# On the Tightness and Computational Tractability of Higher-Dimensional Confidence Sequences

arXiv: https://arxiv.org/abs/2610.03727 · HTML: https://mlsanity.com/paper/2610.03727/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 23 references that are in the map.
- First posted on arXiv 21 Aug 2026.

## Builds on (most cited first, 12 of 12)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Always Valid Inference: Bringing Sequential Analysis to A/B Testing](https://mlsanity.com/paper/1512.04922/) (2015)
- [Time-Uniform Self-Normalized Concentration for Vector-Valued Processes](https://mlsanity.com/paper/2310.09100/) (2023)
- [Peeking with PEAK: Sequential, Nonparametric Composite Hypothesis Tests for Means of Multiple Data Streams](https://mlsanity.com/paper/2402.06122/) (2024)
- [Time-Uniform Confidence Spheres for Means of Random Vectors](https://mlsanity.com/paper/2311.08168/) (2023)
- [Probabilistic Artificial Intelligence](https://mlsanity.com/paper/2502.05244/) (2025)
- [Gambling-Based Confidence Sequences for Bounded Random Vectors](https://mlsanity.com/paper/2402.03683/) (2024)
- [Efficient Post-Hoc Uncertainty Calibration via Variance-Based Smoothing](https://mlsanity.com/paper/2503.15583/) (2025)

## Built on by (most cited first, 1 of 1)

- [When Has a Bayesian Neural Network Sampled Enough? Adaptive Inference Time with Statistical Guarantees](https://mlsanity.com/paper/2610.12212/) (2026)

## Abstract (arXiv)

Modern sequential monitoring problems often involve multiple metrics, where we monitor several data streams simultaneously and may act once the evidence is strong enough. Confidence sequences (CSs) are a natural tool for such continuous monitoring. However, for bounded vector means, existing multivariate CSs are either tight but computationally intractable, or fast to compute but conservative. To address this, we study three lifts of one-dimensional betting-based CSs to higher dimensions: a weighted Bonferroni region, an equivalent max-wealth form, and a portfolio region. The portfolio is typically much tighter, especially in higher dimensions, but its boundary and properties such as volume are not available in closed form. To make this tighter construction usable, we propose tractable outer approximations of the portfolio region that preserve statistical validity: a bounding box, an ℓp-ellipsoid, and their intersection. We prove set relations among all constructions and show empirically that these approximations (i) achieve regions close to the intractable portfolio, (ii) substantially outperform existing tractable multivariate CSs, and (iii) enable practical use cases such as multi-metric A/B testing and model comparison.