
Language Modeling with Reduced Densities
We present a framework for modeling words, phrases, and longer expressio...
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Modeling Sequences with Quantum States: A Look Under the Hood
Classical probability distributions on sets of sequences can be modeled ...
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Quantum and Classical Information Theory with Disentropy
In the present work, some concepts of quantum and classical information ...
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Visualizing and comparing distributions with halfdisk density strips
We propose a userfriendly graphical tool, the halfdisk density strip (...
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Rapid and deterministic estimation of probability densities using scalefree field theories
The question of how best to estimate a continuous probability density fr...
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Negative probabilities: What they are and what they are for
In quantum mechanics, the probability distributions of position and mome...
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Inverses, Conditionals and Compositional Operators in Separative Valuation Algebra
Compositional models were introduce by Jirousek and Shenoy in the genera...
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At the Interface of Algebra and Statistics
This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals recover classical marginal probabilities. In general, these reduced densities will have rank higher than one, and their eigenvalues and eigenvectors will contain extra information that encodes subsystem interactions governed by statistics. We decode this information, and show it is akin to conditional probability, and then investigate the extent to which the eigenvectors capture "concepts" inherent in the original joint distribution. The theory is then illustrated with an experiment that exploits these ideas. Turning to a more theoretical application, we also discuss a preliminary framework for modeling entailment and concept hierarchy in natural language, namely, by representing expressions in the language as densities. Finally, initial inspiration for this thesis comes from formal concept analysis, which finds many striking parallels with the linear algebra. The parallels are not coincidental, and a common blueprint is found in category theory. We close with an exposition on free (co)completions and how the freeforgetful adjunctions in which they arise strongly suggest that in certain categorical contexts, the "fixed points" of a morphism with its adjoint encode interesting information.
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