On Moebius duality and Coarse-Graining

TL;DR

This paper explores duality relations for zeta and Möbius matrices, focusing on positivity, coarse-graining, and applications in genetics, providing conditions for preserving stochasticity and examples in biological models.

Contribution

It introduces conditions for coarse-graining of zeta and Möbius matrices and demonstrates their applicability in genetic models, extending duality theory.

Findings

Conditions for positivity of dual kernels via Möbius cone

Coarse-graining is possible for sets and partitions

Applications to genetics models like Cannings models

Abstract

We study duality relations for zeta and M\"{o}bius matrices and monotone conditions on the kernels. We focus on the cases of family of sets and partitions. The conditions for positivity of the dual kernels are stated in terms of the positive M\"{o}bius cone of functions, which is described in terms of Sylvester formulae. We study duality under coarse-graining and show that an $h-$transform is needed to preserve stochasticity. We give conditions in order that zeta and M\"{o}bius matrices admit coarse-graining, and we prove they are satisfied for sets and partitions. This is a source of relevant examples in genetics on the haploid and multi-allelic Cannings models.