# Relative Information and the Dual Numbers#

## Abstract#

Relative information (Kullback-Leibler divergence) is a fundamental concept in statistics, machine learning and information theory.

In the first half of the talk, I will define conditional relative information, list its axiomatic properties, and describe how it is used in machine learning. For example, the generalization error of a learning algorithm depends on the structure of algebraic geometric singularities of relative information.

In the second half of the talk, I will define the rig category InfoRig of random variables and their conditional maps, as well as the rig category R(e) of dual numbers. Relative information can then be constructed, up to a scalar multiple, via rig functors from InfoRig to R(e). If time permits, I may discuss how this construction relates to the information cohomology of Baudot, Bennequin and Vigneaux, and to the operad derivations of Bradley.

## Pre-requisites#

Random variables, probability distribution, conditional probability; category, object, morphism, functor; rig (semiring).