# Overdrawing Urns using Categories of Signed Probabilities

arXiv: https://arxiv.org/abs/2312.12453 · HTML: https://mlsanity.com/paper/2312.12453/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 24 references that are in the map.
- First posted on arXiv 14 Dec 2023.

## Builds on (most cited first, 5 of 5)

- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [A Channel-Based Perspective on Conjugate Priors](https://mlsanity.com/paper/1707.00269/) (2017)
- [From Multisets over Distributions to Distributions over Multisets](https://mlsanity.com/paper/2105.06908/) (2021)
- [Negative probabilities: What they are and what they are for](https://mlsanity.com/paper/2009.10552/) (2020)

## Abstract (arXiv)

A basic experiment in probability theory is drawing without replacement from an urn filled with multiple balls of different colours. Clearly, it is physically impossible to overdraw, that is, to draw more balls from the urn than it contains. This paper demonstrates that overdrawing does make sense mathematically, once we allow signed distributions with negative probabilities. A new (conservative) extension of the familiar hypergeometric ('draw-and-delete') distribution is introduced that allows draws of arbitrary sizes, including overdraws. The underlying theory makes use of the dual basis functions of the Bernstein polynomials, which play a prominent role in computer graphics. Negative probabilities are treated systematically in the framework of categorical probability and the central role of datastructures such as multisets and monads is emphasised.