A Theory of Uncertainty Variables for State Estimation and Inference

TL;DR

This paper introduces a new framework of uncertainty variables that generalizes probability measures, allowing for modeling uncertainty with sets, and develops corresponding Bayesian networks and inference tools.

Contribution

It proposes a novel set-based framework for uncertainty variables, extending Bayesian networks and inference methods beyond traditional probability measures.

Findings

Proves Bayes' law and total probability for uncertainty variables

Defines and preserves independence notions in the set-based framework

Develops Bayesian uncertainty networks with natural conditional independence properties

Abstract

We develop a new framework of uncertainty variables to model uncertainty. An uncertainty variable is characterized by an uncertainty set, in which its realization is bound to lie, while the conditional uncertainty is characterized by a set map, from a given realization of a variable to a set of possible realizations of another variable. We prove Bayes' law and the law of total probability equivalents for uncertainty variables. We define a notion of independence, conditional independence, and pairwise independence for a collection of uncertainty variables, and show that this new notion of independence preserves the properties of independence defined over random variables. We then develop a graphical model, namely Bayesian uncertainty network, a Bayesian network equivalent defined over a collection of uncertainty variables, and show that all the natural conditional independence properties, expected out of a Bayesian network, hold for the Bayesian uncertainty network. We also define the notion of point estimate, and show its relation with the maximum a posteriori estimate. Probability theory starts with a distribution function (equivalently a probability measure) as a primitive and builds all other useful concepts, such as law of total probability, Bayes' law, independence, graphical models, point estimate, on it. Our work shows that it is perfectly possible to start with a set, instead of a distribution function, and retain all the useful ideas needed for state estimation and inference.