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

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Expected information gain under a wrong operator does not identify the operator-parameter pair

Statement refuted

The following implication is false: in every finite dominated inverse experiment, if expected information gain for a parameter is positive under a fixed nominal forward operator, then an unknown truth-operator pair is identifiable from the observation law, where identifiable means that distinct pairs induce distinct conditional observation laws.

Statement references

MacKay, §6.1 and Conclusion, model-correctness caveat

Chaloner and Verdinelli, §§1.2 and 2.2

Example

Take one scalar observation $Y = a θ + ϵ$ on $Y = {1, 2, 3}$ with counting measure and independent noise $ϵ$ uniform on ${0, 1}$ . In the actual model the unknown pair $H = (θ, a)$ is uniform on ${(1, 1), (2, \frac{1}{2})}$ . In the nominal model the operator is fixed at $a = 1$ and $θ$ is uniform on ${1, 2}$ . Under the actual model both pairs give the same law, uniform on ${1, 2}$ ; under the nominal model the expected information gain for $θ$ is $\frac{1}{2} \log 2 > 0$ .

Example references

A finite parameter space, counting observation measure, proper prior, and dominated likelihood define an indexed experiment.

EIG averages posterior-to-prior KL over the prior-predictive law.

MacKay, §6.1, misspecified-model example

Facts & Assumptions

Given: AC; the finite actual and nominal experiments in the Counterexample; the parameter priors and the counting measure on $Y$ .

[A1]

AC is inherited from the library KL convention. The Axiom of Choice

[F1]

Expected information gain equals $D (P_{θ, Y} ∥ P_{θ} \otimes P_{Y})$ for the specified dominated experiment. Expected information gain is joint relative entropy

[F2]

The expected information gain is the prior-predictive integral of posterior-to-prior KL divergence. Expected information gain

[F3]

KL divergence is nonnegative and vanishes exactly when the two probability measures agree. Kullback–Leibler divergence is nonnegative

[F4]

The natural logarithm is strictly increasing and $\log 1 = 0$ . Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm

[F5]

On a finite probability space, $D (P ∥ Q) = \sum_{x} P ({x}) \log (P ({x}) / Q ({x}))$ , with zero-mass terms equal to zero. Kullback–Leibler divergence

Verification

Given: The finite models in the Counterexample.

1.1A1F1F3givenalgebra

In the actual model, $a θ = 1$ for both possible pairs. Hence $P (Y \in {1, 2} ∣ H = (1, 1)) = P (Y \in {1, 2} ∣ H = (2, \frac{1}{2})) = 1$ , and both conditional laws assign probability $\frac{1}{2}$ to each of $1$ and $2$ . Thus $H$ and $Y$ are independent, so [F1] and [F3] give actual-pair EIG $D (P_{H, Y} ∥ P_{H} \otimes P_{Y}) = 0$ . The two distinct pairs have identical observation laws and are not identifiable.

1.2givenalgebra

In the nominal model, $θ = 1$ gives $Y \in {1, 2}$ equiprobably and $θ = 2$ gives $Y \in {2, 3}$ equiprobably. Therefore the predictive probabilities are $P (Y = 1) = \frac{1}{4}$ , $P (Y = 2) = \frac{1}{2}$ , and $P (Y = 3) = \frac{1}{4}$ .

2.1A1F2F4F5step 1.2algebra

The joint probabilities are $P (θ = 1, Y = 1) = \frac{1}{4}$ , $P (θ = 1, Y = 2) = \frac{1}{4}$ , $P (θ = 2, Y = 2) = \frac{1}{4}$ , and $P (θ = 2, Y = 3) = \frac{1}{4}$ . Dividing by the marginals in step 1.2 gives posterior $δ_{1}$ at $Y = 1$ , the original uniform prior at $Y = 2$ , and $δ_{2}$ at $Y = 3$ . By [F5], the corresponding KL values are $\log 2$ , $0$ , and $\log 2$ . Thus [F2] gives nominal EIG $\frac{1}{4} \log 2 + \frac{1}{2} \cdot 0 + \frac{1}{4} \log 2 = \frac{1}{2} \log 2 > 0$ by [F4].

3.1F1F3step 1.1step 2.1algebra∎

By [F1] and [F3] as applied in step 1.1, the actual-pair EIG is zero although the pairs are distinct and observationally identical; by step 2.1 the nominal fixed-operator EIG is positive. Thus the claimed implication fails.

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Sources

David J. C. MacKay, Information-Based Objective Functions for Active Data Selection, Neural Computation 4(4) (1992), 590–604 (standard reference, not scraped)
Kathryn Chaloner and Isabella Verdinelli, Bayesian Experimental Design: A Review, Statistical Science 10(3) (1995), 273–304 (standard reference, not scraped)