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Exact linear feature estimability through the adjoint range

Definition

Assume Countable Choice. Let $X$ be a real Hilbert space and let $H$ be a separable real Hilbert space. Let $K : X \to H$ be bounded and linear, let $ϵ > 0$ , and let $Z$ be a jointly Gaussian centered family indexed by $H$ , linear into $L^{2}$ with covariance $E [Z (g) Z (k)] = ⟨ g, k ⟩_{H}$ . For each $x \in X$ define $Y_{x} (g) = ⟨ K x, g ⟩_{H} + ϵ Z (g)$ . A continuous linear feature is a map $ϕ \in X^{*}$ , where $X^{*}$ is the continuous dual of $X$ . By the Riesz representation theorem there is a unique $h_{ϕ} \in X$ with $ϕ (x) = ⟨ x, h_{ϕ} ⟩_{X}$ for every $x \in X$ . The feature is exactly linearly estimable by one probe if some $Φ \in H$ makes the statistic $Y_{x} (Φ)$ unbiased for $ϕ (x)$ for every $x \in X$ . Writing $R (K^{*}) = {K^{*} Φ : Φ \in H}$ for the adjoint range, this holds exactly when $h_{ϕ} \in R (K^{*})$ . Every such probe has variance $ϵ^{2} ∥ Φ ∥_{H}^{2}$ . This definition concerns one-probe unbiased linear estimators; it does not identify membership in $\overline{R (K^{*})}$ with exact estimability.

Definition references

Isonormal white-noise inverse experiment

Hilbert-space adjoint

Riesz representation theorem for Hilbert spaces

Kretschmann and Werner, Maximum a posteriori testing in statistical inverse problems, §1.1 and §1.2

Kretschmann, Wachsmuth, and Werner, Optimal regularized hypothesis testing in statistical inverse problems, §1.1

Facts & Assumptions

Given: Real Hilbert spaces $X, H$ , a bounded linear map $K : X \to H$ , an isonormal white-noise experiment with $ϵ > 0$ , and a continuous linear feature $ϕ (x) = ⟨ x, h_{ϕ} ⟩_{X}$ .

[F1]

In the isonormal experiment, $E_{x} Y_{x} (Φ) = ⟨ K x, Φ ⟩_{H}$ and ${Var}_{x} (Y_{x} (Φ)) = ϵ^{2} ∥ Φ ∥_{H}^{2}$ for every fixed $x \in X$ and $Φ \in H$ . Isonormal Gaussian white-noise inverse experiment

[F2]

The adjoint satisfies $⟨ K x, Φ ⟩_{H} = ⟨ x, K^{*} Φ ⟩_{X}$ . The Hilbert-space adjoint of a bounded operator

[F3]

Every continuous linear functional on a real Hilbert space has a unique representing vector $h_{ϕ}$ with $ϕ (x) = ⟨ x, h_{ϕ} ⟩_{X}$ . Riesz representation for Hilbert spaces

[A1]

Countable Choice is assumed by [F2] and [F3]. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Proof

1.1givenA1F1F2F3algebra

Fix a probe $Φ \in H$ . By [F1] and [F2], whose supplier hypotheses are recorded in [A1], its mean at parameter $x$ is $⟨ x, K^{*} Φ ⟩_{X}$ and its variance is finite, equal to $ϵ^{2} ∥ Φ ∥_{H}^{2}$ . If it is unbiased for every $x$ , then $⟨ x, K^{*} Φ - h_{ϕ} ⟩_{X} = 0$ for every $x \in X$ . Taking $x = K^{*} Φ - h_{ϕ}$ gives $K^{*} Φ = h_{ϕ}$ , so $h_{ϕ} \in R (K^{*})$ .

2.1givenA1F1F2F3step 1.1choosealgebra∎

Conversely, if $h_{ϕ} \in R (K^{*})$ , use [A1] to choose $Φ \in H$ with $K^{*} Φ = h_{ϕ}$ . Equations [F1] and [F2] then give $E_{x} Y_{x} (Φ) = ⟨ x, h_{ϕ} ⟩_{X} = ϕ (x)$ for every $x$ , and [F1] gives its finite variance $ϵ^{2} ∥ Φ ∥_{H}^{2}$ . Hence this probe is an exact one-probe estimator, proving both directions and the variance assertion.

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Sources

Remo Kretschmann and Frank Werner, Maximum a posteriori testing in statistical inverse problems, §1.1 and §1.2 (standard reference, not scraped)
Remo Kretschmann, Daniel Wachsmuth, and Frank Werner, Optimal regularized hypothesis testing in statistical inverse problems, §1.1 (standard reference, not scraped)