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Bernstein–von Mises for Adaptively Collected Data
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Proves a Bernstein–von Mises theorem for adaptively collected data (linear Gaussian, bandits, exponential-family bandits, contextual bandits), showing Bayesian UQ matches Wald-type frequentist UQ, and that it can be asymptotically frequentist-invalid when stability fails.
Problem
Whether Bayesian uncertainty quantification remains asymptotically equivalent to Wald-type frequentist UQ when data are collected adaptively (bandits, RL, adaptive experiments), where standard BvM for i.i.d. or bounded eigenvalue-ratio data fails.
Approach
Extends the Bernstein–von Mises theorem to adaptive sampling by noting the sampling-rule terms do not depend on the parameter and are absorbed into normalization, and by truncating posteriors to anisotropic local ellipsoids so that unbounded eigenvalue ratios (e.g., UCB's log-n suboptimal pulls) are allowed; BvM holds without the stability condition, so Bayesian UQ can match Wald intervals that ar
Key results
- For two-arm Gaussian UCB bandits with horizon 10^4, average BvM TV distance decreases toward zero across five parameter configurations, fastest in the zero-margin (equal means) case.
- For Thompson sampling in a two-batch two-arm Gaussian bandit, the 95% credible interval for the margin matches nominal coverage except near the zero-margin case, where it is not frequentist-valid, consistent with theory.
- For NCEC on LQR, the posterior TV distance to the representative normal decreases over time in determined, stabilizable, and unstabilizable configurations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use equal arm means rather than a positive margin between arms | UCB on Gaussian bandits · Two-arm Gaussian bandits | BvM TV distance | Helped |
| Use an undominated parameter configuration rather than a dominated or duplicate configuration | Lin-UCB · Gaussian linear contextual bandits | BvM TV distance | Helped |
| Use a duplicate-arm configuration rather than an undominated configuration | Lin-UCB · Gaussian linear contextual bandits | BvM TV distance | Hurt |
| Use the unstabilizable configuration rather than the determined or stabilizable configuration | Noisy Certainty Equivalent Control · LQR | BvM TV distance | Helped |
| Apply Thompson sampling at zero margin rather than nonzero margin | Thompson sampling · Two-batch two-arm Gaussian bandit | BvM TV distance | Helped |
| Use Bayesian credible intervals in the two-arm batched setting at zero margin | Thompson sampling · Two-batch two-arm Gaussian bandit | Frequentist coverage | Hurt |
Surprising findings
- Bayesian UQ can be asymptotically equivalent to Wald-type frequentist UQ while both are asymptotically frequentist-invalid (e.g., Lai-Wei non-normal MLE example and Thompson sampling zero-margin case).
- BvM convergence is fastest in the zero-margin bandit case even though that case violates stability.
Limitations
- Convergence rate of BvM may be logarithmically slow for optimal bandit algorithms, so finite-sample behavior with misspecified priors is unreliable.
- Only fixed, nonrandom priors are considered; results do not give an asymptotically frequentist-valid Bayesian procedure.
- Theory covers Gaussian and exponential family models and specific linear settings; general parametric models are not covered.
Experimental setup
Metrics: total variation distance between posterior and representativ empirical credible interval coverage
Baselines: UCB Thompson sampling lin-UCB noisy certainty equivalent control
Builds on: Bernstein–von Mises theorem Lai and Wei stability condition Thompson sampling UCB lin-UCB noisy certainty equivalent control
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