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

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A nonunique KL projection set need not select one parameter

Example

On $X = {0, 1}$ with counting measure, let $p_{0} (0) = 3 / 5$ , $p_{0} (1) = 2 / 5$ ; $p_{a} (0) = p_{a} (1) = p_{b} (0) = p_{b} (1) = 1 / 2$ ; and $p_{c} (0) = 1 / 5$ , $p_{c} (1) = 4 / 5$ . Give prior probabilities $Π (a) = Π (b) = 1 / 4$ and $Π (c) = 1 / 2$ . On the labels ${a, b, c}$ use $ρ (u, v) = 0$ for $u = v$ and $1$ otherwise. Take $X_{1}, X_{2}, \dots$ IID with law $P_{0}$ .

Facts & Assumptions

Given: The finite model, prior, and IID sampling law in the Example.

[F1]

KL projection sets and relative KL neighborhoods: the KL projection is the full set of parameters minimizing $K L (P_{0} ∥ P_{θ})$ .

[A1]

Assume AC through the imported KL convention The Axiom of Choice. No additional choice is used in this finite model.

[F2]

Kolmogorov iid l1 strong law: integrable IID real variables have sample means converging almost surely to their expectation.

Verification

1.1A1F1algebra

Since $p_{a} = p_{b}$ , their KL losses are equal. Direct calculation gives $K (c) - K (a) = \frac{3}{5} \log \frac{1 / 2}{1 / 5} + \frac{2}{5} \log \frac{1 / 2}{4 / 5} = \frac{3}{5} \log \frac{5}{2} + \frac{2}{5} \log \frac{5}{8} = \frac{1}{5} \log \frac{3125}{512} > 0.$ Thus the minimizer set is $Θ^{*} = {a, b}$ .

2.1F1F2algebra∎

The likelihoods at $a$ and $b$ agree for every data vector, so their posterior odds remain $Π (a) / Π (b) = 1$ . The posterior odds of $c$ against $a$ have log equal to $\log (Π (c) / Π (a)) + \sum_{i = 1}^{n} \log (p_{c} (X_{i}) / p_{a} (X_{i}))$ . By [F2], the sum divided by $n$ tends almost surely to $- (K (c) - K (a)) < 0$ . Hence the posterior mass of $c$ , the complement of $Θ^{*}$ , tends to zero while the posterior never selects one of $a, b$ over the other.

Depends on

Used by

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Sources

B. J. K. Kleijn and A. W. van der Vaart, The Bernstein-Von-Mises Theorem under Misspecification, Corollary 4.2, p. 379 (standard reference, not scraped)