Hyper Normalisation and Conditioning for Discrete Probability Distributions

TL;DR

This paper introduces a novel, mathematically well-behaved total function for normalisation in probability theory, enabling better reasoning about distributions and conditioning, with applications to information flow refinement.

Contribution

It presents a new hyper normalisation operation as a total function, improving the theoretical understanding and reasoning about probability normalisation and conditioning.

Findings

Hyper normalisation is a total function producing distributions of distributions.

The approach simplifies reasoning about normalisation and conditioning.

Application to refinement in quantitative information flow.

Abstract

Normalisation in probability theory turns a subdistribution into a proper distribution. It is a partial operation, since it is undefined for the zero subdistribution. This partiality makes it hard to reason equationally about normalisation. A novel description of normalisation is given as a mathematically well-behaved total function. The output of this `hyper' normalisation operation is a distribution of distributions. It improves reasoning about normalisation. After developing the basics of this theory of (hyper) normalisation, it is put to use in a similarly new description of conditioning, producing a distribution of conditional distributions. This is used to give a clean abstract reformulation of refinement in quantitative information flow.