A Theory of Uncertainty Variables for State Estimation and Inference
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Probability theory forms an overarching framework for modeling uncertainty, and by extension, also in designing state estimation and inference algorithms. While it provides a good foundation to system modeling, analysis, and an understanding of the real world, its application to algorithm design suffers from computational intractability. In this work, we develop a new framework of uncertainty variables to model uncertainty. A simple 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.
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