A simple geometric proof for the characterisation of e-merging functions

TL;DR

This paper provides a simple geometric proof for the characterization of e-merging functions, clarifying their properties and showing that non-monotone merging rules are dominated by monotone ones, thus simplifying the theoretical framework.

Contribution

It offers an intuitive geometric proof of the characterization of e-merging functions and demonstrates that monotonicity is inherently implied, removing the need for it as an assumption.

Findings

Geometric proof based on supporting hyperplanes and concave envelopes

Non-monotone merging rules are dominated by monotone ones

Characterization of e-merging functions holds without monotonicity assumption

Abstract

E-values offer a powerful framework for aggregating evidence across different (possibly dependent) statistical experiments. A fundamental question is to identify e-merging functions, namely mappings that merge several e-values into a single valid e-value. A simple and elegant characterisation of this function class was recently obtained by Wang(2025), though via technically involved arguments. This note gives a short and intuitive geometric proof of the same characterisation, based on a supporting hyperplane argument applied to concave envelopes. We also show that the result holds even without imposing monotonicity in the definition of e-merging functions, which was needed for the existing proof. This shows that any non-monotone merging rule is automatically dominated by a monotone one, and hence extending the definition beyond the monotone case brings no additional generality.