# A Simple Algorithm for Exact Multinomial Tests

arXiv: https://arxiv.org/abs/2008.12682 · HTML: https://mlsanity.com/paper/2008.12682/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Aug 2020.

## Built on by (most cited first, 3 of 3)

- [Selecting a number of voters for a voting ensemble](https://mlsanity.com/paper/2104.11833/) (2021)
- [Geometry of the Minimum Volume Confidence Sets](https://mlsanity.com/paper/2202.08180/) (2022)
- [Exact Minimum-Volume Confidence Set Intersection for Multinomial Outcomes](https://mlsanity.com/paper/2601.18145/) (2026)

## Abstract (arXiv)

This work proposes a new method for computing acceptance regions of exact multinomial tests. From this an algorithm is derived, which finds exact p-values for tests of simple multinomial hypotheses. Using concepts from discrete convex analysis, the method is proven to be exact for various popular test statistics, including Pearson's chi-square and the log-likelihood ratio. The proposed algorithm improves greatly on the naive approach using full enumeration of the sample space. However, its use is limited to multinomial distributions with a small number of categories, as the runtime grows exponentially in the number of possible outcomes. The method is applied in a simulation study and uses of multinomial tests in forecast evaluation are outlined. Additionally, properties of a test statistic using probability ordering, referred to as the "exact multinomial test" by some authors, are investigated and discussed. The algorithm is implemented in the accompanying R package ExactMultinom.