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

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Empirical mean and scale distance control Gaussian-regression likelihoods

Statement

Let $0 < a < b < \infty$ , let $σ_{0}, σ \in [a, b]$ , and fix design points $x_{1}, \dots, x_{n}$ . Under the true independent laws $Y_{i} \sim N (w_{0} (x_{i}), σ_{0}^{2})$ , the model has independent laws $Y_{i} \sim N (w (x_{i}), σ^{2})$ . Write $δ_{i} = w (x_{i}) - w_{0} (x_{i})$ and $∥ δ ∥_{n}^{2} = n^{- 1} \sum_{i} δ_{i}^{2}$ . There is a finite constant $C_{a, b}$ , depending only on $a, b$ , such that whenever $\max_{i} ∣ δ_{i} ∣ \leq 1$ and $∣ σ - σ_{0} ∣ \leq 1$ , $\frac{1}{n} \sum_{i = 1}^{n} K L (P_{i 0} ∥ P_{i}) \leq C_{a, b} (∥ δ ∥_{n}^{2} + ∣ σ - σ_{0} ∣^{2}),$ $\frac{1}{n} \sum_{i = 1}^{n} E_{i 0} ⁣ [{(\log \frac{p_{i} (Y_{i})}{p_{i 0} (Y_{i})})}^{2}] \leq C_{a, b} (∥ δ ∥_{n}^{2} + ∣ σ - σ_{0} ∣^{2}) .$

Facts & Assumptions

Given: The independent fixed-design normal model with true and candidate scales in $[a, b]$ , $a > 0$ , and the displayed local bounds on mean and scale differences.

[A1]

Assume AC for Kullback–Leibler divergence and the imported normal laws.

[F1]

Standard normal and normal laws: a $N (m, σ^{2})$ variable with $σ > 0$ has the translated and scaled standard-normal law.

[F2]

Gaussian even moments for Brownian increments: a centered $N (0, σ_{0}^{2})$ variable has second and fourth moments $σ_{0}^{2}$ and $3 σ_{0}^{4}$ ; symmetry makes its third moment zero.

[F3]

A uniform derivative bound gives a uniform Taylor remainder bound: a uniform bound on the second derivative bounds the first-order Taylor remainder by a constant times the squared increment.

Proof

1.1F1F2givenalgebra

Put $e_{i} = Y_{i} - w_{0} (x_{i})$ and $δ_{i} = w (x_{i}) - w_{0} (x_{i})$ . Direct substitution of the normal densities from [F1] gives $\log \frac{p_{i} (Y_{i})}{p_{i 0} (Y_{i})} = \log \frac{σ_{0}}{σ} - \frac{(e_{i} - δ_{i})^{2}}{2 σ^{2}} + \frac{e_{i}^{2}}{2 σ_{0}^{2}} .$ Taking its negative expectation and using [F2] yields $K_{i} : = K L (P_{i 0} ∥ P_{i}) = \log \frac{σ}{σ_{0}} + \frac{σ_{0}^{2} + δ_{i}^{2}}{2 σ^{2}} - \frac{1}{2} .$

2.1A1F3step 1.1givenalgebra

Let $s = σ - σ_{0}$ and $g (σ) = \log (σ / σ_{0}) + σ_{0}^{2} / (2 σ^{2}) - 1 / 2$ . Then $g (σ_{0}) = g^{'} (σ_{0}) = 0$ . Its second derivative is bounded uniformly when $σ, σ_{0} \in [a, b]$ , so [F3] gives $∣ g (σ) ∣ \leq C_{a, b} s^{2}$ . Explicitly, $g^{'} (σ) = 1 / σ - σ_{0}^{2} / σ^{3}$ vanishes at $σ_{0}$ , and $∣ g^{''} (σ) ∣ = ∣ - 1 / σ^{2} + 3 σ_{0}^{2} / σ^{4} ∣ \leq a^{- 2} + 3 b^{2} a^{- 4}$ . Also $δ_{i}^{2} / (2 σ^{2}) \leq δ_{i}^{2} / (2 a^{2})$ . The formula in step 1.1 therefore gives $∣ K_{i} ∣ \leq C_{a, b} (δ_{i}^{2} + s^{2})$ and, in particular, the first claimed upper bound after averaging.

3.1F2step 1.1step 2.1givenalgebra∎

The log ratio in step 1.1 equals $m_{i} + B_{i} e_{i} + A (e_{i}^{2} - σ_{0}^{2})$ , where $A = 1 / (2 σ_{0}^{2}) - 1 / (2 σ^{2})$ , $B_{i} = δ_{i} / σ^{2}$ , and $m_{i} = - K_{i}$ . On $[a, b]$ , $∣ A ∣ \leq C_{a, b} ∣ s ∣$ and $∣ B_{i} ∣ \leq a^{- 2} ∣ δ_{i} ∣$ . By [F2], $E e_{i}^{2} = σ_{0}^{2}$ , $E (e_{i}^{2} - σ_{0}^{2})^{2} = 2 σ_{0}^{4}$ , and $E [e_{i} (e_{i}^{2} - σ_{0}^{2})] = 0$ . Therefore the second moment is $K_{i}^{2} + B_{i}^{2} σ_{0}^{2} + 2 A^{2} σ_{0}^{4}$ . Step 2.1 gives $K_{i}^{2} \leq C_{a, b} (δ_{i}^{2} + s^{2})^{2}$ , which is at most a constant times $δ_{i}^{2} + s^{2}$ under the local bounds. Averaging proves the log-square inequality.

Source notes

The 2008 source states the fixed-covariate model but leaves its contraction proof as a sketch. The complete local KL and log-square calculations above make the scale, direction, empirical norm and moment bounds explicit; the independent-observation source is read as separate context, not substituted for the proof.

Depends on

Used by

Gaussian-process posterior contraction in density and regression models Theorem

Dependency tree · two levels

59 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, §3.3, Theorem 3.3 (standard reference, not scraped)
Subhashis Ghosal and Aad W. van der Vaart, Convergence rates of posterior distributions for noniid observations, §3, Theorem 4 and Lemma 2 (standard reference, not scraped)