Definition
Fix a family Φ = (φ_i) of admissible probes: procedures that, applied to an object, yield an outcome with a law. Two objects P and Q are observationally equivalent under Φ when (φ_i)#P = (φ_i)#Q for every i. A13: “Two models or states are equivalent when every admissible probe produces the same observable law.”
What the record holds of it (CLAIM-264):
- It is coarser than identity unless the family is rich enough: two objects can agree on every probe in a restricted family without being isomorphic.
- Rich families separate. All one-dimensional linear projections determine a distribution (Cramér–Wold). Homomorphism counts from every finite structure determine a finite relational structure (Lovász). The hom-functor with its action identifies an object (restricted Yoneda, THEORY-032).
- Marginals against the joint. Matching each probe’s marginal “leaves the joint across probes unconstrained”, so equivalence on probes run one at a time is weaker than on probes run together.
Special cases the record already holds:
- Response trajectories as the probes. TERM-033, fidelity as
observational equivalence over response trajectories, compares
P_o(Y_1:k | a_1:k) with P_t(Y′_1:k | a′_1:k) under corresponding
interventions. TERM-033 names TERM-015 as its successor (
superseded_by, though its status is Active), and TERM-043’s causal component keeps its comparison. It is the narrower term, so the relation runs from TERM-033 to this one, and is left in prose. - A communicative object (TERM-014) is “an equivalence class of situations relative to a chosen family of measurements”, a class of this relation. That is a use, not a narrowing, and also stays in prose.
- The coarse variable’s law as the probe. CLAIM-389’s coarse variable is observationally equivalent, under every probe of its own law, to an autonomous AR(1), while one receives 0.120 nats per step from the rest of its system and the other nothing (CLAIM-397’s “What it does not say”).
What it is not
- Not identity, nor isomorphism (CLAIM-264).
- Not statistical equivalence when the probes are resource-bounded. CLAIM-302: “Equivalence under feasible tests can be much coarser than statistical equivalence”.
- Not evidence that structure was preserved, unless the probes were chosen independently of the objects (CLAIM-322, CLAIM-142).
- Not TERM-037’s observable, which is one probe in the communicative setting.
- Not interventional equivalence unless the family contains the interventions. Two systems equivalent under every passive probe can respond differently to an intervention on their mechanism (CASE-125, CLAIM-407).