Giry and the Machine

Dahlqvist, Fredrik; Danos, Vincent; Garnier, Ilias

doi:10.1016/j.entcs.2016.09.033

Cited by 6 publications

(9 citation statements)

References 2 publications

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“…Note first that the infinite product (2 H ) ω can be defined as the limit of the ccd given by the maps q n+1,n : (2 H ) n+1 → (2 H ) n dropping the last component. By Bochner's theorem ( [7]) this also holds of G((2 H ) ω ). Next, consider any program r. We turn 2 H into a cone for the diagram with limit G((2 H ) ω ) via the inductively defined maps:…”

Section: Approximating the Kleene Star Of Probnetkatmentioning

confidence: 80%

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Borel Kernels and their Approximation, Categorically

Dahlqvist

Silva²,

Danos³

et al. 2018

Electronic Notes in Theoretical Computer Science

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This paper introduces a categorical framework to study the exact and approximate semantics of probabilistic programs. We construct a dagger symmetric monoidal category of Borel kernels where the dagger-structure is given by Bayesian inversion. We show functorial bridges between this category and categories of Banach lattices which formalize the move from kernel-based semantics to predicate transformer (backward) or state transformer (forward) semantics. These bridges are related by natural transformations, and we show in particular that the Radon-Nikodym and Riesz representation theorems -two pillars of probability theory -define natural transformations.With the mathematical infrastructure in place, we present a generic and endogenous approach to approximating kernels on standard Borel spaces which exploits the involutive structure of our category of kernels. The approximation can be formulated in several equivalent ways by using the functorial bridges and natural transformations described above. Finally, we show that for sensible discretization schemes, every Borel kernel can be approximated by kernels on finite spaces, and that these approximations converge for a natural choice of topology.We illustrate the theory by showing two examples of how approximation can effectively be used in practice: Bayesian inference and the Kleene * operation of ProbNetKAT. . We introduce the category BL σ of Banach lattices and σorder continuous positive operators as well as the Köthe dual functor (−) σ : BL op σ → BL σ ( §3). These will play a central role in studying convergence of our approximation schemes. 3. We provide the first 2 categorical understanding of the Radon-Nikodym and the Riesz representation theorems. These arise as natural transformations between two functors relating kernels and Banach lattices ( §4). 4. We show how the †-structure of Krn can be exploited to approximate kernels by averaging ( §5). Due to an important structural feature of Krn (Th. 1) every kernel in Krn can be approximated by kernels on finite spaces. 5. We show a natural class of approximations schemes where the sequence of approximating kernels converges to the kernel to be approximated. The notion of convergence is given naturally by moving to BL σ and considering convergence in the Strong Operator Topology ( §6). 6. We apply our theory of kernel approximations to two practical applications ( §7). First, we show how Bayesian inference can be performed approximately by showing that the †-operation commutes with taking approximations. Secondly, we consider the case of ProbNetKAT, a language developed in [14,22] to probabilistically reason about networks. ProbNetKAT includes a Kleene star operator (−) * with a complex semantics which has proved hard to approximate. We show that (−) * can be approximated, and that the approximation converges.

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Section: Approximating the Kleene Star Of Probnetkatmentioning

confidence: 80%

“…= f (x)(A) • p † m • p m = f m (x)(A) where (1) is by definition (?? ), (2) is by Thm (10) and (3) is by Theorem (7). We have thus shown that f n (−)(A) is a martingale for the filtration generated by the discretization scheme, and the result now follows from Lévy's upward convergence Theorem ([25, Th.…”

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confidence: 87%

Borel Kernels and their Approximation, Categorically

Dahlqvist

Silva²,

Danos³

et al. 2018

Electronic Notes in Theoretical Computer Science

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“…Actually, an instance of a Vietoris functor, which we call compact Vietoris functor, has already been studied multiple times in the coalgebraic setting (e.g. Bezhanishvili et al 2010;Dahlqvist et al 2016;Kupke et al 2004;Venema and Vosmaer 2014), and will appear in a book on coalgebras that is currently in preparation (Adámek 2018). In particular, Kupke et al (2004) shows that compact Vietoris polynomial functors in the category Stone of Stone spaces and continuous maps admit a final coalgebra.…”

Section: Contributions and Related Workmentioning

confidence: 99%

“…In particular, Kupke et al (2004) shows that compact Vietoris polynomial functors in the category Stone of Stone spaces and continuous maps admit a final coalgebra. Also, document (Dahlqvist et al 2016) presents a theorem that can be generalised to show that the compact Vietoris functor in the category CompHaus of compact Hausdorff spaces and continuous maps, admits a final coalgebra. In fact, this generalised result is also implicitly mentioned in Engelking (1989, page 245).…”

Section: Contributions and Related Workmentioning

confidence: 99%

Limits in categories of Vietoris coalgebras

Hofmann¹,

Neves²,

Nora³

2018

Math. Struct. Comp. Sci.

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Motivated by the need to reason about hybrid systems, we study limits in categories of coalgebras whose underlying functor is a Vietoris polynomial one -intuitively, the topological analogue of a Kripke polynomial functor. Among other results, we prove that every Vietoris polynomial functor admits a final coalgebra if it respects certain conditions concerning separation axioms and compactness.When the functor is restricted to some of the categories induced by these conditions the resulting categories of coalgebras are even complete.As a practical application, we use these developments in the specification and analysis of nondeterministic hybrid systems, in particular to obtain suitable notions of stability, and behaviour.

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“…The inspiration for the present work comes from two recent developments. The first is the beginning of a categorical understanding of Bayesian inversion and learning [DG15, DDG16,CDDG17] the second is a categorical reconstruction of relative entropy [BFL11, BF14,Lei]. The present paper provides a categorical treatment of entropy in the spirit of Baez and Fritz in the setting of standard Borel spaces, thus setting the stage to explore the role of entropy in learning.…”

Section: Introductionmentioning

confidence: 99%

A Categorical Characterization of Relative Entropy on Standard Borel Spaces

Gagné

Panangaden

2018

Electronic Notes in Theoretical Computer Science

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We give a categorical treatment, in the spirit of Baez and Fritz, of relative entropy for probability distributions defined on standard Borel spaces. We define a category called SbStat suitable for reasoning about statistical inference on standard Borel spaces. We define relative entropy as a functor into Lawvere's category [0, ∞] and we show convexity, lower semicontinuity and uniqueness.Recently there have been some exciting developments that bring some categorical insights to probability theory and specifically to learning theory. These are reported in some recent papers by Clerc, Dahlqvist, Danos and Garnier [DG15, DDG16, CDDG17]. The first of these papers showed how to view the Dirichlet distribution as a natural transformation thus opening the way to an understanding of higher-order probabilities, while the second gave a powerful framework for constructing several natural transformations. In [DG15] the hope was expressed that one could use these ideas to understand Bayesian inversion, a core concept in machine learning. In [CDDG17] this was realized in a remarkably novel way. These papers carry out their investigations in the setting of standard Borel spaces and are based on the Giry monad [Gir81, Law64].

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Giry and the Machine

Cited by 6 publications

References 2 publications

Borel Kernels and their Approximation, Categorically

Borel Kernels and their Approximation, Categorically

Limits in categories of Vietoris coalgebras

A Categorical Characterization of Relative Entropy on Standard Borel Spaces

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