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Invited Paper

**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

The process of inverting Markov kernels relates to the important subject of Bayesian modelling and learning. In fact, Bayesian update is exactly kernel inversion. In this paper, we investigate how and when Markov kernels (aka stochastic relations, or probabilistic mappings, or simply kernels) can be inverted. We address the question both directly on the category of measurable spaces, and indirectly by interpreting kernels as Markov operators:
- For the direct option, we introduce a typed version of the category of Markov kernels and use the so-called "disintegration of measures". Here, one has to specialise to measurable spaces borne from a simple class of topological spaces -e.g. Polish spaces (other choices are possible). Our method and result greatly simplify a recent development in Ref. [4].
- For the operator option, we use a cone version of the category of Markov operators (kernels seen as predicate transformers). That is to say, our linear operators are not just continuous, but are required to satisfy the stronger condition of being $\om$-chain-continuous. Prior work shows that one obtains an adjunction in the form of a pair of contravariant and inverse functors between the categories of $L_1$- and $L_\infty$-cones [3]. Inversion, seen through the operator prism, is just adjunction. No topological assumption is needed.
- We show that both categories (Markov kernels and $\om$-chain-continuous Markov operators) are related by a family of contravariant functors $T_p$ for $1\leq p\leq\infty$. The $T_p$'s are Kleisli extensions of (duals of) conditional expectation functors introduced in Ref. [3].
- With this bridge in place, we can prove that both notions of inversion agree when both defined: if $f$ is a kernel, and $f\dg$ its direct inverse, then $T_\infty(f)\dg=T_1(f\dg)$.

Fredrik Dahlqvist, Vincent Danos, Ilias Garnier, and Ohad Kammar. Bayesian Inversion by Omega-Complete Cone Duality (Invited Paper). In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dahlqvist_et_al:LIPIcs.CONCUR.2016.1, author = {Dahlqvist, Fredrik and Danos, Vincent and Garnier, Ilias and Kammar, Ohad}, title = {{Bayesian Inversion by Omega-Complete Cone Duality}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {1:1--1:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.1}, URN = {urn:nbn:de:0030-drops-61909}, doi = {10.4230/LIPIcs.CONCUR.2016.1}, annote = {Keywords: probabilistic models, bayesian learning, markov operators} }

Document

**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

We present a method for constructing robustly parameterised families of higher-order probabilistic models. Parameter spaces and models are represented by certain classes of functors in the category of Polish spaces. Maps from parameter spaces to models (parameterisations) are continuous and natural transformations between such functors. Naturality ensures that parameterised models are invariant by change of granularity -- ie that parameterisations are intrinsic. Continuity ensures that models are robust with respect to their parameterisation. Our method allows one to build models from a set of basic functors among which the Giry probabilistic functor, spaces of cadlag trajectories (in continuous and discrete time), multisets and compact powersets. These functors can be combined by guarded composition, product and coproduct. Parameter spaces range over the polynomial closure of Giry-like functors. Thus we obtain a class of robust parameterised models which includes the Dirichlet process, various point processes (random sequences with values in Polish spaces) and other classical objects of probability theory. By extending techniques developed in prior work, we show how to reduce the questions of existence, uniqueness, naturality, and continuity of a parameterised model to combinatorial questions only involving finite spaces.

Fredrik Dahlqvist, Vincent Danos, and Ilias Garnier. Robustly Parameterised Higher-Order Probabilistic Models. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dahlqvist_et_al:LIPIcs.CONCUR.2016.23, author = {Dahlqvist, Fredrik and Danos, Vincent and Garnier, Ilias}, title = {{Robustly Parameterised Higher-Order Probabilistic Models}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {23:1--23:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.23}, URN = {urn:nbn:de:0030-drops-61737}, doi = {10.4230/LIPIcs.CONCUR.2016.23}, annote = {Keywords: Probability, category theory, Giry monad} }