A channel-based perspective on conjugate priors

Jacobs, Bart

doi:10.1017/s0960129519000082

Cited by 18 publications

(9 citation statements)

References 16 publications

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“…The former has been extensively studied, going back to (Lawvere, 1962) and spanning dozens of papers (Corfield, 2021). The latter is less popular, but has seen a number of papers in recent years studying causality (Fong, 2013), conjugate priors (Jacobs, 2017) and general aspects of Bayesian learning Sturtz, 2014, 2013).…”

Section: Applications Successes and Motivationmentioning

confidence: 99%

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Category Theory in Machine Learning

Shiebler,

Gavranović,

Wilson

2021

Preprint

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Over the past two decades machine learning has permeated almost every realm of technology. At the same time, many researchers have begun using category theory as a unifying language, facilitating communication between different scientific disciplines. It is therefore unsurprising that there is a burgeoning interest in applying category theory to machine learning. We aim to document the motivations, goals and common themes across these applications. We touch on gradient-based learning, probability, and equivariant learning.

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Section: Applications Successes and Motivationmentioning

confidence: 99%

“…• Probabilistic Machine Learning. The study of updating a distribution with samples from a dataset and reasoning about uncertainty arising from noisy measurements Sturtz, 2014, 2013;Jacobs, 2017).…”

Section: Applications Successes and Motivationmentioning

confidence: 99%

Category Theory in Machine Learning

Shiebler,

Gavranović,

Wilson

2021

Preprint

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“…All of our examples FinStoch, BorelStoch and Gauss have conditionals [5], [11]. For Gauss, this captures the selfconjugacy of Gaussians [19]. An explicit formula generalizing (4) is given in [11], but we shall only require the existence of conditionals and work with their universal property.…”

Section: Definition Iii3 ([11]mentioning

confidence: 99%

Compositional Semantics for Probabilistic Programs with Exact Conditioning

Stein

Staton

2021

Preprint

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We define a probabilistic programming language for Gaussian random variables with a first-class exact conditioning construct. We give operational, denotational and equational semantics for this language, establishing convenient properties like exchangeability of conditions. Conditioning on equality of continuous random variables is nontrivial, as the exact observation may have probability zero; this is Borel's paradox. Using categorical formulations of conditional probability, we show that the good properties of our language are not particular to Gaussians, but can be derived from universal properties, thus generalizing to wider settings. We define the Cond construction, which internalizes conditioning as a morphism, providing general compositional semantics for probabilistic programming with exact conditioning.

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“…The Bayesian approach to learning can handle additional data via conditioning. This will be made precise below, using a logical formulation that is characteristic for conjugate priors, see [13]. For numbers n and i ∈ n we write 1 {i} : n → [0, 1] for the singleton predicate that is 1 on i ∈ n and 0 elsewhere.…”

Section: Prerequisites On Continuous Probabilitymentioning

confidence: 99%

Categorical Aspects of Parameter Learning

Jacobs

2018

Preprint

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Parameter learning is the technique for obtaining the probabilistic parameters in conditional probability tables in Bayesian networks from tables with (observed) data -where it is assumed that the underlying graphical structure is known. There are basically two ways of doing so, referred to as maximal likelihood estimation (MLE) and as Bayesian learning. This paper provides a categorical analysis of these two techniques and describes them in terms of basic properties of the multiset monad M, the distribution monad D and the Giry monad G. In essence, learning is about the reltionships between multisets (used for counting) on the one hand and probability distributions on the other. These relationsips will be described as suitable natural transformations.

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A channel-based perspective on conjugate priors

Cited by 18 publications

References 16 publications

Category Theory in Machine Learning

Category Theory in Machine Learning

Compositional Semantics for Probabilistic Programs with Exact Conditioning

Categorical Aspects of Parameter Learning

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