# Exact Minimum-Volume Confidence Set Intersection for Multinomial Outcomes

arXiv: https://arxiv.org/abs/2601.18145 · HTML: https://mlsanity.com/paper/2601.18145/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 6 of its 14 references that are in the map.
- First posted on arXiv 26 Jan 2026.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 6 of 6)

- [lil' UCB : An Optimal Exploration Algorithm for Multi-Armed Bandits](https://mlsanity.com/paper/1312.7308/) (2013)
- [On Finding the Largest Mean Among Many](https://mlsanity.com/paper/1306.3917/) (2013)
- [Optimal Confidence Regions for the Multinomial Parameter](https://mlsanity.com/paper/2002.01044/) (2020)
- [A Simple Algorithm for Exact Multinomial Tests](https://mlsanity.com/paper/2008.12682/) (2020)
- [Contamination Estimation via Convex Relaxations](https://mlsanity.com/paper/1506.04257/) (2015)
- [Geometry of the Minimum Volume Confidence Sets](https://mlsanity.com/paper/2202.08180/) (2022)

## Abstract (arXiv)

Computation of confidence sets is central to data science and machine learning, serving as the workhorse of A/B testing and underpinning the operation and analysis of reinforcement learning algorithms. Among all valid confidence sets for the multinomial parameter, minimum-volume confidence sets (MVCs) are optimal in that they minimize average volume, but they are defined as level sets of an exact p-value that is discontinuous and difficult to compute. Rather than attempting to characterize the geometry of MVCs directly, this paper studies a practically motivated decision problem: given two observed multinomial outcomes, can one certify whether their MVCs intersect? We present a certified, tolerance-aware algorithm for this intersection problem. The method exploits the fact that likelihood ordering induces halfspace constraints in log-odds coordinates, enabling adaptive geometric partitioning of parameter space and computable lower and upper bounds on p-values over each cell. For three categories, this yields an efficient and provably sound algorithm that either certifies intersection, certifies disjointness, or returns an indeterminate result when the decision lies within a prescribed margin. We further show how the approach extends to higher dimensions. The results demonstrate that, despite their irregular geometry, MVCs admit reliable certified decision procedures for core tasks in A/B testing.