A Theory of Uncertainty Variables for State Estimation and Inference
This work provides a foundational alternative to probability theory for modeling uncertainty, potentially impacting fields like machine learning and AI, though it is incremental as it adapts existing probabilistic concepts to a set-based approach.
The authors introduced a new framework called uncertainty variables, which uses sets instead of probability distributions to model uncertainty, and proved that key concepts like Bayes' law, independence, and graphical models can be retained in this framework for state estimation and inference.
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.