Proposition: Adaptive equality from complete process equality

Consider two finite adaptive models on identified accessible spaces. Suppose their initial accessible states agree and, after every permitted setting and history, their interval channels and complete selective instruments agree. Then all permitted adaptive record laws agree, including with an untouched reference. For an equal-prior binary randomized test, the type-I and type-II errors satisfy

For arbitrary record laws $P_0,P_1$,

Proof. Induction through the finite experiment tree gives equality of every unnormalized accessible conditional state. Taking its trace gives equality of history weights and hence record laws. Tensoring the assumed map identities with the reference identity gives the same induction with a reference.

If $\phi \in [0,1]$ is the probability of deciding for model $1$, then

The difference of expectations has magnitude at most total variation.