A Comprehensive Analysis of Open Linear Systems in Category Theory

Temel Kavramlar

The authors explore the formalization of open stochastic systems using category theory, combining Gaussian probability with nondeterminism to model various phenomena.

Özet

This content delves into the formalization of open stochastic systems using category theory, focusing on extended Gaussian distributions. It discusses the interplay between probabilistic and nondeterministic uncertainty, emphasizing Willems' open stochastic systems. The article introduces copartiality as a novel concept to model lack of information through coarse σ-algebras. It also explores the compositionality of systems under multiple kinds of uncertainty, such as noisy physical laws and uninformative priors in statistical inference. The authors provide concrete examples like the noisy resistor to illustrate their theoretical framework.

İstatistikler

A noisy resistor is better modeled by V = RI + ϵ where ϵ ∼ N(0, σ2).
Lack of information is modeled through coarse σ-algebras in Willems' approach.
Extended Gaussian distributions combine Gaussian probability with nondeterminism.
Linear relations can be represented as copartial maps in Copar(Vec).
Total linear relations are equivalent to decorated copartial maps in GaussEx.
Transformation rules from Definition 2.10 are valid in GaussEx.
Joint distribution PV I for the noisy resistor can be visualized using string diagrams.

Alıntılar

"In practice, one can sometimes pretend (using the method of improper priors) that X is sampled from the Lebesgue measure."
"We describe them both as measure-theoretic and abstract categorical entities."
"A successful unification of these concepts is achieved by Willems’ open stochastic systems."
"The only events which are assigned a probability are of the form EA = {(V, I) ∈ R2 | V − RI ∈ A}."
"Category theory has emerged as a unifying language for studying systems and their composition."

Önemli Bilgiler Şuradan Elde Edildi

A Categorical Treatment of Open Linear Systems

by Dario Stein,... : arxiv.org 03-07-2024

https://arxiv.org/pdf/2403.03934.pdf

Daha Derin Sorular

How do copartial maps relate to relational nondeterminism?

Copartial maps in the context of categorical theory provide a way to formalize lack of information through maps into quotients. In the case of linear relations, copartial maps can be seen as capturing the essence of relational nondeterminism. Specifically, a copartial map represents a function from one set into a quotient space of another set. This concept aligns with how total linear relations can be represented using kernel representations within the framework of cospans. The relationship between copartial maps and relational nondeterminism is intricate yet significant, showcasing how uncertainty and lack of information can be modeled effectively in categorical settings.

Can Copar be turned into a Markov category?

While Copar (the category based on cospan constructions) has inherent limitations when it comes to being transformed directly into a Markov category due to issues related to copy and delete morphisms, there is an alternative approach that allows for this transformation under specific conditions. If we consider categories with biproducts like Vec (finite-dimensional vector spaces), then Copar(Vec) indeed becomes equivalent to TLinRel (the category representing total linear relations). By leveraging the structure provided by biproducts in Vec, we can establish an equivalence between Copar(Vec) and TLinRel, thereby achieving a form where Copar transitions effectively into a Markov category representation.

What implications does Proposition 4.5 have on modeling systems under multiple kinds of uncertainty?

Proposition 4.5 holds profound implications for modeling systems under various uncertainties by showcasing how different types of uncertainties can be elegantly combined within an abstract framework such as GaussEx (a Markov category dealing with extended Gaussian distributions). By demonstrating that TLinRel (total linear relations) is equivalent to Copar(Vec), which further translates seamlessly into GaussEx, we gain insights into how both Gaussian probability and relational nondeterminism can coexist harmoniously within this unified model.
This proposition essentially provides us with a powerful toolset for rigorously describing complex systems affected by probabilistic fluctuations and non-deterministic behaviors simultaneously—enabling comprehensive analyses encompassing diverse forms of uncertainty prevalent in real-world scenarios like noisy physical laws or uninformative priors in Bayesian statistics.