Non-Source Projection and Internal Identifiability

Joint probes and source identifiability

Jeremy RodgersSection 7 of 14Original PDF ↗Publication DOI ↗

Joint probes and source identifiability

Definition 6.1 (Probe family and joint map).

Let $I$ be an index set and $P_i\colon S\to Y_i$ , $i\in I$ , a family of admissible probes , maps whose values an observer can in principle obtain. Define the joint probe map $P_I\colon S\longrightarrow \prod_{i\in I}Y_i,\qquad P_I(s)=\bigl(P_i(s)\bigr)_{i\in I},$ and corestrict it to its image $\mathop{\mathrm{Im}}P_I$ , so that $P_I\colon S\twoheadrightarrow\mathop{\mathrm{Im}}P_I$ is surjective. Write $s\sim_{P_i}s'\iff P_i(s)=P_i(s')$ and $\sim_I\;:=\;\bigcap_{i\in I}\sim_{P_i},$ the joint observational equivalence relation. The family identifies the source if $P_I(s)=P_I(s')\Rightarrow s=s'$ .

Theorem 6.2 (Joint-probe identifiability).

The following are equivalent.

The probe family identifies the source.
$P_I$ is injective.
The probes separate the points of $S$ : for all $s\neq s'$ there is $i\in I$ with $P_i(s)\neq P_i(s')$ .
$\sim_I\;=\;\Delta_S$ , the diagonal (i.e. equality).
There is a left inverse on the image: a map $R_I\colon\mathop{\mathrm{Im}}P_I\to S$ with $R_I\circ P_I=\mathop{\mathrm{id}}_S$ .

Proof. (I1) $\Leftrightarrow$ (I2) is the definition of injectivity. (I2) $\Leftrightarrow$ (I3): $P_I(s)=P_I(s')$ holds iff $P_i(s)=P_i(s')$ for all $i$ ; so $P_I$ is injective iff no distinct pair agrees on every probe, which is (I3). (I3) $\Leftrightarrow$ (I4): $s\sim_I s'$ means $P_i(s)=P_i(s')$ for all $i$ ; so $\sim_I$ reduces to equality iff distinct points are separated by some probe.

(I2) $\Rightarrow$ (I5). $P_I\colon S\to\mathop{\mathrm{Im}}P_I$ is injective by (I2) and surjective by corestriction, hence bijective; take $R_I:=P_I^{-1}$ , which satisfies $R_I\circ P_I=\mathop{\mathrm{id}}_S$ .

(I5) $\Rightarrow$ (I2). If $P_I(s)=P_I(s')$ then $s=R_I(P_I(s))=R_I(P_I(s'))=s'$ . ◻

Corollary 6.3 (Residual ambiguity).

If $\sim_I\;\neq\;\Delta_S$ , then the maximally identifiable object from the probe family is the quotient $S/\!\sim_I\;\;\xrightarrow{\ \cong\ }\;\;\mathop{\mathrm{Im}}P_I ,\tag{25}$ the bijection being induced by $P_I$ . No rule whose input is the probe values alone reconstructs individual source states within a nontrivial $\sim_I$ -class.

Proof. $\sim_I$ is precisely the kernel pair of $P_I$ : $s\sim_I s'\iff P_I(s)=P_I(s')$ . Hence $[s]\mapsto P_I(s)$ is well defined and injective on $S/\!\sim_I$ , and surjective onto $\mathop{\mathrm{Im}}P_I$ by corestriction, giving (25). For the second claim, let $D\colon\mathop{\mathrm{Im}}P_I\to Z$ be any map (a “rule”) and let $s\sim_I s'$ . Then $D(P_I(s))=D(P_I(s'))$ : the rule returns the same output on both, so it cannot name which of $s,s'$ was realized. ◻

Proposition 6.4 (Monotonicity under added probes).

If $I\subseteq J$ (the probes of $I$ being among those of $J$ ), then $\sim_J\;\subseteq\;\sim_I ,$ and there is a canonical surjection $S/\!\sim_J\;\twoheadrightarrow\;S/\!\sim_I,\qquad [s]_J\longmapsto[s]_I .\tag{26}$

Proof. If $s\sim_J s'$ then $P_i(s)=P_i(s')$ for every $i\in J\supseteq I$ , hence for every $i\in I$ , i.e. $s\sim_I s'$ . This inclusion of equivalence relations makes (26) well defined ( $[s]_J=[s']_J\Rightarrow s\sim_J s'\Rightarrow s\sim_I s'\Rightarrow [s]_I=[s']_I$ ), and it is surjective because every class $[s]_I$ is hit by $[s]_J$ . ◻

Remark 40 (Direction of (26)). The map goes from the finer quotient to the coarser one. More probes means a finer partition, hence a larger quotient set, which surjects onto the smaller. There is in general no map back: information present in $S/\!\sim_J$ cannot be manufactured from $S/\!\sim_I$ .

Thus tomography is the intersection of probe fibers: one adds probes, and the joint fiber $\{s':s'\sim_I s\}$ shrinks. Source recovery occurs precisely when that intersection reaches the diagonal. But we must be exact about what is and is not being claimed. Theorem 6.2 says when a family separates; it does not say that any particular physically available family does. Whether the probes an observer can actually run separate the source is a contingent fact about the model and the observer, not a theorem. The next section studies the case where, by hypothesis, they do not.