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Atrium > Mathematics, Physics & Philosophy: Markov chains

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    • CommentRowNumber1.
    • CommentAuthorIan_Durham
    • CommentTimeMar 20th 2010
    Based on a question I fielded from a physicist today and some work I've been doing with a mathematician, there is some interest in looking at Markov chains from a category theoretic viewpoint. Does anyone know of existing work in this area or in stochastic processes in general? I can't add anything at the moment since I have a plane to catch.
    • CommentRowNumber2.
    • CommentAuthorEric
    • CommentTimeMar 20th 2010
    • (edited Mar 20th 2010)
    Interesting! Two days ago, I received an email from a student (wonder if it was the same? :)) interested in category theory for stochastic calculus (which is closely related to Markov chains) in response to a comment I made here on the nCafe. In a followup comment, I gave a link to a book that looked promising which is directly relevant to your question:

    Stochastic relations: foundations for Markov transition systems
    By Ernst-Erich Doberkat

    Book Overview:
    Collecting information previously scattered throughout the vast literature, including the author’s own research, Stochastic Relations: Foundations for Markov Transition Systems develops the theory of stochastic relations as a basis for Markov transition systems.After an introduction to the basic mathematical tools from topology, measure theory, and categories, the book examines the central topics of congruences and morphisms, applies these to the monoidal structure, and defines bisimilarity and behavioral equivalence within this framework. The author views developments from the general theory of coalgebras in the context of the subprobability functor. These tools show that bisimilarity and behavioral and logical equivalence are the same for general modal logics and for continuous time stochastic logic with and without a fixed point operator.With numerous problems and several case studies, this book is an invaluable study of an important aspect of computer science theory.
    • CommentRowNumber3.
    • CommentAuthorEric
    • CommentTimeMar 20th 2010
    PS: Here is a bunch of papers.
    • CommentRowNumber4.
    • CommentAuthorIan_Durham
    • CommentTimeMar 20th 2010
    Actually the question was from the session chair, Mark Coffey of the Colorado School of Mines (despite it's rather unusual name it is a standard university).

    Thanks for that link! That is great! (And I really am checking out of my hotel and heading for the airport this time...)
    • CommentRowNumber5.
    • CommentAuthorEric
    • CommentTimeMar 20th 2010
    Since posts are cheap, I'll highlight a paper or two I find particularly interesting from that list. Here is the first one:

    A Remark on A. Edalat's Paper Semi-Pullbacks and Bisimulation in Categories of Markov-Processes. Juli 2002.
    • CommentRowNumber6.
    • CommentAuthorEric
    • CommentTimeMar 20th 2010
    • (edited Mar 20th 2010)
    Stochastic Relations Interpreting Modal Logic. Oktober 2003

    Note: The word "diamond" in the abstract caught my eye, e.g. diamonation (ericforgy)
    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 20th 2010

    I entered one characterisation of Markov chain as a coalgebra.

    • CommentRowNumber8.
    • CommentAuthorEric
    • CommentTimeMar 20th 2010
    I changed the page name to Markov chain (singular)
    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeMar 23rd 2010
    • (edited Mar 23rd 2010)

    I was doing some research on Lawvere-Giry monad with Roland Friedrich few years ago, and I am a bit disappointed, that this formulation seems just to scratch the surface of the stochastic theory. I read some of Doberkat's papers on the subject, they are written very formally, lots of notation and text for my taste and much less content. I did not know he published the book in the meantime. I stll hope to resume and finish the work with Roland, we had some generalization of Giry monad related to projective measures and coherent states in physics. This is conceptually nice, and one needs to know a little bit of spectral theory to do it properly, but it is not very deep so far. Roland was prompted to think about this starting after he heard a talk of Voevodsky on this approach, who wanted to sue the formalism in applciations in mathematical biology. Gromov was in the audience and complained that it is just a reformulation and Voevodsky agreed but said that it fist his way of concentrating to put it into categorical framework. I don't find thinking about stochastic processes via Giry monad simpler but more difficult. It was however our hope that in some continuous limits where the study of various spaces in probability things are nontrivial, some variant of Giry monad could help with nontrivial insight. Also I was expecting some generalizations of Giry monad for a class of abstract lattices...and I think that in the meantime some paper roughly in similar direction appeared by other authors.

    • CommentRowNumber10.
    • CommentAuthorIan_Durham
    • CommentTimeMar 24th 2010
    God I wish I had more time for my research. I really think there are some very interesting things that could be tackled in relation to Markov chains via categories but I can barely keep up at the moment with all the administrative junk I have to do. Anyway, thanks for all the interesting links and discussion.

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