Manuscript Psy02-A@1 · Psy02-A.tex · theorem · d92ce402

Theorem: Mechanism non-identifiability

Causal support and mechanism identification › Observational equivalence · lines 2099–2123 · proof included or linked

Psy02-A@1#thm:nonidentifiable

If $P_{0} = P_{1}$ , every randomized test $φ \in [0, 1]$ , declaring model $1$ , satisfies

α + β = 1, α = E_{0} φ, β = E_{1} (1 - φ) .

More generally,

α + β \geq 1 - d_{TV} (P_{0}, P_{1}) .

Proof. Write

α + β = 1 + E_{0} φ - E_{1} φ .

The difference of expectations of a $[0, 1]$ -valued function is at most total variation.

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