LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The L² link norm controls binary-regression likelihood distances

Statement

Let $G$ be a probability measure on $(X, A)$ , let $Ψ : R \to (0, 1)$ be differentiable, and suppose $L : = \sup_{u \in R} ∣ Ψ^{'} (u) ∣ < \infty, R : = \sup_{u \in R} ∣ \frac{Ψ^{'} (u)}{Ψ (u) (1 - Ψ (u))} ∣ < \infty .$ For measurable real functions $w, w_{0}$ , define Bernoulli densities relative to $G \otimes #$ , where $#$ is counting measure on ${0, 1}$ , by $q_{w} (x, y) = Ψ (w (x))^{y} (1 - Ψ (w (x)))^{1 - y}, q_{w_{0}} (x, y) = Ψ (w_{0} (x))^{y} (1 - Ψ (w_{0} (x)))^{1 - y} .$ If $w - w_{0} \in L^{2} (G)$ , then, with $d_{2} (q_{w}, q_{w_{0}})^{2} = \int ∣ q_{w} - q_{w_{0}} ∣^{2} d (G \otimes #)$ , $d_{2} (q_{w}, q_{w_{0}})^{2} \leq 2 L^{2} ∥ w - w_{0} ∥_{L^{2} (G)}^{2},$ $K L (Q_{w_{0}} ∥ Q_{w}) \leq \frac{L R}{2} ∥ w - w_{0} ∥_{L^{2} (G)}^{2}, E_{Q_{w_{0}}} ⁣ [{(\log \frac{q_{w} (X, Y)}{q_{w_{0}} (X, Y)})}^{2}] \leq R^{2} ∥ w - w_{0} ∥_{L^{2} (G)}^{2} .$

Facts & Assumptions

Given: A probability covariate law $G$ , a differentiable link into $(0, 1)$ with the stated bounds, and the displayed Bernoulli joint densities.

[A1]

Assume AC for the directional Kullback–Leibler convention The Axiom of Choice.

[F1]

A function differentiable at $c$ is continuous at $c$ and The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : differentiability gives continuity on each closed interval, and the mean-value theorem bounds increments by an intermediate derivative.

[F2]

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: derivatives of $\log Ψ (u)$ and $\log (1 - Ψ (u))$ are respectively $Ψ^{'} (u) / Ψ (u)$ and $- Ψ^{'} (u) / (1 - Ψ (u))$ .

Proof

1.1F1givenalgebra

For each $x$ , [F1] gives $∣ Ψ (w (x)) - Ψ (w_{0} (x)) ∣ \leq L ∣ w (x) - w_{0} (x) ∣$ . The two values of $y$ contribute the same squared difference to $d_{2}^{2}$ , so integration gives $d_{2}^{2} = 2 \int ∣ Ψ (w) - Ψ (w_{0}) ∣^{2} d G \leq 2 L^{2} ∥ w - w_{0} ∥_{2}^{2}$ .

1.2A1F1F2givenalgebra

Fix $x$ and write $u_{0} = w_{0} (x)$ . The Bernoulli directional divergence at $x$ is $k_{x} (u) = Ψ (u_{0}) \log \frac{Ψ (u_{0})}{Ψ (u)} + (1 - Ψ (u_{0})) \log \frac{1 - Ψ (u_{0})}{1 - Ψ (u)} .$ By [F2] and the chain rule, $k_{x}^{'} (u) = \frac{Ψ^{'} (u)}{Ψ (u) (1 - Ψ (u))} (Ψ (u) - Ψ (u_{0}))$ . Thus $∣ k_{x}^{'} (u) ∣ \leq R L ∣ u - u_{0} ∣$ by [F1], and integration from $u_{0}$ to $w (x)$ yields $∣ k_{x} (w (x)) ∣ \leq (R L / 2) ∣ w (x) - w_{0} (x) ∣^{2}$ . Integrating over $G$ and using $K L (Q_{w_{0}} ∥ Q_{w}) = \int k_{x} (w (x)) G (d x)$ proves the stated upper bound.

2.1F1F2givenalgebra∎

The derivatives in [F2] have absolute values at most $R$ : indeed $Ψ^{'} / Ψ = (Ψ^{'} / [Ψ (1 - Ψ)]) (1 - Ψ)$ and $Ψ^{'} / (1 - Ψ) = (Ψ^{'} / [Ψ (1 - Ψ)]) Ψ$ . The mean-value theorem therefore bounds both Bernoulli log-probability ratios at $x$ by $R ∣ w (x) - w_{0} (x) ∣$ . Their squared conditional average under $Y ∣ X = x \sim Bernoulli (Ψ (w_{0} (x)))$ is at most $R^{2} ∣ w (x) - w_{0} (x) ∣^{2}$ ; integrating over $G$ proves the log-square bound.

Source notes

The source derives the same comparison through the function $Ψ^{'} / [Ψ (1 - Ψ)]$ . The proof here supplies the pointwise derivative and integration bounds, and states the joint-density $L^{2} (G \otimes #)$ convention explicitly.

Depends on

Used by

Gaussian-process posterior contraction in density and regression models Theorem

Dependency tree · two levels

43 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. W. van der Vaart and J. H. van Zanten, Rates of contraction of posterior distributions based on Gaussian process priors, §3.2, Theorem 3.2 and Lemma 3.2 (standard reference, not scraped)
Bas J. K. Kleijn, Aad W. van der Vaart and Harry van Zanten, Lectures on Nonparametric Bayesian Statistics, Chapter 9 and §12.2 (standard reference, not scraped)