A definition of conditional probability distribution with non-stochastic information

TL;DR

This paper introduces a new definition for conditional probability distributions that incorporate non-stochastic information, extending traditional concepts by linking information to outcomes through a loss function and emphasizing the role of Kullback-Leibler divergence.

Contribution

It proposes a novel framework for conditional probabilities with non-stochastic information based on decision theory and divergence measures, expanding the theoretical foundation.

Findings

Defines conditional probability with non-stochastic info via decision theory

Highlights the role of Kullback-Leibler divergence in the new framework

Provides illustrative examples of the new definition

Abstract

The current definition of a conditional probability distribution enables one to update probabilities only on the basis of stochastic information. This paper provides a definition for conditional probability distributions with non-stochastic information. The definition is derived as a solution of a decision theoretic problem, where the information is connected to the outcome of interest via a loss function. We shall show that the Kullback-Leibler divergence plays a central role. Some illustrations are presented.