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

Theorem: Oriented complete-record separation

Causal support and mechanism identification › A robust identifying statistic · lines 2149–2192 · proof included or linked

Psy02-A@1#thm:statistical

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Let a predeclared statistic $f$ take values in $[0, 1]$ . Suppose ideal null and alternative families satisfy

P \in P_{0} sup E_{P} f \leq a, P \in P_{1} in f E_{P} f \geq b, b > a .

Suppose actual complete laws are within total variation $e_{0}, e_{1}$ of their corresponding ideal families. Define

g = b - a - e_{0} - e_{1} .

If $g > 0$ , the midpoint threshold between $a + e_{0}$ and $b - e_{1}$ gives, after $L$ independent identically distributed attempted runs from a fixed family member,

α, β \leq e^{- L g^{2} /2} .

Thus

L \geq \frac{2}{g ^{2}} lo g \frac{1}{α _{*}}

suffices for each error to be at most $α_{*}$ . A simultaneous calibration failure probability at most $β_{cal}$ gives unconditional error at most $α_{*} + β_{cal}$ .

Proof. Total variation changes the expectation of $f$ by at most $e_{h}$ . The common threshold is therefore at least $g /2$ from each allowed expectation. Hoeffding's one-sided inequality Hoeffding gives

exp [- 2 L (g /2)^{2}] .

A union bound includes calibration failure.

Displayed formulas in this passage

unlabeled · line 2153
$P \in P_{0} sup E_{P} f \leq a, P \in P_{1} in f E_{P} f \geq b, b > a .$
unlabeled · line 2162
$g = b - a - e_{0} - e_{1} .$
unlabeled · line 2169
$α, β \leq e^{- L g^{2} /2} .$
unlabeled · line 2173
$L \geq \frac{2}{g ^{2}} lo g \frac{1}{α _{*}}$
unlabeled · line 2188
$exp [- 2 L (g /2)^{2}] .$

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section: A robust identifying statistic