Discussion of Homotopy Type Theory and Univalent Foundations
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From: Bas Spitters <b.a.w.spitters@gmail.com>
To: homotopytypetheory <homotopytypetheory@googlegroups.com>,
	 Constructive News <constructivenews@googlegroups.com>
Subject: [HoTT] Synthetic topology in Homotopy Type Theory for probabilistic programming
Date: Mon, 30 Dec 2019 19:05:40 +0100
Message-ID: <CAOoPQuR4ttw71bPEA+xtXZ__Q5kKFGRqgJVgTA=2PmLc5=d5Lw@mail.gmail.com> (raw)

We have a new paper on the arxiv.

Comments, suggestions and questions are very welcome.

Synthetic topology in Homotopy Type Theory for probabilistic programming
Martin E. Bidlingmaier, Florian Faissole, Bas Spitters

    The ALEA Coq library formalizes measure theory based on a variant
of the Giry monad on the category of sets. This enables the
interpretation of a probabilistic programming language with primitives
for sampling from discrete distributions. However, continuous
distributions have to be discretized because the corresponding
measures cannot be defined on all subsets of their carriers.
    This paper proposes the use of synthetic topology to model
continuous distributions for probabilistic computations in type
theory. We study the initial σ-frame and the corresponding induced
topology on arbitrary sets. Based on these intrinsic topologies we
define valuations and lower integrals on sets, and prove versions of
the Riesz and Fubini theorems. We then show how the Lebesgue
valuation, and hence continuous distributions, can be constructed.

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Discussion of Homotopy Type Theory and Univalent Foundations

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