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Oliver E. Richardson

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3 papers

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UAI Conference 2025 Conference Paper

Learning with Confidence

Oliver E. Richardson

We characterize a notion of confidence that arises in learning or updating beliefs: the amount of trust one has in incoming information and its impact on the belief state. This *learner’s confidence* can be used alongside (and is easily mistaken for) probability or likelihood, but it is fundamentally a different concept—one that captures many familiar concepts in the literature, including learning rates and number of training epochs, Shafer’s weight of evidence, and Kalman gain. We formally axiomatize what it means to learn with confidence, give two canonical ways of measuring confidence on a continuum, and prove that confidence can always be represented in this way. Under additional assumptions, we derive more compact representations of confidence-based learning in terms of vector fields and loss functions. These representations induce an extended language of compound "parallel" observations. We characterize *Bayesian* learning as the special case of an optimizing learner whose loss representation is a linear expectation.

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NeurIPS Conference 2024 Conference Paper

Qualitative Mechanism Independence

Oliver E. Richardson
Spencer Peters
Joseph Y. Halpern

We define what it means for a joint probability distribution to be compatible with aset of independent causal mechanisms, at a qualitative level—or, more precisely with a directed hypergraph $\mathcal A$, which is the qualitative structure of a probabilistic dependency graph (PDG). When A represents a qualitative Bayesian network, QIM-compatibility with $\mathcal A$ reduces to satisfying the appropriate conditional independencies. But giving semantics to hypergraphs using QIM-compatibility lets us do much more. For one thing, we can capture functional dependencies. For another, we can capture important aspects of causality using compatibility: we can use compatibility to understand cyclic causal graphs, and to demonstrate structural compatibility, we must essentially produce a causal model. Finally, compatibility has deep connections to information theory. Applying compatibility to cyclic structures helps to clarify a longstanding conceptual issue in information theory.

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UAI Conference 2023 Conference Paper

Inference for probabilistic dependency graphs

Oliver E. Richardson
Joseph Y. Halpern
Christopher De Sa

Probabilistic dependency graphs (PDGs) are a flexible class of probabilistic graphical models, subsuming Bayesian Networks and Factor Graphs. They can also capture inconsistent beliefs, and provide a way of measuring the degree of this inconsistency. We present the first tractable inference algorithm for PDGs with discrete variables, making the asymptotic complexity of PDG inference similar that of the graphical models they generalize. The key components are: (1) the observation that PDG inference can be reduced to convex optimization with exponential cone constraints, (2) a construction that allows us to express these problems compactly for PDGs of boundeed treewidth, for which we needed to further develop the theory of PDGs, and (3) an appeal to interior point methods that can solve such problems in polynomial time. We verify the correctness and time complexity of our approach, and provide an implementation of it. We then evaluate our implementation, and demonstrate that it outperforms baseline approaches.

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