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On the symmetry of evidential support

Grant Molnar

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.

On the symmetry of evidential support

Abstract

For events and , we have whenever all four quantities are defined. In other words, is evidence for if and only if is evidence for . 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 and are fixed, a table has only one degree of freedom, so every scalar notion of positive association must be governed by the same signed parameter.

Paper Structure

This paper contains 43 sections, 6 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. The Algebraic Proof
  3. Perspective
  4. Proof
  5. The determinant and rank one
  6. Signed area and geometry
  7. The Covariance Proof
  8. Perspective
  9. Proof
  10. Regression slope
  11. Walsh--Fourier expansion
  12. Hilbert-space and canonical-correlation perspective
  13. The Coupling-Parameter Proof
  14. Perspective
  15. Proof
  16. ...and 28 more sections

Key Result

Theorem 1.1

The following are equivalent. Replacing every "$>$" with "$=$" or with "$<$" gives two further equivalences, so the three possible signs of $\Delta$ classify the three possible qualitative regimes of binary association. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (28)

  • Theorem 1.1: Symmetry of evidential support
  • Remark 1: Two binary encodings
  • Example 1: Medical testing
  • Example 2: Correlation and prediction
  • Example 3: Paired performance tests
  • proof : Proof of \ref{['thmmain']}, algebraic version
  • Proposition 1
  • proof
  • Remark 2: Algebraic statistics
  • Proposition 2
  • ...and 18 more