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Asymptotically Log-Optimal Bayes-Assisted Confidence Sequences for Bounded Means

NeurIPS 2026 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Regression uncertainty quantification · 0 citations (OpenAlex)

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Proposes Bayes-assisted confidence sequences for bounded means that use Bayesian predictives (mixture Dirichlet-process, BETEL, RETEL) to choose predictable betting coefficients, giving anytime validity for any predictive and asymptotic log-optimality under Wasserstein consistency.

Problem

Confidence sequences for the mean of bounded IID observations are efficient only if the betting martingale coefficients are well chosen, and existing constructions ignore prior information about plausible distributions or means.

Approach

Use a working predictive distribution (Bayesian or pseudo-Bayesian) to choose, for each candidate mean and time, the one-step linear betting factor maximising predictive expected log-growth; validity follows from the bounded-mean martingale and Ville's inequality regardless of predictive quality, and efficiency approaches the oracle when the predictive is Wasserstein-consistent.

Key results

• On LLM best-arm identification, BETEL needed 85.28 average LUCB calls versus 98.48 for the empirical predictive and 106.68 for universal portfolio, stopping earlier than empirical in 35 of 50 paired repetitions.
• On attributedQA PPI test, MDP rejected with 52.79 average labels versus 65.17 for the empirical predictive, winning 65 of 100 paired comparisons.
• Theory: asymptotic log-optimality holds when the predictive is Wasserstein-consistent; the RETEL predictive is shown to be Wasserstein-consistent, and the MDP predictive has expected W1 error O(n^{-1/2}).

Surprising findings

• Universal portfolio requires more LUCB calls on average than the empirical predictive in the LLM experiment.
• MDP predictive is close to the empirical predictive in the LLM best-arm experiment, suggesting its beta-based prior component may be unsuitable there.

Limitations

• Computing the confidence sets is more expensive than closed-form baselines.
• Inverted sets are not guaranteed to be single intervals.
• Efficiency gains depend on the prior or auxiliary predictive being sufficiently informative.

Experimental setup

Datasets: galaxies attributedQA

Benchmarks: galaxies (PPI) attributedQA (PPI) LLM sorting best-arm identification

Metrics: confidence sequence width oracle-normalised width empirical cumulative miscoverage stopping time (LUCB average calls) average rank win/tie/loss counts labels to rejection

Baselines: Empirical predictive (GRAPA-style) Universal portfolio (Cover-Ordentlich) Hedged-Capital PRECiSE-CO96 Parametric beta predictive LUCB Classical labelled-only CS

Builds on: GRAPA Kelly log-growth criterion Ville's inequality BETEL prediction-powered inference

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