ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Asymmetric loss can change the posterior-optimal action

Example

Let $Θ = A = {0, 1}$ and let $Π ({1}) = 0.7$ . Under zero–one loss the maximal-mass action is $1$ . Under the asymmetric loss with false-positive cost $3$ and false-negative cost $1$ , the unique Bayes action is instead $0$ .

Facts & Assumptions

Given: The posterior masses are $Π ({1}) = 0.7$ and $Π ({0}) = 0.3$ ; the actions are $0, 1$ ; false-positive loss is $L (0, 1) = 3$ , false-negative loss is $L (1, 0) = 1$ , and correct classifications have loss zero.

[F1]

Posterior risk is the posterior integral of the loss when finite. Finite posterior risk and Bayes action rule

[F2]

Under finite-class zero–one loss, a maximal-posterior-mass class minimizes posterior risk. A maximal-mass action is Bayes under finite-class zero–one loss

Verification

Given: The hypotheses in the Example and Facts & Assumptions.

1.1givenF1F2algebra

Under zero–one loss, the risks are $1 - Π ({0}) = 0.7$ for action $0$ and $1 - Π ({1}) = 0.3$ for action $1$ . Thus action $1$ is the unique zero–one Bayes action, in agreement with [F2].

1.2givenF1algebra

Under the asymmetric loss, the risk of action $1$ is $3 Π ({0}) = 0.9$ , while the risk of action $0$ is $Π ({1}) = 0.7$ . Both risks are finite and the lower one is attained uniquely by action $0$ .

2.1step 1.1step 1.2algebra∎

The two loss functions therefore select different actions under the same posterior: zero–one loss selects $1$ , while the stated asymmetric loss selects $0$ . This demonstrates that posterior modes alone do not determine a Bayes action without a specified loss.

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Sources

Andrew Gelman et al., Bayesian Data Analysis, 3rd ed., §9.1 (standard reference, not scraped)
Ryan Martin, Lecture Notes on Advanced Statistical Theory, §5.3.1, PDF pp. 104–105 (standard reference, not scraped)