Manuscript Psy04-A@1 · Psy04-A-v1.tex · lemma · d5800e2a

Lemma: Bounded-score stability

Robust model and apparatus allowances › Complete-law stability · lines 967–983 · proof included or linked

Psy04-A@1#lem:tv

For probability laws $P, P^{'}$ on the same complete record space and any score $f \in [0, 1]$ ,

\abs E_{P} f - E_{P^{'}} f \leq d_{TV} (P, P^{'}) .

Proof. Use the layer-cake representation

E_{P} f - E_{P^{'}} f = \int_{0}^{1} (P (f > t) - P^{'} (f > t)) d t,

and the defining supremum for total variation.

Displayed formulas in this passage

unlabeled · line 971
$\abs E_{P} f - E_{P^{'}} f \leq d_{TV} (P, P^{'}) .$
unlabeled · line 978
$E_{P} f - E_{P^{'}} f = \int_{0}^{1} (P (f > t) - P^{'} (f > t)) d t,$