The Polytope of Probability Functions on a Finite Poset

TL;DR

This paper explores the structure of probability functions on finite posets by constructing a related poset and a polytope that captures all such functions, revealing geometric properties and limitations.

Contribution

It introduces the probability functions poset and the probability functions polytope, linking them to order polytopes and analyzing their geometric features.

Findings

The probability functions polytope can be realized as an intersection of an order polytope with an affine space.

Vertices of the probability functions polytope are partially characterized.

The polytope is not always a lattice polytope, unlike the order polytope.

Abstract

Kim, Kim, and Neggers (2019) defined probability functions on a poset, by listing some very natural conditions that a function \(\pi: P \times P \to [0,1]\) should satisfy in order to capture the intuition of "the likelihood that \(a\) precedes \(b\) in \(P\)". In particular, this generalizes the common notion of poset probability for finite posets, where \(\pi(a,b)\) is the proportion of linear extensions of \(P\) in which \(a\) precedes \(b\). They constructed a family of such functions for posets embedded in the ordered plane; that is two say, for posets of order dimension at most two. We study probability functions of a finite poset \(P\) by constructing an ancillary poset \(\tilde{P}\), that we call *probability functions posets*. The relations of this new poset encodes the restrictions imposed on probability functions of the original poset by the conditions of the definition. Then, we define the probability functions polytope, which parameterizes the probability functions on \(P\), and show that it can be realized as the order polytope of \(\tilde{P}\) intersected by a certain affine subspace. We give a partial description of the vertices of probability functions polytope and show that, in contrast to the order polytope, it is not always a lattice polytope.