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In coordination games and speculative over-the-counter financial markets, solutions depend on higher-order average expectations: agents' expectations about what counterparties, on average, expect their counterparties to think, etc. We offer…
We introduce categories of extended Gaussian maps and Gaussian relations which unify Gaussian probability distributions with relational nondeterminism in the form of linear relations. Both have crucial and well-understood applications in…
Word embeddings have been found to capture a surprisingly rich amount of syntactic and semantic knowledge. However, it is not yet sufficiently well-understood how the relational knowledge that is implicitly encoded in word embeddings can be…
These are lecture notes written at the University of Zurich during spring 2014 and spring 2015. The first part of the notes gives an introduction to probability theory. It explains the notion of random events and random variables,…
Bayesian networks provide an elegant formalism for representing and reasoning about uncertainty using probability theory. Theyare a probabilistic extension of propositional logic and, hence, inherit some of the limitations of propositional…
Typically, when evaluating Theory of Mind, we consider the beliefs of others to be binary: held or not held. But what if someone is unsure about their own beliefs? How can we quantify this uncertainty? We propose a new suite of tasks,…
We present a framework to formally describe probabilistic system behavior and symbolically reason about it. In particular we aim at reasoning about possible failures and fault tolerance. We regard systems which are composed of different…
We introduce a general theory of epistemic random fuzzy sets for reasoning with fuzzy or crisp evidence. This framework generalizes both the Dempster-Shafer theory of belief functions, and possibility theory. Independent epistemic random…
Ordinal classification problems, where labels exhibit a natural order, are prevalent in high-stakes fields such as medicine and finance. Accurate uncertainty quantification, including the decomposition into aleatoric (inherent variability)…
Quantum uncertainty relations are formulated in terms of relative entropy between distributions of measurement outcomes and suitable reference distributions with maximum entropy. This type of entropic uncertainty relation can be applied…
The uncertainty relation is a distinguishing feature of quantum theory, characterizing the incompatibility of noncommuting observables in the preparation of quantum states. Recently, many uncertainty relations were proposed with improved…
Causal inference is a central goal across many scientific disciplines. Over the past several decades, three major frameworks have emerged to formalize causal questions and guide their analysis: the potential outcomes framework, structural…
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that typically arise from applying decision rules…
By representing words with probability densities rather than point vectors, probabilistic word embeddings can capture rich and interpretable semantic information and uncertainty. The uncertainty information can be particularly meaningful in…
Quantitative characterizations and estimations of uncertainty are of fundamental importance in optimization and decision-making processes. Herein, we propose intuitive scores, which we call certainty and doubt, that can be used in both a…
In this paper, a unified framework for representing uncertain information based on the notion of an interval structure is proposed. It is shown that the lower and upper approximations of the rough-set model, the lower and upper bounds of…
Recent work in cognitive science has uncovered a diversity of explanatory values, or dimensions along which we judge explanations as better or worse. We propose a Bayesian account of how these values fit together to guide explanation. The…
Expert systems applications that involve uncertain inference can be represented by a multidimensional contingency table. These tables offer a general approach to inferring with uncertain evidence, because they can embody any form of…
Uncertainty principle is one of the cornerstones of quantum theory. In the literature, there are two types of uncertainty relations, the operator form concerning the variances of physical observables and the entropy form related to entropic…
The complexity of the operating environment and required technologies for highly automated driving is unprecedented. A different type of threat to safe operation besides the fault-error-failure model by Laprie et al. arises in the form of…