ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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A posterior mean can lie outside a nonconvex signal class

Example

Assume the Axiom of Choice. Let the admissible signal set be $S = {- 1, 1} \subset R$ , which is nonconvex because it omits the midpoint $0$ , let the target be the signal itself, and let the data space consist of one point $Y = {0}$ . Use the constant forward map $F (t) = 0$ and the prior assigning probability $1 / 2$ to each signal. The posterior law at the sole datum is therefore $Π_{0} = \frac{1}{2} δ_{- 1} + \frac{1}{2} δ_{1}$ . Under squared loss on the full action space $R$ , the posterior mean is $0$ , which is outside $S$ ; its posterior risk at action $a$ is $1 + a^{2}$ , so its unique unconstrained minimum is $1$ at $a = 0$ . If actions are restricted to $S$ , both admissible actions have risk $2$ . The proposition concerns the full Hilbert action space and does not assert that the posterior mean is an admissible signal.

Example references

Forward map, admissible truths, and target map

Point reconstruction rules and their target space

Squared-error risk under a joint law

The posterior mean uniquely minimizes full-space squared risk

Conditional law given the constant observation

Facts & Assumptions

Given: AC, the two-point signal set, constant observation, equal prior weights, and squared loss.

[A1]

The Axiom of Choice is assumed by the general posterior-mean proposition. The Axiom of Choice

[F1]

A conditional law is a probability kernel satisfying the conditional-event integral identities. Conditional law given a random element

[F2]

In a real Hilbert target with finite second moment, the posterior mean uniquely minimizes posterior squared risk over the full target space. The posterior mean is the unique squared-loss Bayes action in a Hilbert target

[F3]

Reconstruction risk is the expected target-action loss under the specified joint law. Inverse reconstruction risk under train and test laws

Verification

1.1F1given

Since $Y = 0$ almost surely, the constant kernel $K (0, \cdot) = Π_{0}$ satisfies the conditional-law identity both for the empty event and for the whole sample space; hence it is a posterior version for the stated joint law.

1.2givenalgebra

Direct integration gives $μ (0) = \frac{1}{2} (- 1) + \frac{1}{2} (1) = 0$ . Since $S = {- 1, 1}$ , this posterior mean is not an admissible signal, and $S$ is nonconvex because it omits the midpoint of its two elements.

2.1A1F2F3givenalgebra∎

For every $a \in R$ , $\int (t - a)^{2} Π_{0} (d t) = \frac{1}{2} (- 1 - a)^{2} + \frac{1}{2} (1 - a)^{2} = 1 + a^{2}$ . The unique minimizer over $R$ is $0$ ; on the restricted set $S$ , both values $a = \pm 1$ have risk $2$ . This also agrees with the full-space proposition [F2], whose AC hypothesis is stated in the Example, while the restricted-action calculation is separate.

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