On the symmetry of evidential support

TL;DR

This paper proves the symmetry in evidential support between two events, showing that each supports the other if and only if the reverse is true, using seven different proofs and interpretations.

Contribution

It provides seven distinct proofs of the symmetry of evidential support and explores the implications for scalar measures of association.

Findings

Seven proofs of the symmetry property.

The symmetry holds under fixed marginals in a 2x2 table.

Scalar measures of positive association are governed by a single signed parameter.

Abstract

For events $A$ and $B$, we have \[ \mathbb{P}(A\mid B) > \mathbb{P}(A\mid \neg B) \qquad\Longleftrightarrow\qquad \mathbb{P}(B\mid A) > \mathbb{P}(B\mid \neg A) \] whenever all four quantities are defined. In other words, $B$ is evidence for $A$ if and only if $A$ is evidence for $B$. This note gives seven different proofs of this fact -- by cross-multiplication, covariance, coupling parameters, odds ratios, pointwise mutual information, combinatorial double counting, and mixed discrete derivatives -- and develops a surrounding web of interpretations. Once the marginals $\mathbb{P}(A)$ and $\mathbb{P}(B)$ are fixed, a $2\times 2$ table has only one degree of freedom, so every scalar notion of positive association must be governed by the same signed parameter.