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A finite misspecified model with a unique KL projection

Example

On $X = {0, 1}$ with counting measure, let the true density be $p_{0} (0) = 3 / 5$ , $p_{0} (1) = 2 / 5$ , and let the model densities be $p_{a} (0) = p_{a} (1) = 1 / 2$ and $p_{b} (0) = 1 / 5$ , $p_{b} (1) = 4 / 5$ . Give $a$ and $b$ prior probability $1 / 2$ each and use the metric $ρ (u, v) = 0$ 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 set consists of the minimizers of the directional contrast $K (θ) = K L (P_{0} ∥ P_{θ})$ .

[A1]

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

[F2]

Likelihood-tilted measures at a KL projection: for a selected minimizer $a$ , the likelihood-tilt mass at $b$ is $E_{P_{0}} [p_{b} (X) / p_{a} (X)]$ and need not equal one.

[F4]

Likelihood tilts are sigma-finite measures: the likelihood tilt is a measure; its total mass is the integral of its density.

[F3]

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

Verification

1.1A1F1algebra

The relative KL contrast is $K (b) - K (a) = E_{P_{0}} \log \frac{p_{a} (X)}{p_{b} (X)} = \frac{2}{5} \log \frac{5}{8} + \frac{3}{5} \log \frac{5}{2} = \frac{1}{5} \log \frac{3125}{512} > 0.$ Thus $a$ is the unique KL minimizer.

1.2F1F3algebra

The posterior odds are $\frac{Π (b ∣ X_{1 : n})}{Π (a ∣ X_{1 : n})} = \prod_{i = 1}^{n} \frac{p_{b} (X_{i})}{p_{a} (X_{i})} .$ The log factors are bounded and IID, with expectation $- (K (b) - K (a)) < 0$ . By [F3], their sample mean converges almost surely to that negative number, so the odds tend to zero and $Π (b ∣ X_{1 : n}) \to 0$ .

2.1F2F4algebra∎

The tilted measure has total mass $Q_{b} (X) = E_{P_{0}} \frac{p_{b} (X)}{p_{a} (X)} = \frac{2}{5} \cdot \frac{8}{5} + \frac{3}{5} \cdot \frac{2}{5} = \frac{22}{25} < 1.$ This example has a unique KL projection and a finite likelihood tilt that is not a probability law.

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, §4.1 (standard reference, not scraped)