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

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An exact posterior sample need not minimize point risk

Example

Assume the Axiom of Choice. Let $T$ take values $- 1$ and $1$ with equal probabilities, let $Y = 0$ deterministically, and use the squared-error loss on $R$ . The posterior is $Π_{0} = \frac{1}{2} δ_{- 1} + \frac{1}{2} δ_{1}$ . Given $Y = 0$ , draw $\tilde{T}$ independently of $T$ from $Π_{0}$ . Here exact posterior calibration means $L (\tilde{T} ∣ Y = 0) = Π_{0}$ , which holds because the sampler kernel is $S_{0} = Π_{0}$ . The posterior mean action $0$ has posterior risk $1$ , while the expected squared error between the true posterior draw and the independent posterior sample is $2$ . Thus even an exactly calibrated posterior sample is not the Bayes-optimal point action and is not itself the posterior mean.

Example references

Posterior samplers are probability kernels and their draws are random

Risk of a randomized rule under a joint law

The posterior mean uniquely minimizes full-space squared risk

Conditional law and its almost-sure version convention

Facts & Assumptions

Given: AC; the symmetric two-point target law; the deterministic one-point data value; the posterior sampling kernel; and squared-error 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; values at marginal-null data remain version choices. Conditional law given a random element

[F2]

A posterior sampler is a kernel from data to target space, and it is exact when its law equals the posterior kernel. Learned inverse estimators and reconstruction targets

[F3]

The posterior mean uniquely minimizes squared posterior risk over the full real Hilbert target. The posterior mean is the unique squared-loss Bayes action in a Hilbert target

[F4]

A randomized reconstruction rule's risk averages its loss over the joint target-data law and its output kernel. Inverse reconstruction risk under train and test laws

Verification

1.1F1given

Because $Y = 0$ with probability one, the constant kernel $Π_{0} = \frac{1}{2} δ_{- 1} + \frac{1}{2} δ_{1}$ satisfies the conditional-law identity for the empty event and the whole sample space, so it is a posterior version.

1.2F2givenalgebra

The sampler kernel is $S_{0} = Π_{0}$ , hence $L (\tilde{T} ∣ Y = 0) = Π_{0}$ and exact posterior calibration holds by [F2]. Its mean is $\frac{1}{2} (- 1) + \frac{1}{2} (1) = 0$ .

1.3A1F3givenalgebra

Under posterior risk, the action $0$ has loss expectation $\frac{1}{2} (- 1 - 0)^{2} + \frac{1}{2} (1 - 0)^{2} = 1$ . By [F3], whose AC hypothesis is stated in the Example, it is the unique Bayes point action.

1.4F1F4givenalgebra

Conditional independence gives probability $1 / 2$ to equal draws, with squared difference $0$ , and probability $1 / 2$ to unequal draws, with squared difference $4$ . Therefore $E [(T - \tilde{T})^{2} ∣ Y = 0] = \frac{1}{2} (0) + \frac{1}{2} (4) = 2$ , which is the randomized rule risk under [F4].

2.1F3step 1.2step 1.3step 1.4∎

The exact posterior sample has conditional law $Π_{0}$ , but its random squared-error risk $2$ exceeds the posterior mean action's risk $1$ ; therefore posterior calibration of a sample does not make the sample the point Bayes action.

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