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Isonormal Gaussian white-noise inverse experiment

Definition

Let $H$ be a separable real Hilbert space, where separable means having a countable dense subset, and let $(Ω, F, P)$ be a probability space. A real isonormal Gaussian process on $H$ is a family $Z (g)$ of real random variables with finite second moments, indexed by $g \in H$ , such that $g \mapsto [Z (g)]$ is linear into $L^{2} (P)$ , $E Z (g) = 0$ , and $E [Z (g) Z (h)] = ⟨ g, h ⟩_{H}$ . It is also required that for every finite orthonormal tuple $(e_{1}, \dots, e_{m})$ the variables $Z (e_{1}), \dots, Z (e_{m})$ are independent and each has density $φ (t) = (2 π)^{- 1 / 2} e^{- t^{2} / 2}$ . For any finite list of vectors, expressing them in an orthonormal basis of their finite-dimensional span shows that their process values have the corresponding linear-image Gaussian law. Here $L^{2} (P)$ identifies square-integrable random variables that agree almost surely. When $H \neq {0}$ , all unit-vector marginals have the same law; write $N (a, σ^{2})$ for the law of $a + σ Z (e)$ for any unit vector $e \in H$ and $σ \geq 0$ .

Let $X$ be a set, let $Θ \subseteq X$ be a nonempty parameter set, let $K : X \to H$ be a signal map, and let $ϵ > 0$ . For $x \in Θ$ and $g \in H$ , define the observation functional

$Y_{x} (g) = ⟨ K x, g ⟩_{H} + ϵ Z (g) .$

The associated statistical inverse experiment has sample space $R^{H}$ with the cylinder sigma-algebra generated by the evaluation maps $y \mapsto y (g)$ , and its sampling law $P_{x}$ is the law of the random function $g \mapsto Y_{x} (g)$ . Each sample is indexed by test functionals on $H$ ; it is not asserted to be a vector in $H$ or to have a continuous path. If $H = {0}$ , linearity forces $Z (0) = 0$ and the experiment is deterministic.

Definition references

A statistical experiment is a measurable sample space with a parameter-indexed family of sampling laws; the parameter set need not be measurable.

Real Hilbert spaces and their inner products

Random elements are measurable maps on a probability space

The L2 equivalence-class convention for square-integrable random variables

Independence of random elements

The law of a measurable random element is a probability measure

Kretschmann and Werner, Maximum a posteriori testing in statistical inverse problems, §1.1, PDF pp. 1–2

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §1.2.2, equations (1.6)–(1.8), PDF pp. 21–22

Depends on

Used by

SOLIT parameter choice on a geometric-variance grid Algorithm
Sharp Gaussian asymptotics under a vanishing largest weight Corollary
Exact linear feature estimability through the adjoint range Definition
Gaussian sequence inverse model Definition
Finite truncation with its omitted white-noise tail Example
Gaussian coefficient sequence Example
The lambda-zero sparse detection boundary Example
Two direct white-noise functionals Example
A one-coordinate projection risk bound Lemma
Risk decomposition and monotone bias for ordered spectral filters Lemma
Gaussian posterior law in an isonormal inverse experiment Theorem
Level, power, and weighted regularization of Gaussian MAP feature tests Theorem
q=2 Gaussian-sequence ellipsoid detection boundary Theorem
SOLIT oracle risk bound and spectral rates Theorem
Sparse q<2 detection regime split Theorem
White-noise spectral sequence representation Theorem

Dependency tree · two levels

21 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

Kretschmann and Werner, Maximum a posteriori testing in statistical inverse problems, §1.1, PDF pp. 1–2 (standard reference, not scraped)
Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §1.2.2, equations (1.6)–(1.8), PDF pp. 21–22 (standard reference, not scraped)