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Gaussian-process posterior contraction

1 · Prerequisites

2 · Summary

Objective

This page proves posterior contraction for three separate experiments using Gaussian prior concentration, local likelihood comparisons, exponential tests and a posterior denominator bound. The log-Gaussian density and binary-regression branches retain the stated concentration rate. The fixed-design Gaussian regression branch includes an independent unknown scale with a lower-bounded density on a fixed positive compact interval.

For every allowed scale prior, its proved rate is $ρ_{n} = \max {ϵ_{n}, \sqrt{\log n / n}}$ . It retains the sharper $ϵ_{n}$ rate for nonzero Gaussian laws by a nonzero-functional small-ball argument. For the zero Gaussian law it retains that sharper rate when the scale density is locally bounded above near the truth, using a separate parametric likelihood proof. The explicit spiky-prior counterexample on the companion page explains why the unrestricted sharper rate would be false. Generic concentration bounds are not themselves posterior theorems; the three model-specific sampling and measurability hypotheses remain essential.

3 · Logical flowchart

4 · Definitions, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

A tight Gaussian law is carried by a separable closed subspace

Statement

Let $B$ be a real Banach space and let $W$ be a Borel measurable $B$ -valued random element with tight law $μ$ . Tight means that for every $η > 0$ some compact $K \subset B$ satisfies $μ (K) > 1 - η$ . Assume $W$ is centered Gaussian, meaning that every continuous real linear functional $ℓ : B \to R$ makes $ℓ (W)$ a centered scalar normal random variable. Then there is a separable closed linear subspace $E \subset B$ with $μ (E) = 1$ . The law of $W$ viewed as an $E$ -valued random element is tight and centered Gaussian, and its support in $E$ is exactly the support of $μ$ in $B$ , viewed as a subset of $B$ . In particular, every ambient support point belongs to $E$ .

Facts & Assumptions

Given: A real Banach space $B$ , a tight Borel probability law $μ$ on $B$ , and a centered Gaussian random element $W$ with law $μ$ .

[A1]

The Axiom of Choice: AC permits the countable selection of compact sets and finite nets used below.

[A2]

The real dominated-extension principle as an additional hypothesis over ZF: the real dominated-extension principle extends a linear functional dominated by a sublinear functional to the ambient vector space, preserving domination.

Proof

1.1givenA1construct

For each integer $m \geq 1$ , tightness and [A1] give a compact set $K_{m} \subset B$ with $μ (K_{m}) > 1 - 2^{- m}$ . Each compact metric space $K_{m}$ has a countable dense subset: for each integer $r \geq 1$ , compactness gives a finite $1 / r$ -net, and the union of these finite nets is countable and dense. The union over $m$ of these dense subsets is countable; let $S$ be its closure. Then $S$ is separable, contains every $K_{m}$ , and $μ (S) \geq μ (K_{m}) > 1 - 2^{- m}$ for every $m$ , hence $μ (S) = 1$ .

2.1A2step 1.1givenconstruct

Let $E$ be the closed real linear span of $S$ . Rational linear combinations of a countable dense subset of $S$ form a countable dense subset of its real linear span, so $E$ is separable; it is closed by construction and therefore Banach, and $μ (E) = 1$ . Define $W_{E} = W$ on the measurable event ${W \in E}$ and $W_{E} = 0$ on its null complement. Then $W_{E}$ is $E$ -valued and Borel measurable: each Borel set of $E$ is $E \cap A$ for some ambient Borel $A$ , and its preimage is ${W \in E} \cap W^{- 1} (A)$ together with the null complement if the set contains zero. Its law is the restriction of $μ$ to $E$ . For any $ℓ \in E^{*}$ , let $p (x) = ∥ ℓ ∥_{E^{*}} ∥ x ∥_{B}$ on $B$ ; $ℓ$ is dominated by $p$ on $E$ . By [A2] it extends to a linear $\tilde{ℓ}$ on $B$ dominated by $p$ , and applying domination to $x$ and $- x$ gives $∣ \tilde{ℓ} (x) ∣ \leq ∥ ℓ ∥_{E^{*}} ∥ x ∥_{B}$ . Since $W_{E} = W$ almost surely, $ℓ (W_{E}) = \tilde{ℓ} (W)$ almost surely, so $ℓ (W_{E})$ is centered normal. Thus the restricted law is Gaussian.

3.1step 1.1step 2.1givenalgebra∎

For tightness in $E$ , replace each compact $K_{m}$ by $K_{m} \cap E$ , which is compact and has the same $μ$ -mass because $μ (E) = 1$ . If $x \notin E$ , closedness gives a $B$ -open neighborhood disjoint from $E$ , of $μ$ -mass zero, so $x$ is outside the ambient support. If $x \in E$ , every ambient open ball about $x$ intersects $E$ in an $E$ -open neighborhood and has the same measure as that intersection; hence all such neighborhoods have positive measure exactly when all relative neighborhoods do. The two supports therefore agree.

Source notes

The cited sources use tight Borel Gaussian random elements and their support/RKHS framework. The separable closed carrier and the transfer of Gaussianity to it are proved here; neither source excerpt is treated as a substitute proof.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

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.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

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.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Linear and quadratic Gaussian statistics have exponential tails

Statement

Assume AC. Let $n \geq 1$ be an integer. Let $Z_{1}, \dots, Z_{n}$ be independent $N (0, 1)$ variables, let $c_{1}, \dots, c_{n}, δ_{1}, \dots, δ_{n} \in R$ , and let $0 < σ \leq b$ . Put $∥ c ∥_{n}^{2} = n^{- 1} \sum_{i} c_{i}^{2}$ and $∥ δ ∥_{n}^{2} = n^{- 1} \sum_{i} δ_{i}^{2}$ . For $t > 0$ , $\Pr ⁣ (∣ n^{- 1} \sum_{i} c_{i} σ Z_{i} ∣ \geq t) \leq 2 \exp ⁣ (- \frac{n t^{2}}{2 σ^{2} ∥ c ∥_{n}^{2}}),$ with zero probability when $∥ c ∥_{n} = 0$ . If $∥ δ ∥_{n} \leq 1$ and $0 < t \leq b^{2} + 2$ , then $\Pr ⁣ (∣ n^{- 1} \sum_{i} (δ_{i} + σ Z_{i})^{2} - (σ^{2} + ∥ δ ∥_{n}^{2}) ∣ \geq t) \leq 2 \exp ⁣ (- \frac{n t^{2}}{4 V_{b}}), V_{b} = 2 b^{4} + 4 b^{2} .$

Facts & Assumptions

Given: The independent standard-normal variables and the stated deterministic coefficients.

[A1]

Assume AC for the standard-normal law Standard normal and normal laws.

[F1]

Expectations factor over finite products of independent random variables: expectations of integrable products of functions of independent variables factor.

[F2]

A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not gives positive linear scaling of Lebesgue measure, and Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation gives translation invariance. Applying these set identities first to nonnegative simple functions and then using Monotone convergence for the integral gives $\int_{R} f (a z + h) d z = a^{- 1} \int_{R} f (y) d y$ for nonnegative Borel $f$ , $a > 0$ , and $h \in R$ .

[F3]

Markov's inequality for random variables: for a nonnegative random variable $X$ and $a > 0$ , $\Pr (X \geq a) \leq E X / a$ .

[F4]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: for $x > 0$ , $\log x = \int_{1}^{x} d u / u$ .

Proof

1.1A1F1F2F3givenalgebra

By Standard normal and normal laws, $Z$ has density $e^{- z^{2} / 2} / \sqrt{2 π}$ . Completing the square and translating the integration variable by $u$ using [F2] gives $E e^{u Z} = e^{u^{2} / 2}$ for every real $u$ . Independence and [F1] therefore give the moment generating function of $n^{- 1} \sum_{i} c_{i} σ Z_{i}$ as $\exp (λ^{2} σ^{2} \sum_{i} c_{i}^{2} / (2 n^{2}))$ . Exponential Markov [F3] with $λ = n^{2} t / (σ^{2} \sum_{i} c_{i}^{2})$ bounds each one-sided tail by $\exp (- n t^{2} / (2 σ^{2} ∥ c ∥_{n}^{2}))$ ; if all $c_{i} = 0$ , the statistic is identically zero.

1.2A1F1F2F4givenalgebra

For $∣ λ ∣ \leq 1 / (4 b^{2})$ , completing the square in the same density integral gives $E \exp {λ [(δ_{i} + σ Z_{i})^{2} - (δ_{i}^{2} + σ^{2})]} = (1 - 2 λ σ^{2})^{- 1 / 2} \exp ⁣ (- λ σ^{2} + \frac{2 λ^{2} σ^{2} δ_{i}^{2}}{1 - 2 λ σ^{2}}) .$ Here $∣ 2 λ σ^{2} ∣ \leq 1 / 2$ . By [F4], for $∣ x ∣ \leq 1 / 2$ , $- x - \log (1 - x) \leq x^{2}$ : for $x \geq 0$ integrate $u / (1 - u) \leq 2 u$ on $[0, x]$ , and for $x < 0$ integrate $u / (1 + u) \leq u$ on $[0, ∣ x ∣]$ . Taking logarithms in the display, summing over $i$ , and using [F1] therefore bounds the centered-sum log moment generating function above by $λ^{2} (2 n b^{4} + 4 b^{2} \sum_{i} δ_{i}^{2}) \leq n V_{b} λ^{2}$ .

2.1F3step 1.2givenalgebra∎

For $0 < t \leq b^{2} + 2 = V_{b} / (2 b^{2})$ , choose $λ = t / (2 V_{b})$ , which satisfies $0 < λ \leq 1 / (4 b^{2})$ . Exponential Markov [F3] and step 1.2 give the upper tail $\exp (- n t^{2} / (4 V_{b}))$ ; applying the same argument to the negative centered sum gives the lower tail. Adding the two bounds proves the claimed quadratic-statistic inequality.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Local KL and log-square mass bounds a non-IID posterior denominator

Statement

Assume AC. Let $C \geq 0$ , and for each integer $n \geq 1$ let $ϵ_{n} > 0$ . Let $Y_{1}, \dots, Y_{n}$ be independent, with true densities $p_{i 0}$ and candidate densities $p_{i θ}$ relative to supplied sigma-finite measures $μ_{i}$ . Assume $(θ, y) \mapsto p_{i θ} (y)$ is jointly measurable and let $Π$ be a prior. If a measurable set $K_{n}$ satisfies $Π (K_{n}) \geq e^{- C n ϵ_{n}^{2}}, \sup_{θ \in K_{n}} \frac{1}{n} \sum_{i = 1}^{n} K L (P_{i 0} ∥ P_{i θ}) \leq ϵ_{n}^{2}, \sup_{θ \in K_{n}} \frac{1}{n} \sum_{i = 1}^{n} E_{i 0} ⁣ [{(\log \frac{p_{i θ} (Y_{i})}{p_{i 0} (Y_{i})})}^{2}] \leq ϵ_{n}^{2},$ then the denominator $D_{n} = \int \prod_{i = 1}^{n} \frac{p_{i θ} (Y_{i})}{p_{i 0} (Y_{i})} Π (d θ)$ is finite $P_{0}^{(n)}$ -almost surely and satisfies $P_{0}^{(n)} {D_{n} \geq e^{- (C + 2) n ϵ_{n}^{2}}} \geq 1 - \frac{1}{n ϵ_{n}^{2}} .$

Facts & Assumptions

Given: A jointly measurable dominated independent-observation model, prior $Π$ , the measurable local set $K_{n}$ , and its displayed average directional KL/log-square bounds.

[A1]

Assume AC for Kullback–Leibler divergence and finite-valued density representatives.

[F1]

Independent random elements and Measurable coordinatewise functions preserve independence: measurable functions of distinct independent observations remain independent.

[F2]

Expectations factor over finite products of independent random variables: expectations of nonnegative products of independent functions factor.

[F3]

Jensen's integral inequality for a probability measure: for a probability average, the square of the average is at most the average of the square, and the average of a logarithm is at most the logarithm of the average.

[F4]

Variance of a finite sum as the sum of all variances and covariances: the variance of a sum of independent square-integrable variables is the sum of their variances.

[F5]

Chebyshev's inequality for random variables: if $X$ has finite variance, $\Pr (∣ X - E X ∣ \geq t) \leq Var (X) / t^{2}$ .

[F6]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative product-measurable functions may be integrated in either order.

Proof

1.1A1F1F2F6given

On the $P_{0}^{(n)}$ -full set where every $p_{i 0} (Y_{i})$ is positive and finite, put $R_{n} (θ) = \prod_{i} p_{i θ} (Y_{i}) / p_{i 0} (Y_{i})$ . For fixed $θ$ , independence [F1]–[F2] gives $E_{0} R_{n} (θ) = \prod_{i} \int_{{p_{i 0} > 0}} p_{i θ} d μ_{i} \leq 1.$ Tonelli [F6] then gives $E_{0} D_{n} \leq 1$ , so $D_{n} < \infty$ almost surely.

1.2A1F1F3F4F6givenalgebra

Let $Π_{K_{n}}$ be $Π$ conditioned on $K_{n}$ and set $Z_{n} = \int_{K_{n}} \sum_{i} \log \frac{p_{i θ} (Y_{i})}{p_{i 0} (Y_{i})} Π_{K_{n}} (d θ) .$ Tonelli [F6] applied to the log-ratio squares gives integrability over the conditional prior and sample space; the inequality $∣ u ∣ \leq 1 + u^{2}$ also makes the log ratios jointly integrable. Thus their conditional-prior integrals have measurable finite versions almost surely, and Jensen [F3] makes them square-integrable. Directional KL gives $E_{0} Z_{n} \geq - n ϵ_{n}^{2}$ . Writing $Z_{n} = \sum_{i} V_{i} (Y_{i})$ , independence [F1] and [F4] give ${Var}_{0} Z_{n} \leq \sum_{i} E_{0} V_{i} (Y_{i})^{2}$ . Jensen [F3] applied to the conditional prior integral yields $E_{0} V_{i} (Y_{i})^{2} \leq \int_{K_{n}} E_{i 0} ⁣ [{(\log \frac{p_{i θ} (Y_{i})}{p_{i 0} (Y_{i})})}^{2}] Π_{K_{n}} (d θ),$ so ${Var}_{0} Z_{n} \leq n ϵ_{n}^{2}$ .

2.1F3F5step 1.1step 1.2givenalgebra∎

Chebyshev [F5] implies $P_{0}^{(n)} {Z_{n} < - 2 n ϵ_{n}^{2}} \leq 1 / (n ϵ_{n}^{2})$ , since $E_{0} Z_{n} \geq - n ϵ_{n}^{2}$ and ${Var}_{0} Z_{n} \leq n ϵ_{n}^{2}$ . Jensen for $\log$ gives $D_{n} \geq \int_{K_{n}} R_{n} (θ) Π (d θ) \geq Π (K_{n}) \exp (Z_{n}) .$ On the complementary event, this is at least $e^{- (C + 2) n ϵ_{n}^{2}}$ . Together with step 1.1 this proves the claim.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Tests and a sieve tail imply posterior contraction from a denominator bound

Statement

For each $n$ , let $P_{θ, n}$ and $P_{0, n}$ have jointly measurable densities $p_{θ, n}$ and $p_{0, n}$ relative to a common sigma-finite measure, and let $Π$ be a prior. Let $Θ_{n}$ be a measurable sieve, let $A_{n} = {θ : d_{n} (θ, θ_{0}) > r_{n}},$ and let $D_{n} = \int p_{θ, n} (Y) / p_{0, n} (Y) Π (d θ)$ be the likelihood-ratio denominator on the $P_{0, n}$ -full set where $p_{0, n} (Y) > 0$ . Suppose $ϵ_{n} > 0$ , $n ϵ_{n}^{2} \to \infty$ , and constants $A < B$ , $T > A$ satisfy $P_{0, n} {D_{n} < e^{- A n ϵ_{n}^{2}}} \to 0, Π (Θ_{n}^{c}) \leq e^{- B n ϵ_{n}^{2}},$ and there are measurable tests $ϕ_{n}$ with $P_{0, n} (ϕ_{n} = 1) \leq e^{- T n ϵ_{n}^{2}}, \sup_{θ \in A_{n} \cap Θ_{n}} P_{θ, n} (ϕ_{n} = 0) \leq e^{- T n ϵ_{n}^{2}} .$ Then the posterior mass of $A_{n}$ tends to zero in $P_{0, n}$ -probability.

Facts & Assumptions

Given: The displayed dominated model, measurable sieve and tested alternative, denominator lower bound, sieve-tail bound, and test errors.

[F1]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative likelihood-ratio integrands may be integrated in either order.

[F2]

Markov's inequality for random variables: for nonnegative $X$ and $u > 0$ , $P (X > u) \leq E X / u$ .

Proof

1.1F1givenalgebra

Define $N_{n} = \int_{A_{n} \cap Θ_{n}} \frac{p_{θ, n} (Y)}{p_{0, n} (Y)} Π (d θ), T_{n} = \int_{A_{n} \cap Θ_{n}^{c}} \frac{p_{θ, n} (Y)}{p_{0, n} (Y)} Π (d θ) .$ By change of density on ${p_{0, n} > 0}$ and Tonelli [F1], $E_{0, n} [(1 - ϕ_{n}) N_{n}] \leq \int_{A_{n} \cap Θ_{n}} P_{θ, n} (ϕ_{n} = 0) Π (d θ) \leq e^{- T n ϵ_{n}^{2}} .$ For every $θ$ , $E_{0, n} [p_{θ, n} (Y) / p_{0, n} (Y)] = \int_{{p_{0, n} > 0}} p_{θ, n} \leq 1$ , so Tonelli also gives $E_{0, n} T_{n} \leq Π (Θ_{n}^{c}) \leq e^{- B n ϵ_{n}^{2}}$ . Integrating the same bound over all $Θ$ gives $E_{0, n} D_{n} \leq 1$ , hence $D_{n} < \infty$ almost surely.

2.1F2step 1.1givenalgebra∎

Fix $δ > 0$ . By step 1.1 the denominator is finite almost surely. On fibers with $D_{n} = 0$ , assign the posterior mass any value in $[0, 1]$ ; these fibers lie in ${D_{n} < e^{- A n ϵ_{n}^{2}}}$ . On the event $D_{n} \geq e^{- A n ϵ_{n}^{2}}$ and $ϕ_{n} = 0$ , posterior mass $Π (A_{n} ∣ Y) > δ$ implies $N_{n} + T_{n} > δ e^{- A n ϵ_{n}^{2}}$ . The union bound and Markov [F2] therefore give $P_{0, n} {Π (A_{n} ∣ Y) > δ} \leq P_{0, n} {D_{n} < e^{- A n ϵ_{n}^{2}}} + e^{- T n ϵ_{n}^{2}} + δ^{- 1} ⁣ (e^{- (T - A) n ϵ_{n}^{2}} + e^{- (B - A) n ϵ_{n}^{2}}) .$ Every term tends to zero because $T > A$ , $B > A$ and $n ϵ_{n}^{2} \to \infty$ . Since this holds for every $δ > 0$ , posterior mass on $A_{n}$ converges to zero in probability.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

A nonzero Gaussian direction forces logarithmic small-ball cost

Statement

Let $W$ be a centered Gaussian random element in a real normed space $B$ . Let $ℓ \in B^{*}$ satisfy $Var (ℓ (W)) = τ^{2} > 0$ , and let $H$ be a Hilbert subspace of $B$ with nonnegative squared norm. For $w_{0} \in B$ , define $ϕ_{w_{0}} (ϵ) = \inf_{\begin{array}{c} h \in H \\ ∥ h - w_{0} ∥_{B} < ϵ \end{array}} {∥ h ∥_{H}^{2}} - \log \Pr (∥ W ∥_{B} < ϵ),$ with $\inf \emptyset = + \infty$ . Then for every $ϵ > 0$ , $ϕ_{w_{0}} (ϵ) \geq \log (1 / ϵ) - C_{ℓ}, C_{ℓ} = \log ⁣ (\frac{2 ∥ ℓ ∥_{B^{*}}}{\sqrt{2 π} τ}) .$

Facts & Assumptions

Given: A centered Gaussian random element, a continuous linear functional with positive variance, and the displayed nonnegative RKHS approximation penalty.

[F1]

Standard normal and normal laws: a centered scalar Gaussian with variance $τ^{2} > 0$ has density bounded above by $(\sqrt{2 π} τ)^{- 1}$ .

[F2]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log y = \int_{1}^{y} d t / t$ for $y > 0$ .

Proof

1.1F1givenalgebra

On the event ${∥ W ∥_{B} < ϵ}$ , continuity and linearity give $∣ ℓ (W) ∣ \leq ∥ ℓ ∥_{B^{*}} ∥ W ∥_{B} < ∥ ℓ ∥_{B^{*}} ϵ$ . The interval $(- ∥ ℓ ∥ ϵ, ∥ ℓ ∥ ϵ)$ has length $2 ∥ ℓ ∥ ϵ$ , so its scalar Gaussian probability is at most its length times the density bound in [F1]. Therefore $\Pr (∥ W ∥_{B} < ϵ) \leq \frac{2 ∥ ℓ ∥_{B^{*}}}{\sqrt{2 π} τ} ϵ .$

2.1F2step 1.1givenalgebra∎

The approximation penalty in the displayed definition is nonnegative. Taking negative logarithms of the bound in step 1.1 and adding this penalty yields $ϕ_{w_{0}} (ϵ) \geq \log (1 / ϵ) - \log (2 ∥ ℓ ∥_{B^{*}} / (\sqrt{2 π} τ))$ , which is the asserted constant.

Source notes

The source defines the Gaussian concentration function as an RKHS approximation cost plus centered small-ball exponent. The one-dimensional projection bound is proved here and does not use a source inequality.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Supremum distance controls local log-Gaussian density distances

Statement

Let $(X, A, ν)$ be a measure space, and let $w, w_{0} : X \to R$ be measurable with finite positive normalizers $Z_{w} = \int e^{w} d ν$ and $Z_{0} = \int e^{w_{0}} d ν$ . Set $p_{w} = e^{w} / Z_{w}$ , $p_{0} = e^{w_{0}} / Z_{0}$ , and $r = ∥ w - w_{0} ∥_{\infty} \leq 1$ . With root-L2 Hellinger distance as in Root-L2 Hellinger distance between density laws, and the directional convention of Kullback–Leibler divergence, $h (P_{w}, P_{0}) \leq e^{r} r, K L (P_{0} ∥ P_{w}) \leq \frac{1}{2} e^{2 r} r^{2}, \int {(\log \frac{p_{w}}{p_{0}})}^{2} p_{0} d ν \leq 4 r^{2} .$

Facts & Assumptions

Given: The stated measurable functions, finite positive normalizers and supremum-distance bound.

[A1]

Assume AC for the imported directional KL convention The Axiom of Choice.

[F1]

Root-L2 Hellinger distance between density laws: $h^{2} (P_{w}, P_{0}) = \int (\sqrt{p_{w}} - \sqrt{p_{0}})^{2} d ν$ .

[F2]

A uniform derivative bound gives a uniform Taylor remainder bound: if a scalar function has derivatives through order two on the closed interval between the two inputs, with the usual endpoint continuity, its first-order Taylor remainder is bounded by half a bound on the absolute second derivative times the squared increment.

[F3]

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 $f (b) - f (a) = f^{'} (c) (b - a)$ : for a scalar function continuous on the closed interval between distinct inputs and differentiable on its interior, the increment equals its derivative at an intermediate point times the input increment. By The exponential function is smooth and $(\exp)^{'} = \exp$ , the exponential is $C^{\infty}$ with derivative equal to itself, so it satisfies these hypotheses on every compact interval.

[F4]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log x = \int_{1}^{x} d t / t$ for $x > 0$ ; hence $\log (1 + x) \leq x$ for $x \geq 0$ .

Proof

1.1givenalgebra

Put $g = w - w_{0}$ , $c = \log \int e^{g} p_{0} d ν$ . Since $e^{- r} \leq e^{g} \leq e^{r}$ , we have $∣ c ∣ \leq r$ and $\log (p_{w} / p_{0}) = g - c$ , so $∣ \log (p_{w} / p_{0}) ∣ \leq 2 r$ . Squaring and integrating against the probability density $p_{0}$ proves the log-square bound.

1.2A1F2F3F4givenalgebra

The directional divergence is $K L (P_{0} ∥ P_{w}) = c - \int g p_{0} d ν = \log \int e^{g - \int g p_{0} d ν} p_{0} d ν$ . Let $U = g - \int g p_{0} d ν$ . Then $E_{0} U = 0$ , $∣ U ∣ \leq 2 r$ , and $E_{0} U^{2} \leq E_{0} g^{2} \leq r^{2}$ . Taylor's remainder [F2], applied to $e^{u}$ at zero, gives $e^{U} \leq 1 + U + \frac{1}{2} e^{2 r} U^{2}$ ; hence $E_{0} e^{U} \leq 1 + \frac{1}{2} e^{2 r} r^{2}$ . The integral formula in [F4] gives $\log (1 + x) \leq x$ for $x \geq 0$ , so the KL bound follows.

2.1F1F3step 1.1givenalgebra∎

Write $R = p_{w} / p_{0} = e^{g - c}$ . Then $∣ \sqrt{R} - 1 ∣ = ∣ e^{(g - c) / 2} - 1 ∣ \leq e^{r} ∣ g - c ∣ / 2 \leq e^{r} r$ , by [F3]. Therefore $h^{2} (P_{w}, P_{0}) = E_{0} (\sqrt{R} - 1)^{2} \leq e^{2 r} r^{2}$ , which yields the Hellinger bound.

Source notes

The cited treatments compare Hellinger, directional KL and log-likelihood variance to the supremum norm. The displayed constants here are derived directly under the finite-normalizer hypotheses.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Supremum sieves admit tests in fixed-design Gaussian regression

Statement

Let $x_{1}, \dots, x_{n}$ be deterministic, $Y_{i} = w_{0} (x_{i}) + σ_{0} Z_{i}$ under the truth with independent $Z_{i} \sim N (0, 1)$ , and let candidate parameters be $(w, σ)$ with $σ \in [a, b]$ , $0 < a < b$ , and $σ_{0} \in [a, b]$ . Put $∥ u ∥_{n}^{2} = n^{- 1} \sum_{i} u (x_{i})^{2}$ . Suppose $ϵ_{n} \to 0$ , $n ϵ_{n}^{2} \to \infty$ , and a measurable function sieve $B_{n}$ has a finite supremum-norm cover of radius $3 ϵ_{n}$ with at most $\exp (6 C n ϵ_{n}^{2})$ centers, for a fixed $C > 0$ . For every $T > 0$ , there is a sufficiently large fixed $M$ and a measurable test $ϕ_{n}$ such that, for all sufficiently large $n$ , $P_{0}^{n} (ϕ_{n} = 1) \leq e^{- T n ϵ_{n}^{2}}, \sup_{\begin{array}{c} w \in B_{n}, σ \in [a, b] \\ ∥ w - w_{0} ∥_{n} + ∣ σ - σ_{0} ∣ > M ϵ_{n} \end{array}} P_{w, σ}^{n} (ϕ_{n} = 0) \leq e^{- T n ϵ_{n}^{2}} .$

Facts & Assumptions

Given: The deterministic design, true independent Gaussian errors, compact positive scale interval, rate sequence and finite measurable supremum cover.

[A1]

Assume AC The Axiom of Choice to select a finite cover for each n.

[F1]

Linear and quadratic Gaussian statistics have exponential tails: if $∥ c ∥_{n}$ is fixed, $n^{- 1} \sum_{i} c_{i} σ Z_{i}$ has tails at scale $∥ c ∥_{n} / \sqrt{n}$ ; if $∥ δ ∥_{n} \leq 1$ , the centered residual-energy statistic $n^{- 1} \sum_{i} (δ_{i} + σ Z_{i})^{2} - (σ^{2} + ∥ δ ∥_{n}^{2})$ has tails bounded by $2 e^{- n t^{2} / (4 V_{b})}$ for $t \leq b^{2} + 2$ .

Proof

Given: Fix a cover $v_{1}, \dots, v_{N_{n}}$ of $B_{n}$ with $∥ v_{j} - v ∥_{\infty} < 3 ϵ_{n}$ for the covered $v$ , and $N_{n} \leq e^{6 C n ϵ_{n}^{2}}$ .

1.1A1F1givenconstruct

For each center put $δ_{j i} = v_{j} (x_{i}) - w_{0} (x_{i})$ and $ρ_{j} = ∥ δ_{j} ∥_{n}$ . For centers with $ρ_{j} \geq M ϵ_{n} / 4$ , define $U_{j} = n^{- 1} \sum_{i} δ_{j i} (Y_{i} - w_{0} (x_{i}))$ and $ϕ_{j} = 1 {U_{j} \geq ρ_{j}^{2} / 4}$ . Under the truth, $U_{j}$ is centered normal with variance $σ_{0}^{2} ρ_{j}^{2} / n$ , so [F1] gives $P_{0}^{n} (ϕ_{j} = 1) \leq \exp [- n ρ_{j}^{2} / (32 b^{2})]$ .

1.2givenalgebra

Put $Δ = M ϵ_{n} / 2$ and $Q_{n} = n^{- 1} \sum_{i} (Y_{i} - w_{0} (x_{i}))^{2}$ . If $∥ w - w_{0} ∥_{n} \leq Δ$ and $∣ σ - σ_{0} ∣ > Δ$ , then under $(w, σ)$ , $E Q_{n} = σ^{2} + ∥ w - w_{0} ∥_{n}^{2}$ . For an upper scale alternative, this mean is at least $σ_{0}^{2} + a M ϵ_{n}$ ; for a lower scale alternative, it is at most $σ_{0}^{2} - a M ϵ_{n} + Δ^{2} \leq σ_{0}^{2} - a M ϵ_{n} / 2$ once $M ϵ_{n} \leq 2 a$ . Define $ϕ_{+} = 1 {Q_{n} \geq σ_{0}^{2} + a M ϵ_{n} / 4}$ and $ϕ_{-} = 1 {Q_{n} \leq σ_{0}^{2} - a M ϵ_{n} / 4}$ . Under the truth and on this scale-alternative region, the relevant deviations from the respective means are at least $a M ϵ_{n} / 4$ .

2.1F1step 1.1givenalgebra

If $∥ w - w_{0} ∥_{n} > M ϵ_{n} / 2$ , choose a cover center $v_{j}$ within $3 ϵ_{n}$ of $w$ . For $M \geq 12$ , $ρ_{j} \geq (M / 2 - 3) ϵ_{n} \geq M ϵ_{n} / 4$ ; for $M \geq 24$ , $ρ_{j} \geq 6 ϵ_{n}$ . Under $(w, σ)$ , $E U_{j} = ⟨ v_{j} - w_{0}, w - w_{0} ⟩_{n} \geq ρ_{j}^{2} - 3 ϵ_{n} ρ_{j} \geq ρ_{j}^{2} / 2$ . The threshold is therefore at least $ρ_{j}^{2} / 4$ below the mean, and [F1] gives $P_{w, σ}^{n} (ϕ_{j} = 0) \leq \exp [- n ρ_{j}^{2} / (32 b^{2})] \leq \exp [- M^{2} n ϵ_{n}^{2} / (512 b^{2})]$ .

3.1F1step 1.1step 2.1step 1.2givenchoose∎

For sufficiently large n, $Δ \leq 1$ and $a M ϵ_{n} / 4 \leq b^{2} + 2$ , so [F1] bounds each null tail and each uniform scale-alternative error by $2 \exp [- n a^{2} M^{2} ϵ_{n}^{2} / (64 V_{b})]$ . Let $J_{n} = {j : ρ_{j} \geq M ϵ_{n} / 4}$ be the finite set of centers retained in step 1.1, and set $ϕ_{n} = ϕ_{+} \lor ϕ_{-} \lor \max_{j \in J_{n}} ϕ_{j}$ , defining the last maximum to be $0$ when $J_{n} = \emptyset$ . By step 2.1 every mean-separated alternative has a center in $J_{n}$ ; all other alternatives are covered by the scale tests from step 1.2. The null error is at most $e^{6 C n ϵ_{n}^{2}} e^{- M^{2} n ϵ_{n}^{2} / (512 b^{2})} + 4 e^{- n a^{2} M^{2} ϵ_{n}^{2} / (64 V_{b})}$ . Choose fixed $M$ large enough that both bounds are at most $e^{- T n ϵ_{n}^{2}}$ for all sufficiently large n.

Source notes

The 2008 GP article states the fixed-design theorem but gives only a sketch of its testing step. The non-IID article supplies the average-Hellinger product setting. The explicit mean-score and residual-energy tests above provide the complete local argument for the claimed Gaussian model.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Gaussian-process posterior contraction in density and regression models

Statement

Let $ϵ_{n} > 0$ satisfy $ϵ_{n} \to 0$ and $n ϵ_{n}^{2} \to \infty$ . For the centered tight Gaussian prior in each branch, let $μ$ be its law and let $B$ be a separable closed Banach carrier with the norm inherited from that branch's stated function space, as supplied by A tight Gaussian law is carried by a separable closed subspace. Write $H$ for the covariance RKHS of $μ$ on $B$ , and define $ϕ_{w_{0}} (ϵ) = \inf_{\begin{array}{c} h \in H \\ ∥ h - w_{0} ∥_{B} < ϵ \end{array}} {∥ h ∥_{H}^{2} - \log μ (ϵ B^{\circ})}, ϵ B^{\circ} = {u \in B : ∥ u ∥_{B} < ϵ} .$ Suppose $ϕ_{w_{0}} (ϵ_{n}) \leq n ϵ_{n}^{2}$ .

In each branch, parameter-to-data densities are jointly measurable and the posterior $Π_{n}$ is the data-indexed normalization of the prior times the likelihood on fibers with finite positive evidence.

Log-Gaussian density estimation. On a measurable space $(X, A)$ with sigma-finite measure $ν$ , let $W$ be a tight centered Borel Gaussian random element in $ℓ^{\infty} (X)$ , and let $w_{0}$ belong to its support. Assume path evaluation $(w, x) \mapsto w (x)$ is jointly measurable, $Z_{w} = \int e^{w} d ν \in (0, \infty)$ for prior-almost every $w$ and for $w_{0}$ , and set $p_{w} = e^{w} / Z_{w}$ , $p_{0} = p_{w_{0}}$ . For IID data $X_{1}, \dots, X_{n} \sim P_{0}$ , sufficiently large fixed $M$ gives $E_{0} Π_{n} {w : h (P_{w}, P_{0}) > M ϵ_{n} ∣ X^{n}} ⟶ 0.$ When $0 < ν (X) < \infty$ , bounded paths imply the required normalizers are finite and positive.
Binary regression. Let $(X_{i}, Y_{i})$ be IID, with $X_{i} \sim G$ , $Y_{i} \in {0, 1}$ and $P_{0} (Y = 1 ∣ X = x) = f_{0} (x) = Ψ (w_{0} (x)) \in (0, 1)$ . Assume $Ψ : R \to (0, 1)$ is differentiable, $\sup ∣ Ψ^{'} ∣ < \infty$ , and $\sup ∣ Ψ^{'} / [Ψ (1 - Ψ)] ∣ < \infty$ . Suppose the prior is either a tight centered Gaussian law in $L^{2} (G)$ with a jointly measurable evaluation version representing its $L^{2} (G)$ element almost everywhere, or a tight centered Gaussian law in $ℓ^{\infty} (X)$ with jointly measurable evaluation and bounded $w_{0}$ ; in either case $w_{0}$ belongs to the prior support. Then for sufficiently large fixed $M$ , $E_{0} Π_{n} {∥ Ψ (w) - f_{0} ∥_{L^{2} (G)} > M ϵ_{n} ∣ (X_{i}, Y_{i})_{i = 1}^{n}} ⟶ 0.$
Fixed-design Gaussian regression. For deterministic $x_{1}, \dots, x_{n}$ , assume independent $Y_{i} \sim N (w_{0} (x_{i}), σ_{0}^{2})$ , a centered tight Borel Gaussian prior $W$ in $ℓ^{\infty} (X)$ with $w_{0}$ in support, and an independent prior on the standard deviation $σ$ supported on a fixed $[a, b] \subset (0, \infty)$ containing $σ_{0}$ , with Lebesgue density bounded below by a positive constant. Put $ρ_{n} = \max {ϵ_{n}, \sqrt{\log n / n}}$ for $n \geq 2$ . Then for sufficiently large fixed $M$ , $E_{0} Π_{n} {∥ w - w_{0} ∥_{n} + ∣ σ - σ_{0} ∣ > M ρ_{n} ∣ Y_{1 : n}} ⟶ 0.$ The sharper conclusion with $ϵ_{n}$ in place of $ρ_{n}$ holds if the Gaussian law is not the point mass at zero. It also holds when $W = 0$ almost surely and the scale-prior density is bounded above on a neighborhood of $σ_{0}$ ; this extra upper bound is needed only for that sharper degenerate-prior alternative, not for the universal $ρ_{n}$ conclusion. In the nondegenerate case $ρ_{n} = O (ϵ_{n})$ , so the two formulations differ only by the fixed constant $M$ . Here $∥ w - w_{0} ∥_{n}^{2} = n^{- 1} \sum_{i} (w (x_{i}) - w_{0} (x_{i}))^{2}$ .

The concentration-function hypothesis is interpreted on a separable Banach space carrying the prior. For a tight Gaussian law initially given in nonseparable $ℓ^{\infty} (X)$ , the separable-support reduction is A tight Gaussian law is carried by a separable closed subspace; the norm and prior law are restricted to that support space.

Facts & Assumptions

Given: The three model contracts above, a common rate sequence satisfying $ϵ_{n} \to 0$ and $n ϵ_{n}^{2} \to \infty$ , and the displayed Gaussian concentration-function inequality.

[F8]

The support of a Banach Gaussian is the closure of its RKHS: under AC and countable choice, a centered tight Borel Gaussian law on a real separable Banach space has support equal to the Banach-norm closure of its covariance RKHS.

[F9]

Concentration function of a Banach Gaussian prior: the current definition uses $\inf_{∥ h - w_{0} ∥_{B} < ϵ} {∥ h ∥_{H}^{2} - \log μ (ϵ B^{\circ})}$ for $w_{0} \in {\overline{H}}^{B}$ .

[F10]

Gaussian concentration-function sieve bounds: under AC and countable choice, the full-squared concentration hypothesis supplies measurable sieves with $\log N (3 r, B_{n}, ∥ \cdot ∥_{B}) \leq 6 C n r^{2}$ , prior tail at most $e^{- C n r^{2}}$ , and mass at least $e^{- n r^{2}}$ in the $2 r$ -ball about the truth. Its sharp finite-dimensional expansion and Banach expansion suppliers have local proofs.

[F11]

Centered Borel Gaussian random element on a separable Banach space: a centered Borel Gaussian random element is a Borel-measurable map into a real separable Banach space whose continuous linear images are centered normal.

[F12]

Covariance operator and reproducing-kernel Hilbert space of a Banach Gaussian: under AC, the covariance RKHS is the completed covariance-form quotient embedded as a Hilbert space of shifts in the Banach space.

[A2]

In the fixed-design branch the scale prior has density at least $c > 0$ on $[a, b]$ . For $r ↓ 0$ , including when $σ_{0}$ is an endpoint, its $r$ -neighborhood therefore has prior mass at least $c r$ once $r < b - a$ . An upper density bound is assumed only in the sharper degenerate-prior alternative stated above.

[A3]

Assume AC and countable choice for the Gaussian concentration-sieve interface The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ). The separate Hahn–Banach extension principle used in [F5] is not inferred from these assumptions.

[A4]

Assume the separate relative Hahn–Banach extension principle The real dominated-extension principle as an additional hypothesis over ZF for the separable-support reduction [F5]; it is not silently deduced from AC.

[F1]

KL prior mass gives a posterior denominator lower bound in probability: for a jointly measurable dominated IID model, local prior mass at least $e^{- C n δ_{n}^{2}}$ on a set with directional KL and log-square at most $δ_{n}^{2}$ gives a denominator lower bound $e^{- (C + 2) n δ_{n}^{2}}$ except with probability at most $(n δ_{n}^{2})^{- 1}$ .

[F2]

Sieve metric entropy produces uniform exponential tests: same-metric packing entropy at radius $δ_{n}$ bounded by $n δ_{n}^{2}$ gives tests with exponentially small null and uniform alternative errors outside a sufficiently large multiple of $δ_{n}$ ; for density $L^{2}$ the densities must be uniformly bounded.

[F3]

Supremum distance controls local log-Gaussian density distances, The L² link norm controls binary-regression likelihood distances, and Empirical mean and scale distance control Gaussian-regression likelihoods give the local Hellinger, density- $L^{2}$ , directional KL and log-square comparisons used on this page.

[F4]

Local KL and log-square mass bounds a non-IID posterior denominator and Tests and a sieve tail imply posterior contraction from a denominator bound give the non-IID denominator bound and the tested-numerator/sieve-tail split used in the IID and fixed-design arguments.

[F5]

A tight Gaussian law is carried by a separable closed subspace: a tight Borel Gaussian law in a Banach space is carried by a separable closed subspace without changing its support or inherited norm.

[F6]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: joint measurability and Tonelli make the log-Gaussian normalizer a measurable function of the path.

[F7]

A nonzero Gaussian direction forces logarithmic small-ball cost: if the prior has a nonzero Gaussian direction, $ϕ_{w_{0}} (ϵ) \leq n ϵ^{2}$ implies $\log (1 / ϵ) \leq n ϵ^{2} + C$ for a fixed constant.

[F13]

A separable Banach space has a countable norming family under the declared Hahn–Banach and countable-choice assumptions. Thus a centered Gaussian law different from $δ_{0}$ has a continuous linear functional with positive variance. A separable Banach space has a countable norming family.

[F14]

A supremum-norm sieve with covering entropy at radius $3 r_{n}$ bounded by $6 C n r_{n}^{2}$ supplies fixed-design Gaussian tests outside a sufficiently large empirical-mean-plus-scale ball, with arbitrarily prescribed fixed exponential error exponent. Supremum sieves admit tests in fixed-design Gaussian regression.

[F15]

The centered Gaussian residual-energy statistic has an exponential tail at deviations of order $R / \sqrt{n}$ for fixed $R$ . Linear and quadratic Gaussian statistics have exponential tails.

Proof

Given: The hypotheses of the Statement and its three distinct model branches.

1.1A3A4F5F8F9F11F12givenalgebra

The separable-support reduction [F5] leaves the inherited norm, law and support point unchanged. For the first two branches use the rate $r_{n} = ϵ_{n}$ ; for the universal fixed-design branch use $r_{n} = ρ_{n}$ . The concentration function is nonincreasing in its radius, because its feasible RKHS approximation set grows and centered small-ball probabilities increase. Thus $ϕ_{w_{0}} (r_{n}) \leq ϕ_{w_{0}} (ϵ_{n}) \leq n ϵ_{n}^{2} \leq n r_{n}^{2}$ . Both choices satisfy $r_{n} \to 0$ and $n r_{n}^{2} \to \infty$ . By [F8], the support point lies in the RKHS closure.

1.2F3F15givenalgebra

Now suppose $W = 0$ almost surely and its scale density $g$ is bounded above by $U < \infty$ on $∣ σ - σ_{0} ∣ < δ$ within $[a, b]$ . The support assumption forces $w_{0} = 0$ . Put $S_{n} = \sum_{i} Y_{i}^{2}$ , $q = S_{n} / (n σ_{0}^{2})$ , and $L_{n} (σ) = (σ_{0} / σ)^{n} \exp [(1 - σ_{0}^{2} / σ^{2}) S_{n} / (2 σ_{0}^{2})]$ . On $E_{R} = {∣ S_{n} - n σ_{0}^{2} ∣ \leq R \sqrt{n}}$ , write $z = (σ - σ_{0}) / σ_{0}$ . Compactness of the positive scale interval and Taylor's formula give constants $c, C > 0$ independent of $n, R$ with $- C n (σ - σ_{0})^{2} - C R \sqrt{n} ∣ σ - σ_{0} ∣ \leq \log L_{n} (σ) \leq - c n (σ - σ_{0})^{2} + C R \sqrt{n} ∣ σ - σ_{0} ∣ .$ For verification, its deterministic part is $n [- \log (1 + z) + (1 - (1 + z)^{- 2}) / 2]$ , negative away from zero, with second derivative $- 2$ at zero; the remaining term is $(S_{n} - n σ_{0}^{2}) (1 - (1 + z)^{- 2}) / (2 σ_{0}^{2})$ , bounded by $C R \sqrt{n} ∣ z ∣$ . The residual-energy tail in [F15] with zero mean shift gives ${lim sup}_{n} P_{0} (E_{R}^{c}) \leq 2 e^{- c_{0} R^{2}}$ for fixed $R$ .

2.1F1F3F6F10step 1.1givenchoose

In log-Gaussian density estimation, evaluation measurability and Tonelli [F6] make $w \mapsto Z_{w}$ measurable. On its full-prior positive-finite set, $(w, x) \mapsto e^{w (x)} / Z_{w}$ is a jointly measurable density family. The $2 r_{n}$ supremum ball has directional KL at most $2 e^{4 r_{n}} r_{n}^{2}$ and log-square at most $16 r_{n}^{2}$ by [F3]. Choose a fixed sufficiently large $k$ . Its prior mass is at least $e^{- n r_{n}^{2}}$ by [F10], so [F1] at radius $k r_{n}$ gives a denominator at least $e^{- A n r_{n}^{2}}$ for fixed $A = 1 + 2 k^{2}$ except with probability tending to zero.

2.2F1F2F3F10step 1.1givenchoose

In binary regression, [F3] places the $2 r_{n}$ prior ball in a directional KL/log-square set of radius $k^{2} r_{n}^{2}$ for a fixed sufficiently large $k$ . Its mass is at least $e^{- n r_{n}^{2}}$ by [F10], so [F1] gives the same type of denominator bound. The joint Bernoulli densities relative to $G \otimes #$ are bounded by one, as required for the density- $L^{2}$ tests in [F2].

2.3A2F3F4F10step 1.1givenalgebra

For the universal fixed-design branch take $r_{n} = ρ_{n}$ . On $∥ w - w_{0} ∥_{\infty} < 2 r_{n}$ and $∣ σ - σ_{0} ∣ < r_{n}$ , [F3] bounds average directional KL and log-square by $K r_{n}^{2}$ for fixed $K$ . Independence and [A2], together with [F10], give joint local prior mass at least $c r_{n} e^{- n r_{n}^{2}}$ . Since $r_{n} \geq \sqrt{\log n / n}$ , eventually $\log (1 / r_{n}) \leq \frac{1}{2} \log n \leq \frac{1}{2} n r_{n}^{2}$ ; hence this mass is at least $e^{- C_{0} n r_{n}^{2}}$ for fixed $C_{0}$ . The independent-observation denominator lemma in [F4] applies.

2.4A3F8F9F10step 1.1givenchoose

For each branch choose $C > 1$ larger than its denominator exponent; eventually $e^{- C n r_{n}^{2}} < 1 / 2$ . Apply the prior sieve theorem [F10] using step 1.1 to get measurable $B_{n}$ with entropy at radius $3 r_{n}$ at most $6 C n r_{n}^{2}$ and tail at most $e^{- C n r_{n}^{2}}$ . The same theorem supplies the local prior mass used in steps 2.1, 2.2, 2.3.

2.5A2step 1.2givenalgebra

On $E_{R}$ integrate the likelihood over $∣ σ - σ_{0} ∣ \leq 1 / \sqrt{n}$ within $[a, b]$ . This interval has length at least $1 / \sqrt{n}$ for large $n$ , even at an endpoint; [A2] and step 1.2 bound its integral below by $c_{R} / \sqrt{n}$ for some $c_{R} > 0$ . On the local neighborhood where $g \leq U$ , substitute $u = \sqrt{n} (σ - σ_{0})$ in the numerator outside $M ϵ_{n}$ . Dividing by that lower bound gives at most $C_{R} \int_{∣ u ∣ > M \sqrt{n} ϵ_{n}} e^{- c u^{2} + C R ∣ u ∣} d u \to 0$ , since $n ϵ_{n}^{2} \to \infty$ . Outside the local neighborhood, step 1.2 bounds $L_{n}$ by $e^{- c^{'} n}$ for large $n$ , whose integral against the entire probability prior is at most $e^{- c^{'} n}$ ; after dividing by the denominator this also tends to zero. First let $n \to \infty$ and then $R \to \infty$ . This proves the sharper degenerate-prior contraction in probability for every fixed $M > 0$ .

3.1F2F3F4F6F10step 2.1step 2.4givenchoose

In the density branch intersect $B_{n}$ with the measurable positive-finite-normalizer set, changing its prior tail by zero. From a radius- $3 r_{n}$ cover, select a valid center for every cover ball meeting that intersection; finite selection doubles the radius to $6 r_{n}$ without increasing the number. The pairwise norm comparison [F3] sends this to a Hellinger cover of radius at most $7 r_{n}$ for large $n$ . Thus any $14 r_{n}$ -separated set has size at most $e^{6 C n r_{n}^{2}}$ . Taking packing radius at least $\max {14, \sqrt{6 C}} r_{n}$ , [F2] gives tests with null and uniform sieve alternative errors at most $e^{- T n r_{n}^{2}}$ for any fixed desired $T > A$ , by increasing the alternative multiple. The denominator and sieve tail now satisfy the posterior split in [F4], proving contraction in probability.

3.2F2F3F4F10step 2.2step 2.4choosealgebra

In binary regression the norm-ball and sieve cover map by [F3] to density- $L^{2}$ balls and covers of constant times $r_{n}$ . Taking packing radius at least twice the cover radius and at least $\sqrt{6 C} r_{n}$ gives packing entropy at most its squared radius times $n$ . By [F2] and [F4], the tests, denominator and sieve tail prove contraction in probability. Since the joint Bernoulli densities satisfy $d_{2} (q_{w}, q_{w_{0}}) = \sqrt{2} ∥ Ψ (w) - f_{0} ∥_{L^{2} (G)}$ , this is the stated binary distance.

3.3F4F10F14step 2.3step 2.4givenchoose

In fixed-design regression take the product sieve $B_{n} \times [a, b]$ . Its prior tail is at most $e^{- C n r_{n}^{2}}$ , and its function component has the radius- $3 r_{n}$ cover from step 2.4. The local fixed-design testing lemma [F14] gives tests outside $∥ w - w_{0} ∥_{n} + ∣ σ - σ_{0} ∣ > M r_{n}$ with errors at most $e^{- T n r_{n}^{2}}$ for any fixed $T$ by choosing fixed $M$ large. Choose $T$ and $C$ above the denominator exponent in step 2.3. The posterior split [F4] proves the universal $ρ_{n}$ contraction in probability.

4.1A3A4F7F13step 3.3givenalgebra

If the Gaussian law is not $δ_{0}$ , [F13] supplies a continuous linear functional with nonzero Gaussian variance. Indeed, otherwise every member of a countable norming family would vanish almost surely, forcing $∥ W ∥ = 0$ almost surely. By [F7] there is a fixed constant $D$ such that $\log (1 / ϵ_{n}) \leq n ϵ_{n}^{2} + D$ . Put $t_{n} = n ϵ_{n}^{2} \to \infty$ ; then $\log n = \log t_{n} + 2 \log (1 / ϵ_{n}) \leq 3 t_{n} + 2 D$ eventually, since $\log t_{n} \leq t_{n}$ . Consequently $ρ_{n} \leq C_{1} ϵ_{n}$ for fixed $C_{1}$ . The universal conclusion in step 3.3 therefore proves the sharper $ϵ_{n}$ conclusion after enlarging $M$ by $C_{1}$ .

5.1step 3.1step 3.2step 3.3step 4.1step 2.5givenalgebra∎

In each proved branch the posterior tail is a random variable $T_{n} \in [0, 1]$ converging to zero in probability. For $η > 0$ , $E T_{n} \leq η + P (T_{n} > η)$ ; first take the limit superior and then $η ↓ 0$ . Thus all asserted expectation limits follow. The universal fixed-design rate requires no upper bound on the scale density; the sharper degenerate alternative uses only the explicitly local upper bound.

Remarks

A scale density bounded below supplies interval mass of order its radius, but the Gaussian concentration condition does not bound the logarithm of that radius when $W = 0$ . An unrestricted scale density can have spikes that make the originally proposed arbitrary $ϵ_{n}$ rate false; A spiky scale prior defeats an arbitrary degenerate-Gaussian concentration rate ↗ gives an explicit counterexample. The universal $ρ_{n}$ rate retains every allowed prior. A nonzero Gaussian direction or a local upper scale-density bound in the degenerate case supplies the sharper alternative by separate proved arguments.

5 · Examples, counterexamples and false statements

None yet.

Gaussian-process posterior contraction

1 · Prerequisites

2 · Summary

Objective

3 · Logical flowchart

4 · Definitions, theorems and proofs

A tight Gaussian law is carried by a separable closed subspace

Statement

Facts & Assumptions

Proof

Source notes

The L² link norm controls binary-regression likelihood distances

Statement

Facts & Assumptions

Proof

Source notes

Empirical mean and scale distance control Gaussian-regression likelihoods

Statement

Facts & Assumptions

Proof

Source notes

Linear and quadratic Gaussian statistics have exponential tails

Statement

Facts & Assumptions

Proof

Local KL and log-square mass bounds a non-IID posterior denominator

Statement

Facts & Assumptions

Proof

Tests and a sieve tail imply posterior contraction from a denominator bound

Statement

Facts & Assumptions

Proof

A nonzero Gaussian direction forces logarithmic small-ball cost

Statement

Facts & Assumptions

Proof

Source notes

Supremum distance controls local log-Gaussian density distances

Statement

Facts & Assumptions

Proof

Source notes

Supremum sieves admit tests in fixed-design Gaussian regression

Statement

Facts & Assumptions

Proof

Source notes

Gaussian-process posterior contraction in density and regression models

Statement

Facts & Assumptions

Proof

Remarks

5 · Examples, counterexamples and false statements

Sources