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

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A generalized Bayes credible set can have zero coverage

Statement refuted

A positive temperature and a proper generalized Bayes loss update alone guarantee that every posterior $0.95$ -credible set has frequentist coverage at least $0.95$ under the sampling experiment.

Facts & Assumptions

Given: The finite parameter space $Θ = {0, 1}$ , singleton observation space $X = {x_{0}}$ , prior $(1 / 100, 99 / 100)$ , sampling laws $P_{0} = P_{1} = δ_{x_{0}}$ , and the displayed loss at $w = 1$ .

[F1]

A countable discrete measurable space is standard Borel: a nonempty at-most-countable set with its power-set sigma-algebra is standard Borel.

[F2]

Probability measures and probability spaces: a probability measure has total mass one.

[F3]

The Dirac set function at a point: $δ_{x_{0}} (A) = 1$ if $x_{0} \in A$ and is zero otherwise.

[F4]

A Dirac set function is a probability measure: each Dirac set function is a probability measure.

[F5]

Statistical experiments: an experiment consists of a sample measurable space, parameter set and a probability law on the sample space for each parameter.

[F6]

Generalized Bayes loss update and temperature: if $0 < Z = \int e^{- w ℓ_{x}} d π < \infty$ , the update is the normalized measure with that density.

[F7]

A finite-normalizer loss update is a probability measure: each update with a finite positive normalizer is a probability measure.

[F8]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $e^{t} > 0$ and $e^{- t} = 1 / e^{t}$ for real $t$ .

[F9]

The exponential function is strictly increasing: the exponential is strictly increasing on $R$ .

[F10]

Confidence sets and coverage: a confidence-set procedure maps data to parameter subsets with measurable membership sections; coverage at $θ$ is the sampling probability that its set contains $θ$ .

Counterexample

Let $Θ = {0, 1}$ and $X = {x_{0}}$ , each with its power-set sigma-algebra. Use prior $π (0) = 1 / 100$ , $π (1) = 99 / 100$ , and sampling laws $P_{0} = P_{1} = δ_{x_{0}}$ . At the sole observation $x_{0}$ , let $ℓ (0, x_{0}) = 1$ and $ℓ (1, x_{0}) = 0$ , with temperature $w = 1$ . The singleton data-to-set rule $C (x_{0}) = {1}$ is called $0.95$ -credible here when the update $q_{x_{0}}$ satisfies $q_{x_{0}} (C (x_{0})) \geq 0.95$ . It has posterior mass greater than $0.95$ , yet its coverage at true parameter $0$ is zero.

1.1F1F2F3F5F6F8F7F4givenalgebra

The parameter and data spaces are nonempty finite sets, so [F1] makes them standard Borel. The prior masses sum to one, hence [F2] makes $π$ a probability. [F3]–[F5] show the sampling laws define a statistical experiment. [F6]–[F8] give the normalizer $Z = \frac{1}{100} e^{- 1} + \frac{99}{100} e^{0},$ which is finite and positive. By [F7], the resulting update $q$ is a probability measure.

2.1F9F8F6step 1.1algebra

Since $- 1 < 0$ , [F9] gives $e^{- 1} < e^{0}$ . Both are positive by [F8], and $Z = \frac{1}{100} e^{- 1} + \frac{99}{100} e^{0} < e^{0} .$ Therefore $q ({1}) = \frac{(99 / 100) e^{0}}{Z} > \frac{99}{100} > 0.95.$ Thus $C (x_{0}) = {1}$ is a posterior credible set of mass greater than $0.95$ .

Verification

3.1F10step 1.1step 2.1algebra∎

Regard $C$ as the data-to-set procedure with sole value $C (x_{0}) = {1}$ . Its membership section for parameter $0$ is empty, and for parameter $1$ is ${x_{0}}$ ; both are measurable. The coverage definition [F10] therefore applies, and at true parameter $0$ it gives $P_{0} {x : C (x) ∋ 0} = P_{0} (\emptyset) = 0.$ This is below $0.95$ , refuting the proposed universal guarantee.

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Sources

P. G. Bissiri, C. C. Holmes, and S. G. Walker, A general framework for updating belief distributions, JRSS B 78(5), 2016 (standard reference, not scraped)
L. Perrotta, Practical Calibration of the Temperature Parameter in Gibbs Posteriors, arXiv:2004.10522, 2020 (standard reference, not scraped)