Relations among conditional probabilities

TL;DR

This paper develops a mathematical framework using Groebner bases to understand relations among conditional probabilities, connecting algebraic structures to probabilistic concepts and exploring geometric representations.

Contribution

It introduces a Groebner basis for relations among conditional probabilities and links these to geometric objects like generalized permutohedra and simplexes.

Findings

Derived a Groebner basis for conditional probability relations

Connected algebraic relations to geometric structures such as permutohedra

Described a conditional probability simplex for various cases

Abstract

We describe a Groebner basis of relations among conditional probabilities in a discrete probability space, with any set of conditioned-upon events. They may be specialized to the partially-observed random variable case, the purely conditional case, and other special cases. We also investigate the connection to generalized permutohedra and describe a conditional probability simplex.