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A non-eigenfeature can give an unrestricted Gaussian MAP test level one

Statement

Assume the Axiom of Choice. Let $X, H$ be separable real Hilbert spaces, $K : X \to H$ bounded and linear, $ϵ > 0$ , and let $U$ be a Borel Gaussian prior with mean $m_{0}$ and positive self-adjoint, injective, trace-class covariance $C_{0}$ , independent of a jointly Gaussian centered family $Z$ indexed by $H$ , linear into $L^{2}$ with covariance $E [Z (g) Z (k)] = ⟨ g, k ⟩_{H}$ . Put $S = C_{0}^{1 / 2}$ , $B = S K^{*} K S$ , $R = (B + ϵ^{2} I_{X})^{- 1}$ . For a feature vector $h \in X$ , define $Φ_{h} = K S R S h$ , $c_{h} = ⟨ K m_{0}, Φ_{h} ⟩_{H} - ⟨ m_{0}, h ⟩_{X}$ , and the raw test $δ_{h} = 1 {Y_{x} (Φ_{h}) > c_{h}}$ , where $Y_{x} (g) = ⟨ K x, g ⟩_{H} + ϵ Z (g)$ . Suppose $C_{0} K^{*} K = K^{*} K C_{0}$ , $h \neq 0$ , and $B h$ is not a scalar multiple of $h$ . Then the unrestricted null set $H_{0} = {x \in X : ⟨ x, h ⟩_{X} \leq 0}$ satisfies $\sup_{x \in H_{0}} P_{x} {δ_{h} = 1} = 1.$ The supremum need not be attained at any finite parameter value.

Statement references

Gaussian MAP feature test

Level, power, and weighted regularization of Gaussian MAP feature tests

Kretschmann and Werner, Maximum a posteriori testing in statistical inverse problems, Theorem 2.6 and its proof

Facts & Assumptions

Given: The Gaussian inverse model, operators, feature, and test in the Statement, together with the commutation and non-eigenfeature hypotheses.

[F1]

Under the commutation hypothesis, $K^{*} Φ_{h} = B (B + ϵ^{2} I)^{- 1} h$ , and the raw test has rejection probability $G ((⟨ x, K^{*} Φ_{h} ⟩ - c_{h}) / (ϵ ∥ Φ_{h} ∥))$ when $Φ_{h} \neq 0$ . Level, power, and weighted regularization of Gaussian MAP feature tests

[F2]

The shifted normal operator $B + ϵ^{2} I$ is invertible, and its inverse commutes with $B$ . The shifted Hilbert-space normal operator is invertible with inverse norm at most $1 / λ$

[F3]

The adjoint identity is $⟨ K x, Φ ⟩_{H} = ⟨ x, K^{*} Φ ⟩_{X}$ , and inner products satisfy Cauchy–Schwarz. The Hilbert-space adjoint of a bounded operator, Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs

[A1]

The Axiom of Choice is inherited from the posterior and polar-decomposition branches; Countable Choice is used for Hilbert adjoints and shifted-operator inversion. The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )

Proof

1.1givenA1F1F2algebra

Put $A = B (B + ϵ^{2} I)^{- 1}$ . By [F1] and its inherited Axiom of Choice hypothesis [A1], $K^{*} Φ_{h} = A h$ . If $A h = η h$ for a real scalar $η$ , multiply the identity by $B + ϵ^{2} I$ and use [F2] to obtain $B h = η (B + ϵ^{2} I) h$ , hence $(1 - η) B h = η ϵ^{2} h$ . The case $η = 1$ forces $h = 0$ , contrary to the hypothesis; otherwise $B h = η ϵ^{2} (1 - η)^{- 1} h$ . Thus any scalar dependence of $A h$ and $h$ would make $B h$ a scalar multiple of $h$ . The assumed non-eigenfeature condition therefore implies $A h \notin span {h}$ .

2.1givenA1F1F2F3step 1.1constructalgebra

Let $w = A h - \frac{⟨ A h, h ⟩_{X}}{∥ h ∥_{X}^{2}} h$ . Then $w \neq 0$ , $⟨ w, h ⟩_{X} = 0$ , and $⟨ A h, w ⟩_{X} = ∥ w ∥_{X}^{2} > 0$ by orthogonal projection. For $t > 0$ , $x_{t} = t w$ belongs to $H_{0}$ . Moreover $A h \neq 0$ , since otherwise $B h = 0$ by invertibility of $B + ϵ^{2} I$ and $h$ would be an eigenvector with eigenvalue zero; so $K^{*} Φ_{h} = A h$ implies $Φ_{h} \neq 0$ .

3.1givenA1F1F3step 2.1algebra∎

By [F3] and the inherited Axiom of Choice hypothesis [A1], the test statistic at $x_{t}$ has mean $⟨ x_{t}, K^{*} Φ_{h} ⟩_{X} = t ⟨ w, A h ⟩_{X} = t ∥ w ∥_{X}^{2}$ and positive standard deviation $ϵ ∥ Φ_{h} ∥_{H}$ . The exact tail formula [F1] gives $P_{x_{t}} {δ_{h} = 1} = G ⁣ (\frac{t ∥ w ∥_{X}^{2} - c_{h}}{ϵ ∥ Φ_{h} ∥_{H}}) ⟶ 1$ as $t \to \infty$ . Every $x_{t}$ is null, so the supremum over $H_{0}$ is at least this limit and at most one; it is therefore one. Since a nondegenerate normal tail is strictly below one at every finite threshold, the limiting sequence does not assert attainment.

Depends on

Used by

A non-eigenfeature gives Gaussian MAP rejection probabilities tending to one Example

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Sources

Remo Kretschmann and Frank Werner, Maximum a posteriori testing in statistical inverse problems, Theorem 2.6 and its proof (standard reference, not scraped)