Pointless Learning

Clerc, Florence; Danos, Vincent; Dahlqvist, Fredrik; Garnier, Ilias

doi:10.1007/978-3-662-54458-7_21

Cited by 28 publications

(64 citation statements)

References 11 publications

(19 reference statements)

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“…However, the Köthe dual will still exist. Thus our semantics is free of some of the 'pointful' technicalities surrounding the existence of disintegrations, and follow the 'pointless' perspective advocated in [13]. Similarly, we do not have to worry about the ambiguity cause by the fact that disintegrations are only defined up to a null set: the Köthe dual of an operator between regular ordered Banach spaces exists completely unambiguously.…”

Section: ) Conditionals and While Loopsmentioning

confidence: 81%

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Semantics of higher-order probabilistic programs with conditioning

Dahlqvist

Kozen

2019

Proc. ACM Program. Lang.

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We present a denotational semantics for higherorder probabilistic programs in terms of linear operators between Banach spaces. Our semantics is rooted in the classical theory of Banach spaces and their tensor products, but bears similarities with the well-known semantics of higher-order programsà la Scott through the use ordered Banach spaces which allow definitions in terms of fixed points. Being based on a monoidal rather than cartesian closed structure, our semantics effectively treats randomness as a resource.

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Section: ) Conditionals and While Loopsmentioning

confidence: 81%

“…As in the case of the Lebesgue integral, we start with simple functions. Given a measurable space (X, F ), we generalise (13) and say that a function f :…”

Section: G Supplementary Materials On Projective Tensor Products 1) Dmentioning

confidence: 99%

Semantics of higher-order probabilistic programs with conditioning

Dahlqvist

Kozen

2019

Proc. ACM Program. Lang.

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“…In [6] the first three authors presented a category of Borel kernels similar in spirit to the construction of this section, but with a major shortcoming. As we will shortly see, our category Krn of Borel kernels can be equipped with an involutive functor -a dagger operation † in the terminology of [21] -which captures the notion of Bayesian inversion and is absolutely crucial to everything that follows.…”

Section: A Category Of Borel Kernelsmentioning

confidence: 99%

“…As we will shortly see, our category Krn of Borel kernels can be equipped with an involutive functor -a dagger operation † in the terminology of [21] -which captures the notion of Bayesian inversion and is absolutely crucial to everything that follows. In [6] this operation had merely been identified as a map, i.e. not even as a functor.…”

Section: A Category Of Borel Kernelsmentioning

confidence: 99%

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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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Causal Inference by String Diagram Surgery

Jacobs

Kissinger

Zanasi

2019

Lecture Notes in Computer Science

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Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors.A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior dependencies. We represent the effect of such an intervention as an endofunctor which performs 'string diagram surgery' within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on a well-known toy example, where we predict the causal effect of smoking on cancer in the presence of a confounding common cause. After developing this specific example, we show this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature.

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Pointless Learning

Cited by 28 publications

References 11 publications

Semantics of higher-order probabilistic programs with conditioning

Semantics of higher-order probabilistic programs with conditioning

Borel Kernels and their Approximation, Categorically

Causal Inference by String Diagram Surgery

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