measurable rectangle $\mathbf{R}\times\{\mathsf{K}=y\}$) and $(r,\omega)\mapsto\ell^{(y-\mathsf{m}e_{c,j}),\omega}(r)$ ($\mathcal{R}\otimes\mathcal{U}$-measurable by claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, hence $\mathcal{R}\otimes\mathcal{F}$-measurable since $\mathcal{U}\subseteq\mathcal{F}$), hence measurable by claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; the countable sum, whose partial sums along the enumeration of (Q2) converge pointwise in $\mathbb{R}$ (at most one term being nonzero), is measurable by claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. The bound $(N\tilde{B})^{k}$ on $\mathbf{R}^{(k)}\times\Omega$ is inherited from the same claim 3.

The removal ratio. $\mathsf{K}_{c,j}$ is $\mathcal{V}$-measurable with values in $\mathbb{N}_0$ (claim 1 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), so $\varrho_{c,j}(\mathsf{K}_{c,j})=\sum_{k\in\mathbb{N}_0}\varrho_{c,j}(k)\mathbf{1}\{\mathsf{K}_{c,j}=k\}$, a pointwise limit in $\mathbb{R}$ of finite sums of $\mathcal{V}$-measurable maps (exactly one term is nonzero), is $\mathcal{V}$-measurable by claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, with finite values. By (Q1), $\mathbb{E}[\varrho_{c,j}(\mathsf{K}_{c,j})]=\sum_k\varrho_{c,j}(k)P(\mathsf{K}_{c,j}=k)$ and $\mathbb{E}[\varrho_{c,j}(\mathsf{K}_{c,j})^{2}]=\sum_k\varrho_{c,j}(k)^{2}P(\mathsf{K}_{c,j}=k)$. Since $\mathsf{K}_{c,j}$ has the Poisson distribution with parameter $\mu_{c,j}$ (claim 2 of Uniform Representation of the Rate-One Poisson Counting Path on a Bounded Interval: Distinct Points, Poisson Increments, Conditional Law Given the Cell Counts, and Point Insertion, $\mathsf{K}_{c,j}$ being a cell count of that lemma by the preamble of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), $P(\mathsf{K}_{c,j}=k)=\mathrm{poi}_{\mu_{c,j}}(k)$, and claims 2 and 3 of The Poisson Removal Ratio for Moves of Several Points: Move Score, Mean, Exact Second Moment, Move Information, and Pointwise Bounds give the two displayed values.

The composed functional. By claim 2(a) of Weighted Cauchy-Schwarz Inequality on a Measure Space and the Symmetrised Score Functional: Bounds and Averaging, $\Phi_w$ is sequentially continuous on $[0,\infty)^{1+d}$. The $1+d$ components $(r,\omega)\mapsto\ell^{\sharp,\omega}(r)$ (claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record) and $(r,\omega)\mapsto\varrho_{c,j}(\mathsf{K}_{c,j}(\omega))\ell^{-(c,j),\omega}(r)$ (products of the measurable maps above, the factor $\varrho_{c,j}(\mathsf{K}_{c,j})$ being $\mathcal{V}$-measurable, hence $\mathcal{F}$-measurable as $\mathcal{V}\subseteq\mathcal{F}$, and composed with the second projection $\mathbf{R}\times\Omega\to\Omega$) are $\mathcal{R}\otimes\mathcal{F}$-measurable with values in $[0,\infty)$, so the composition with $\Phi_w$ is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable (with $E=[0,\infty)^{1+d}$), with values in $[0,\infty)$.

Step 3 (claim 3: the shifted family). Fix $x=y/\sqrt{N}\in\mathsf{S}$, $r\in\mathbf{R}$ and $(c,j)\in\mathsf{L}$. Unshifted. By definition, $\mathsf{p}(x)f(x,r)=P(\mathsf{K}=y)\,\mathbb{E}[\ell^{(y),\cdot}(r)]=\mathbb{E}[\mathbf{1}\{\mathsf{K}=y\}\ell^{(y),\cdot}(r)]$ by the factorisation (the map $\omega\mapsto\ell^{(y),\omega}(r)$ being $\mathcal{U}$-measurable as the section at $r$ of an $\mathcal{R}\otimes\mathcal{U}$-measurable map: the sets whose section at $r$ lies in $\mathcal{U}$ form a $\sigma$-algebra containing the measurable rectangles, hence containing $\mathcal{R}\otimes\mathcal{U}$ by Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra), and on $\{\mathsf{K}=y\}$ one has $\ell^{(y),\omega}=\ell^{\sharp,\omega}$ by claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record; this is the first identity.

Shifted. By Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound, $\mathsf{p}_{c,j}(x)=\mathsf{p}(x-a_{c,j})$ and $f_{c,j}(x,r)=f(x-a_{c,j},r)$ if $x-a_{c,j}\in\mathsf{S}$, $f_{c,j}(x,r)=0$ otherwise. Here $x-a_{c,j}=(y-\mathsf{m}e_{c,j})/\sqrt{N}$, which lies in $\mathsf{S}$ if and only if $y_{c,j}\ge\mathsf{m}$.

Case $y_{c,j}<\mathsf{m}$. Then $x-a_{c,j}\notin\mathsf{S}$, so $\mathsf{p}_{c,j}(x)=0$ ($\mathsf{p}$ vanishes off $\mathsf{S}$) and the left side is $0$; on the right, $\varrho_{c,j}(\mathsf{K}_{c,j})=\varrho_{c,j}(y_{c,j})=0$ on $\{\mathsf{K}=y\}$, so the expectation is $0$.

Case $y_{c,j}\ge\mathsf{m}$. Put $y'=y-\mathsf{m}e_{c,j}\in\mathbb{N}_0^{\mathsf{L}}$. Then $\mathsf{p}_{c,j}(x)f_{c,j}(x,r)=\mathsf{p}(y'/\sqrt{N})f(y'/\sqrt{N},r)=P(\mathsf{K}=y')\,\mathbb{E}[\ell^{(y'),\cdot}(r)]$. In the product formula for $P(\mathsf{K}=y')$ and $P(\mathsf{K}=y)$ all factors agree except the one indexed by $(c,j)$, whence $P(\mathsf{K}=y')=P(\mathsf{K}=y)\,\mathrm{poi}_{\mu_{c,j}}(y_{c,j}-\mathsf{m})/\mathrm{poi}_{\mu_{c,j}}(y_{c,j})=P(\mathsf{K}=y)\varrho_{c,j}(y_{c,j})$. Hence, by the factorisation applied to the $\mathcal{U}$-measurable map $\omega\mapsto\ell^{(y'),\omega}(r)$ and by Linearity and Monotonicity of the Lebesgue Integral for the constant $\varrho_{c,j}(y_{c,j})$, $\mathsf{p}_{c,j}(x)f_{c,j}(x,r)=\varrho_{c,j}(y_{c,j})\,\mathbb{E}\bigl[\mathbf{1}\{\mathsf{K}=y\}\ell^{(y'),\cdot}(r)\bigr]=\mathbb{E}\bigl[\mathbf{1}\{\mathsf{K}=y\}\varrho_{c,j}(\mathsf{K}_{c,j})\ell^{-(c,j),\cdot}(r)\bigr],$ since on $\{\mathsf{K}=y\}$ one has $\varrho_{c,j}(\mathsf{K}_{c,j})=\varrho_{c,j}(y_{c,j})$ and $\ell^{-(c,j),\omega}=\ell^{(\mathsf{K}(\omega)-\mathsf{m}e_{c,j}),\omega}=\ell^{(y'),\omega}$.

Step 4 (claim 4: the reduction). Let $\Psi$ be the pathwise score map of the statement, which is $\mathcal{R}\otimes\mathcal{F}$-measurable and $[0,\infty)$-valued by Step 2; the measurability of $r\mapsto\mathbb{E}[\Psi(r,\cdot)]$ and of $\omega\mapsto\int_{\mathbf{R}}\Psi(r,\omega)\rho(dr)$ asserted in claim 2 is the Tonelli theorem on $\mathbf{R}\times\Omega$.

One lattice point. Fix $x=y/\sqrt{N}\in\mathsf{S}$ and $r\in\mathbf{R}$. Define on $(\Omega,\mathcal{F},P)$ the maps $\ell_0(\omega)=\mathbf{1}\{\mathsf{K}(\omega)=y\}\ell^{\sharp,\omega}(r)$ and $\ell_{c,j}(\omega)=\mathbf{1}\{\mathsf{K}(\omega)=y\}\varrho_{c,j}(\mathsf{K}_{c,j}(\omega))\ell^{-(c,j),\omega}(r)$, $(c,j)\in\mathsf{L}$. They are measurable (sections at $r$ of the measurable maps of Step 2, by the section argument of Step 3, times indicators), nonnegative, and bounded (by $(N\tilde{B})^{k}$ for $r\in\mathbf{R}^{(k)}$, respectively by $\varrho_{c,j}(y_{c,j})(N\tilde{B})^{k}$), hence have finite integrals, which by Step 3 are $A=\mathbb{E}[\ell_0]=\mathsf{p}(x)f(x,r)$ and $B_{c,j}=\mathbb{E}[\ell_{c,j}]=\mathsf{p}_{c,j}(x)f_{c,j}(x,r)$. Claim 3 of Weighted Cauchy-Schwarz Inequality on a Measure Space and the Symmetrised Score Functional: Bounds and Averaging (averaging inequality, on the measure space $(\Omega,\mathcal{F},P)$) gives $\Phi_w\bigl(\mathsf{p}(x)f(x,r),(\mathsf{p}_{c,j}(x)f_{c,j}(x,r))_{(c,j)}\bigr)\le\mathbb{E}\bigl[\Phi_w(\ell_0,(\ell_{c,j})_{(c,j)})\bigr]=\mathbb{E}\bigl[\mathbf{1}\{\mathsf{K}=y\}\,\Psi(r,\cdot)\bigr],$ where the last equality is pointwise in $\omega$: by the homogeneity of claim 2(c) of Weighted Cauchy-Schwarz Inequality on a Measure Space and the Symmetrised Score Functional: Bounds and Averaging with the factor $\mathbf{1}\{\mathsf{K}(\omega)=y\}\in\{0,1\}$, $\Phi_w(\ell_0(\omega),(\ell_{c,j}(\omega))_{(c,j)})=\mathbf{1}\{\mathsf{K}(\omega)=y\}\Psi(r,\omega)$.

Summation. By the definition of $\mathsf{J}^{\mathrm{sym}}$ in Gaussian Smoothing of a Kernel-Weighted Discrete Family with Countable Support: Joint Density, van Trees Regularity, and the Symmetrised Directional Score Bound (whose claim 1 gives the $\mathcal{R}$-measurability in $r$ of its integrand for each $x$), the monotonicity of the integral (Linearity and Monotonicity of the Lebesgue Integral) and of sums of nonnegative terms, and the reindexing (Q2), $\mathsf{J}^{\mathrm{sym}}=\sum_{x\in\mathsf{S}}\int_{\mathbf{R}}\Phi_w\bigl(\mathsf{p}(x)f(x,r),(\mathsf{p}_{c,j}(x)f_{c,j}(x,r))_{(c,j)}\bigr)\rho(dr)\le\sum_{y\in\mathbb{N}_0^{\mathsf{L}}}\int_{\mathbf{R}}\mathbb{E}\bigl[\mathbf{1}\{\mathsf{K}=y\}\Psi(r,\cdot)\bigr]\rho(dr),$ the map $r\mapsto\mathbb{E}[\mathbf{1}\{\mathsf{K}=y\}\Psi(r,\cdot)]$ being $\mathcal{R}$-measurable by the Tonelli theorem on $\mathbf{R}\times\Omega$ ($\rho$ is $\sigma$-finite by the preamble of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, $P$ is finite). Applying (Q1) twice (first for $\rho$, then for $P$) and using $\sum_y\mathbf{1}\{\mathsf{K}=y\}=1$ pointwise ($\mathsf{K}$ takes values in $\mathbb{N}_0^{\mathsf{L}}$), the right side equals $\int_{\mathbf{R}}\mathbb{E}\Bigl[\sum_{y}\mathbf{1}\{\mathsf{K}=y\}\Psi(r,\cdot)\Bigr]\rho(dr)=\int_{\mathbf{R}}\mathbb{E}[\Psi(r,\cdot)]\,\rho(dr)=\mathbb{E}\Bigl[\int_{\mathbf{R}}\Psi(r,\cdot)\,\rho(dr)\Bigr],$ the last equality being Tonelli on $\mathbf{R}\times\Omega$ for the nonnegative $\mathcal{R}\otimes\mathcal{F}$-measurable $\Psi$. $\blacksquare$