Manuscript Hawk01-A@1 · Hawk01-A.tex · proposition · 11979653

Proposition: Testing obstruction

Complete-attempt assessment and observability › Record equality and the passive-astronomy boundary · lines 2178–2200 · proof included or linked

Hawk01-A@1#prop:testing, Hawk01-A@1#eq:testingobstruction

For two complete observable record laws $P_{0}, P_{1}$ , every randomized test with type-I and type-II errors $α, β$ satisfies

α + β \geq 1 - TV (P_{0}, P_{1}) .

If the two laws are equal, then $α + β = 1$ .

Proof. Let $φ \in [0, 1]$ be the probability of declaring model one. Then

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

The variational definition of total variation gives (eq:testingobstruction).

Displayed formulas in this passage

Hawk01-A@1#eq:testingobstruction · line 2183
$α + β \geq 1 - TV (P_{0}, P_{1}) .$
unlabeled · line 2193
$α + β = E_{0} φ + E_{1} (1 - φ) = 1 + E_{0} φ - E_{1} φ .$