TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Equal observation laws force a metric-risk lower bound

Statement

Let $(Q, d_{Q})$ be a metric target space and let $G$ be a sigma-algebra on $Q$ . Let $τ_{i} = (x_{i}, A_{i}, ν_{i})$ , $i \in {0, 1}$ , be two admissible truth/operator/noise specifications on one measurable data space $(Y, Y)$ , with sampling laws $P_{i}$ and targets $q_{i} = q (x_{i})$ . Assume the metric-loss sections $a \mapsto d_{Q} (q_{i}, a)$ are $G$ -measurable. Let $δ$ be one randomized decision rule from $(Y, Y)$ to $(Q, G)$ , used under both specifications, and define its expected metric risk by $R_{i} (δ) = \int_{Y} (\int_{Q} d_{Q} (q_{i}, a) δ (y, d a)) P_{i} (d y) .$ If $P_{0} = P_{1}$ and $Δ = d_{Q} (q_{0}, q_{1})$ , then $\max {R_{0} (δ), R_{1} (δ)} \geq \frac{Δ}{2} .$ Each risk is an extended nonnegative integral; deterministic measurable estimators are included as point-mass decision kernels. This theorem makes no assertion when the observation laws differ, and quantitative robustness requires a separately established stability modulus.

Statement references

Maxim Raginsky, Minimax lower bounds, §1, Lemma 1 and complete proof

John Duchi, Lecture Notes on Statistics and Information Theory, Chapter 8, §§8.1–8.3

Facts & Assumptions

Given: Two admissible specifications $τ_{i} = (x_{i}, A_{i}, ν_{i})$ with sampling laws $P_{i}$ on $(Y, Y)$ , one randomized decision kernel $δ$ from $(Y, Y)$ to $(Q, G)$ , $P_{0} = P_{1} = P$ , and $Δ = d_{Q} (q (x_{0}), q (x_{1}))$ .

[F1]

Operator, noise-law, and prior-support mismatch: each admissible truth/operator/noise specification defines its sampling law as a probability measure on the common data space.

[F2]

Inverse-problem models and solution classes: each admissible truth $x$ has a target $q (x)$ in the metric target space $(Q, d_{Q})$ .

[F3]

Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric: $d_{Q}$ is symmetric and satisfies $d_{Q} (a, c) \leq d_{Q} (a, b) + d_{Q} (b, c)$ for all $a, b, c \in Q$ .

[F4]

Nonnegativity of a metric is a consequence of the other axioms, not an axiom: every metric distance is nonnegative.

[F5]

Loss functions and risk: a randomized decision rule is a probability kernel.

[F6]

Additivity of the nonnegative Lebesgue integral: integrals of sums of nonnegative measurable functions equal the sum of their extended nonnegative integrals.

[F7]

Monotonicity and nonnegative homogeneity of the nonnegative integral: the nonnegative integral is monotone and homogeneous under nonnegative finite scalars.

[F8]

Probability measures and probability spaces: $P (Y) = 1$ .

[F9]

The nonnegative Lebesgue integral The nonnegative integral agrees with the simple integral on simple functions The integral of a nonnegative simple function: for a probability measure $μ$ on a measurable space $(S, S)$ , the constant function $z \mapsto Δ$ is simple and has integral $Δ μ (S) = Δ$ .

[F10]

A randomized procedure has a measurable action law: if $a \mapsto g (a)$ is a nonnegative measurable loss section and $δ$ is a probability kernel, then $y \mapsto \int_{Q} g (a) δ (y, d a)$ is measurable.

[F11]

Loss functions and risk: risk is the outer integral of the kernel-averaged nonnegative measurable loss under the sampling law; a deterministic measurable estimator is included as a point-mass kernel.

Proof

1.1givenF2F3F4F5F6F7F8F9algebra

By [F2], $q_{i} = q (x_{i})$ are points of $Q$ . For every $y \in Y$ and $a \in Q$ , [F3] gives $Δ = d_{Q} (q_{0}, q_{1}) \leq d_{Q} (q_{0}, a) + d_{Q} (a, q_{1})$ . Integrating this pointwise inequality against the probability measure $δ (y, \cdot)$ , using nonnegativity in [F4], monotonicity in [F7], total mass one in [F8], the constant integral identity in [F9], symmetry in [F3] to write $d_{Q} (a, q_{1}) = d_{Q} (q_{1}, a)$ , the probability-kernel property in [F5], and additivity in [F6], yields $Δ \leq \int_{Q} d_{Q} (q_{0}, a) δ (y, d a) + \int_{Q} d_{Q} (q_{1}, a) δ (y, d a)$ . The integrands are measurable by Given and nonnegative by [F4].

2.1givenF1F4F6F7F8F9F10F11step 1.1algebra∎

The inner loss functions of $y$ are measurable by [F10], so we may integrate step 1.1 under the common law $P$ . By [F11], these outer integrals are $R_{0} (δ)$ and $R_{1} (δ)$ . Monotonicity and additivity give $Δ P (Y) \leq R_{0} (δ) + R_{1} (δ)$ . By [F1], $P$ is a sampling law; by [F8] and [F9], the left side is $Δ$ . Each risk is nonnegative by [F4] and [F11]. If $\max {R_{0}, R_{1}} < Δ / 2$ , both risks are finite and their sum is less than $Δ$ , contradicting the preceding inequality. Hence $\max {R_{0} (δ), R_{1} (δ)} \geq Δ / 2$ , including the case of an infinite risk.

Depends on

Used by

Distinct truths and operators with identical noisy data laws Example

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Sources

Maxim Raginsky, Minimax lower bounds, §1, Lemma 1 and complete proof, PDF pp. 3–5 (standard reference, not scraped)
John Duchi, Lecture Notes on Statistics and Information Theory, Chapter 8, §§8.1–8.3 (standard reference, not scraped)