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Gaussian concentration-function sieve bounds

Statement

Assume AC and countable choice. Let $μ$ be a centered tight Borel Gaussian law on a real separable Banach space $B$ , with covariance RKHS $H$ , and let $w_{0} \in {\overline{H}}^{B}$ . Suppose $n \geq 1$ , $ε > 0$ , $C > 1$ , $e^{- C n ε^{2}} < 1 / 2$ , and $ϕ_{w_{0}} (ε) \leq n ε^{2}$ , where $ϕ_{w_{0}}$ is the concentration function. Then there is a measurable set $B_{n} \subseteq B$ such that $\log N (3 ε, B_{n}, ∥ \cdot ∥_{B}) \leq 6 C n ε^{2}, μ (B ∖ B_{n}) \leq e^{- C n ε^{2}},$ and $μ {∥ W - w_{0} ∥_{B} < 2 ε} \geq e^{- n ε^{2}} .$ These are prior-mass and sieve bounds only; they assert no posterior contraction without separate statistical-distance and Kullback–Leibler comparisons.

Facts & Assumptions

[A1]

AC and countable choice are inherited from the Gaussian translation and Banach expansion suppliers (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2]

Under AC, the real dominated-extension theorem supplies the Hahn–Banach principle HB used by the dual-norming interface (Hahn-Banach dominated extension theorem for real vector spaces, The real dominated-extension principle as an additional hypothesis over ZF).

[F1]

The concentration function satisfies $ϕ_{w_{0}} (ε) = \inf_{∥ h - w_{0} ∥_{B} < ε} {∥ h ∥_{H}^{2} - \log μ (ε B^{\circ})}$ (Concentration function of a Banach Gaussian prior).

[F2]

For $h \in H$ , the law of $W + h$ has density $\exp (z_{h} - ∥ h ∥_{H}^{2} / 2)$ with respect to $μ$ (Cameron–Martin translation formula for a separable Banach Gaussian).

[F3]

The Gaussian expansion inequality gives $μ_{*} (A + r H_{1}) \geq Φ (Φ^{- 1} (μ (A)) + r)$ for Borel $A$ (Borell–Sudakov–Tsirelson expansion for a Banach Gaussian law).

[F4]

$H_{1}$ is compact in the Banach norm (The Cameron–Martin unit ball is compact in the Banach norm).

[F5]

A standard normal has density $(2 π)^{- 1 / 2} e^{- z^{2} / 2}$ (Standard normal and normal laws).

[F6]

A Borel subset of a Polish space is inner regular, so its inner measure equals its probability (Assuming countable choice, Borel probability measures on Polish spaces are inner regular).

[F7]

Balls with rational radii and centers in a countable dense subset form a countable base for $B$ ; together with the countable base of $R$ , this makes $B \times R$ second countable and its Borel sigma-algebra the product Borel sigma-algebra (Separability: the existence of an at most countable dense subset, The rationals embed densely in the reals, $Q$ is countably infinite, A product of two at most countable sets is at most countable, Borel sigma-algebra of a product of second-countable spaces). The Borel version of $z_{h}$ from the Cameron–Martin theorem therefore gives a Borel joint law for $(W, z_{h})$ . Appending $z_{h}$ to a countable norming family for $B$ gives a norming family for $B \times R$ , whose finite centered Gaussian coordinate laws are invariant under simultaneous negation (Cameron–Martin translation formula for a separable Banach Gaussian, A separable Banach space has a countable norming family, A countable norming family determines Borel laws, Multivariate normal law, including singular covariance).

[F8]

$N (δ, A, ∥ \cdot ∥_{B})$ is the least number of closed radius- $δ$ balls centered in $A$ that cover $A$ , and metric entropy is its logarithm (Covering, packing, bracketing numbers, and metric entropy).

Proof

Given: $μ, B, H, w_{0}, n, ε, C$ as in the Statement. Set $t = n ε^{2}$ and $p = μ (ε B^{\circ})$ .

1.1A1A2F1F2F7givenalgebra

The hypothesis [F1] implies $p \geq e^{- t} > 0$ , because every candidate $h$ has $∥ h ∥_{H}^{2} \geq 0$ . Fix $h \in H$ . Take a countable norming family for $B$ and append the scalar coordinate $z_{h}$ ; this is a countable norming family for $B \times R$ with the maximum product norm. By [F7], each finite coordinate vector is centered Gaussian and hence invariant under negation, so countable law determination makes the joint law symmetric under $(x, z) \mapsto (- x, - z)$ . Thus, since $ε B^{\circ}$ is symmetric, $\int_{ε B^{\circ}} e^{- z_{h} (x)} d μ (x) = \int_{ε B^{\circ}} \cosh (z_{h} (x)) d μ (x) \geq p .$ The Cameron–Martin identity [F2] for $- h$ now gives $μ (h + ε B^{\circ}) \geq e^{- ∥ h ∥_{H}^{2} / 2} p$ .

2.1F1step 1.1choosealgebra

For each $δ > 0$ , choose $h \in H$ with $∥ h - w_{0} ∥_{B} < ε$ and $∥ h ∥_{H}^{2} - \log p < ϕ_{w_{0}} (ε) + δ \leq t + δ$ . Since $∥ h ∥_{H}^{2} / 2 - \log p \leq ∥ h ∥_{H}^{2} - \log p < t + δ$ , and $h + ε B^{\circ} \subset w_{0} + 2 ε B^{\circ}$ , step 1.1 gives $μ {∥ W - w_{0} ∥_{B} < 2 ε} \geq e^{- t - δ}$ . Letting $δ ↓ 0$ proves the small-ball lower bound.

2.2F3F6step 1.1algebra

Let $b = Φ^{- 1} (e^{- C t}) < 0$ and $M = - 2 b > 0$ , and set $B_{n} = ε B^{\circ} + M H_{1}$ . This is open, hence measurable. Since $p \geq e^{- t}$ and $C > 1$ , monotonicity of $Φ^{- 1}$ gives $Φ^{- 1} (p) > b$ . By [F3], $μ (B_{n}) \geq Φ (Φ^{- 1} (p) + M) > Φ (- b) = 1 - e^{- C t} .$ The set $B_{n}$ is open and therefore Borel, so [F6] identifies its inner measure with $μ (B_{n})$ . Thus $μ (B ∖ B_{n}) \leq e^{- C t}$ .

2.3F1F2F4F8step 1.1construct

Choose a maximal finite set ${h_{1}, \dots, h_{N}} \subset M H_{1}$ with pairwise distances greater than $2 ε$ ; such a set exists and covers $M H_{1}$ by closed $2 ε$ -balls because $M H_{1}$ is compact by [F4]. The open balls $h_{i} + ε B^{\circ}$ are disjoint. Step 1.1 and $∥ h_{i} ∥_{H} \leq M$ imply each has $μ$ -measure at least $e^{- M^{2} / 2} p$ . Hence $N e^{- M^{2} / 2} p \leq 1$ and $\log N \leq M^{2} / 2 - \log p \leq M^{2} / 2 + t$ . Every point of $B_{n}$ is $x + h$ with $x \in ε B^{\circ}$ and $h \in M H_{1}$ , so it lies within $3 ε$ of some $h_{i}$ . Also $h_{i} = 0 + h_{i} \in B_{n}$ because $0 \in ε B^{\circ}$ . Thus these are admissible centers in the covering-number convention [F8], and $N (3 ε, B_{n}, ∥ \cdot ∥_{B}) \leq N$ .

3.1F5step 2.1step 2.2step 2.3algebra∎

Completing the square in the standard normal density [F5] gives $E e^{- λ Z} = e^{λ^{2} / 2}$ . For $x \geq 0$ , on ${Z \leq - x}$ one has $e^{- x Z} \geq e^{x^{2}}$ , so $Φ (- x) = P (Z \leq - x) \leq e^{- x^{2}} E e^{- x Z} = e^{- x^{2} / 2}$ . With $x = \sqrt{2 C t}$ this gives $Φ (- \sqrt{2 C t}) \leq e^{- C t} = Φ (b)$ , hence $- \sqrt{2 C t} \leq b < 0$ and $b^{2} \leq 2 C t$ . Consequently $\log N (3 ε, B_{n}, ∥ \cdot ∥_{B}) \leq 2 b^{2} + t \leq 4 C t + t \leq 5 C t \leq 6 C t$ , using $C > 1$ . Step 2.2 supplies the complement bound and step 2.1 the small-ball bound. No posterior statement follows from these three prior inequalities alone.

Depends on

Used by

Gaussian-process posterior contraction in density and regression models Theorem

Dependency tree · two levels

112 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 (2008) (standard reference, not scraped)
C. Borell, The Brunn–Minkowski inequality in Gauss space (1975) (standard reference, not scraped)