Dario Stein
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View article: Compositional Inference for Bayesian Networks and Causality
Compositional Inference for Bayesian Networks and Causality Open
Inference is a fundamental reasoning technique in probability theory. When applied to a large joint distribution, it involves updating with evidence (conditioning) in one or more components (variables) and computing the outcome in other co…
View article: Dagger categories of relations: The equivalence of dilatory dagger categories and epi-regular independence categories
Dagger categories of relations: The equivalence of dilatory dagger categories and epi-regular independence categories Open
Several categories look like categories of relations, but do not fit the established theory of relations in regular categories. They include the category of surjective multivalued functions, the category of injective partial functions, the…
View article: A Categorical Treatment of Open Linear Systems
A Categorical Treatment of Open Linear Systems Open
An open stochastic system à la Jan Willems is a system affected by two qualitatively different kinds of uncertainty: one is probabilistic fluctuation, and the other one is nondeterminism caused by a fundamental lack of information. We pres…
View article: Random Variables, Conditional Independence and Categories of Abstract Sample Spaces
Random Variables, Conditional Independence and Categories of Abstract Sample Spaces Open
Two high-level "pictures" of probability theory have emerged: one that takes as central the notion of random variable, and one that focuses on distributions and probability channels (Markov kernels). While the channel-based picture has bee…
View article: Combs, Causality and Contractions in Atomic Markov Categories
Combs, Causality and Contractions in Atomic Markov Categories Open
We present a counterexample showing that Markov categories with conditionals (such as BorelStoch) need not validate a natural scheme of axioms which we call contraction identities. These identities hold in every traced monoidal category, s…
View article: A Categorical Treatment of Open Linear Systems
A Categorical Treatment of Open Linear Systems Open
An open stochastic system à la Jan Willems is a system affected by two qualitatively different kinds of uncertainty: one is probabilistic fluctuation, and the other one is nondeterminism caused by a fundamental lack of information. We pres…
View article: Graphical Quadratic Algebra
Graphical Quadratic Algebra Open
Convex analysis and Gaussian probability are tightly connected, as mostly evident in the theory of linear regression. Our work introduces an algebraic perspective on such relationship, in the form of a diagrammatic calculus of string diagr…
View article: Towards a Compositional Framework for Convex Analysis (with Applications to Probability Theory)
Towards a Compositional Framework for Convex Analysis (with Applications to Probability Theory) Open
We introduce a compositional framework for convex analysis based on the notion of convex bifunction of Rockafellar. This framework is well-suited to graphical reasoning, and exhibits rich dualities such as the Legendre-Fenchel transform, w…
View article: Probabilistic Programming with Exact Conditions
Probabilistic Programming with Exact Conditions Open
We spell out the paradigm of exact conditioning as an intuitive and powerful way of conditioning on observations in probabilistic programs. This is contrasted with likelihood-based scoring known from languages such as Stan. We study exact …
View article: Overdrawing Urns using Categories of Signed Probabilities
Overdrawing Urns using Categories of Signed Probabilities Open
A basic experiment in probability theory is drawing without replacement from\nan urn filled with multiple balls of different colours. Clearly, it is\nphysically impossible to overdraw, that is, to draw more balls from the urn\nthan it cont…
View article: Towards a Compositional Framework for Convex Analysis (with Applications to Probability Theory)
Towards a Compositional Framework for Convex Analysis (with Applications to Probability Theory) Open
We introduce a compositional framework for convex analysis based on the notion of convex bifunction of Rockafellar. This framework is well-suited to graphical reasoning, and exhibits rich dualities such as the Legendre-Fenchel transform, w…
View article: Pearl's and Jeffrey's Update as Modes of Learning in Probabilistic Programming
Pearl's and Jeffrey's Update as Modes of Learning in Probabilistic Programming Open
The concept of updating a probability distribution in the light of new evidence lies at the heart of statistics and machine learning. Pearl's and Jeffrey's rule are two natural update mechanisms which lead to different outcomes, yet the si…
View article: Probabilistic Programming with Exact Conditions
Probabilistic Programming with Exact Conditions Open
We spell out the paradigm of exact conditioning as an intuitive and powerful way of conditioning on observations in probabilistic programs. This is contrasted with likelihood-based scoring known from languages such as Stan . We study exact…
View article: Dilations and information flow axioms in categorical probability
Dilations and information flow axioms in categorical probability Open
We study the positivity and causality axioms for Markov categories as properties of dilations and information flow and also develop variations thereof for arbitrary semicartesian monoidal categories. These help us show that being a positiv…
View article: Pearl's and Jeffrey's Update as Modes of Learning in Probabilistic Programming
Pearl's and Jeffrey's Update as Modes of Learning in Probabilistic Programming Open
The concept of updating a probability distribution in the light of new evidence lies at the heart of statistics and machine learning. Pearl's and Jeffrey's rule are two natural update mechanisms which lead to different outcomes, yet the si…
View article: Absolute continuity, supports and idempotent splitting in categorical probability
Absolute continuity, supports and idempotent splitting in categorical probability Open
Markov categories have recently turned out to be a powerful high-level framework for probability and statistics. They accommodate purely categorical definitions of notions like conditional probability and almost sure equality, as well as p…
View article: Dilations and information flow axioms in categorical probability
Dilations and information flow axioms in categorical probability Open
We study the positivity and causality axioms for Markov categories as properties of dilations and information flow in Markov categories, and in variations thereof for arbitrary semicartesian monoidal categories. These help us show that bei…
View article: A Category for unifying Gaussian Probability and Nondeterminism
A Category for unifying Gaussian Probability and Nondeterminism Open
We introduce categories of extended Gaussian maps and Gaussian relations which unify Gaussian probability distributions with relational nondeterminism in the form of linear relations. Both have crucial and well-understood applications in s…
View article: Compositional Semantics for Probabilistic Programs with Exact Conditioning
Compositional Semantics for Probabilistic Programs with Exact Conditioning Open
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 …
View article: Probabilistic programming semantics for name generation
Probabilistic programming semantics for name generation Open
We make a formal analogy between random sampling and fresh name generation. We show that quasi-Borel spaces, a model for probabilistic programming, can soundly interpret the ν-calculus, a calculus for name generation. Moreover, we prove th…
View article: Probabilistic programming semantics for name generation
Probabilistic programming semantics for name generation Open
We make a formal analogy between random sampling and fresh name generation. We show that quasi-Borel spaces, a model for probabilistic programming, can soundly interpret the ν-calculus, a calculus for name generation. Moreover, we prove th…
View article: The Beta-Bernoulli process and algebraic effects
The Beta-Bernoulli process and algebraic effects Open
In this paper we use the framework of algebraic effects from programming language theory to analyze the Beta-Bernoulli process, a standard building block in Bayesian models. Our analysis reveals the importance of abstract data types, and t…
View article: The Beta-Bernoulli process and algebraic effects
The Beta-Bernoulli process and algebraic effects Open
In this paper we use the framework of algebraic effects from programming language theory to analyze the Beta-Bernoulli process, a standard building block in Bayesian models. Our analysis reveals the importance of abstract data types, and t…