Foundations of Inference

TL;DR

This paper develops a unified, minimal-symmetry foundation for finite inference, deriving probability calculus, divergence, entropy, and information from lattice-based logical statement quantification.

Contribution

It introduces a lattice-based approach that unifies and extends Kolmogorov and Cox's methods, deriving key inference measures with minimal assumptions.

Findings

Derivation of probability calculus from lattice symmetries

Unique quantification rules for divergence, entropy, and information

Minimal assumptions without negation, continuity, or differentiability

Abstract

We present a simple and clear foundation for finite inference that unites and significantly extends the approaches of Kolmogorov and Cox. Our approach is based on quantifying lattices of logical statements in a way that satisfies general lattice symmetries. With other applications such as measure theory in mind, our derivations assume minimal symmetries, relying on neither negation nor continuity nor differentiability. Each relevant symmetry corresponds to an axiom of quantification, and these axioms are used to derive a unique set of quantifying rules that form the familiar probability calculus. We also derive a unique quantification of divergence, entropy and information.