A Cartesian closed category for random variables.
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Synopsis
Scott domains, or more generally continuous domains, have been the traditional framework for semantics of programming languages.
It has however been a key open problem since 1980’s to develop a model of probabilistic semantics based on continuous domains---the obstacle being that no appropriate Cartesian closed category (CCC), closed under the probabilistic power domain, is known to exist.
In this talk, bypassing the latter stumbling block, we provide a solution to this long-standing problem.
We show that the probabilistic power domain of a Scott domain can be equally well represented by the Scott domain of random variables from any standard probability space to the given Scott domain: there is an effectively given, surjective and Scott continuous map from this domain of random variables to that of the probabilistic power domain of the underlying Scott domain.
This map simply takes any random variable on the domain to its associated probability distribution in the probabilistic power domain, crucially by preserving canonical basis elements. By enriching the category of Scott domains with a partial equivalence relation---to capture the equivalence of random variables---we obtain a CCC.
We can then develop four canonical commutative monads for constructing random variables from four standard probability spaces to objects of this category. We show that all basic probability distributions on finite dimensional Euclidean spaces can be denoted by their corresponding random variables in this framework.
This is joint work with Pietro Di Gianantonio, University of Udine.
About the speaker
Abbas Edalat
Abbas Edalat is a Professor of Computer Science and Mathematics at Imperial College London.
He came to Imperial having worked previously at Sharif University of Technology in Iran. In 1990’s and 2000’s, he uncovered connections between domain theory and several branches of mathematics, including measure and integration theory, fractal geometry, computational geometry, exact computation, differential calculus and ODE’s, leading to new computational models and algorithms in these subjects.
While he has continued to work in these areas to this day, his main work since early 2010’s is focused on a new research area he has pioneered and called Algorithmic Human Development.
Informed by Attachment Theory in developmental psychology and supported by several mathematical models, AHD reformulates Mother Nature’s 2-agent evolutionary algorithm for producing healthy infants and individuals into a 1-agent algorithm that any individual can practice on their own, including these days by using VR and chatbots, to treat mental illness and enhance well-being, and social and emotional intelligence.
This event is brought to you by: BCS FACS (Formal Aspects of Computing Science) group
