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

arXiv: https://arxiv.org/abs/2605.07964 · HTML: https://mlsanity.com/paper/2605.07964/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 71 references that are in the map.
- First posted on arXiv 8 May 2026.

## Builds on (most cited first, 25 of 25)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Prediction-Powered Inference](https://mlsanity.com/paper/2301.09633/) (2023)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Attributed Question Answering: Evaluation and Modeling for Attributed Large Language Models](https://mlsanity.com/paper/2212.08037/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Mixture Martingales Revisited with Applications to Sequential Tests and Confidence Intervals](https://mlsanity.com/paper/1811.11419/) (2018)
- [Hypothesis testing with e-values](https://mlsanity.com/paper/2410.23614/) (2024)
- [Tight Concentrations and Confidence Sequences from the Regret of Universal Portfolio](https://mlsanity.com/paper/2110.14099/) (2021)
- [Time-uniform central limit theory and asymptotic confidence sequences](https://mlsanity.com/paper/2103.06476/) (2021)
- [Bayes-optimal prediction with frequentist coverage control](https://mlsanity.com/paper/2105.14045/) (2021)
- [Nonparametric learning from Bayesian models with randomized objective functions](https://mlsanity.com/paper/1806.11544/) (2018)
- [Prediction-Powered E-Values](https://mlsanity.com/paper/2502.04294/) (2025)
- [On the near-optimality of betting confidence sets for bounded means](https://mlsanity.com/paper/2310.01547/) (2023)
- [On the optimality of coin-betting for mean estimation](https://mlsanity.com/paper/2412.02640/) (2024)
- [FAB-PPI: Frequentist, Assisted by Bayes, Prediction-Powered Inference](https://mlsanity.com/paper/2502.02363/) (2025)
- [Anytime-valid, Bayes-assisted, Prediction-Powered Inference](https://mlsanity.com/paper/2505.18000/) (2025)
- [Adaptive multigroup confidence intervals with constant coverage](https://mlsanity.com/paper/1612.08287/) (2016)
- [Regularized Exponentially Tilted Empirical Likelihood for Bayesian Inference](https://mlsanity.com/paper/2312.17015/) (2023)
- [Bayes-assisted Confidence Regions: Focal Point Estimator and Bounded-influence Priors](https://mlsanity.com/paper/2410.20169/) (2024)
- [Exact adaptive confidence intervals for linear regression coefficients](https://mlsanity.com/paper/1705.08331/) (2017)
- [Confidence sequences with informative, bounded-influence priors](https://mlsanity.com/paper/2506.22925/) (2025)
- [Closed-form empirical Bernstein confidence sequences for scalars and matrices](https://mlsanity.com/paper/2512.21300/) (2025)

## Abstract (arXiv)

Confidence sequences based on test martingales provide time-uniform uncertainty quantification for the mean of bounded IID observations without parametric distributional assumptions. Their practical efficiency, however, depends strongly on the choice of martingale updates, and many existing constructions do not exploit prior information about plausible data-generating distributions or mean values. We propose a Bayes-assisted framework that uses a Bayesian working predictive model to adaptively construct confidence sequences. For each candidate mean and time point, the predictive distribution selects, among valid one-step martingale factors, the update maximising predictive expected log-growth; validity is therefore preserved even when the prior or working model is misspecified. We prove that if the predictive distribution is Wasserstein-consistent, the resulting procedure is asymptotically log-optimal, matching the per-sample log-growth of an oracle procedure with access to the true distribution. We instantiate the framework using robust predictives based on Dirichlet-process mixtures and Bayesian exponentially tilted empirical likelihood. Experiments on synthetic data, sequential best-arm identification for LLM evaluation, and prediction-powered inference show that informative priors can substantially reduce confidence-sequence width and sampling effort while retaining anytime-valid coverage.