arXiv subjects

Kirk Sturtz

Publications and source records attributed to Kirk Sturtz.

11 recordsLinked to original sources

arXiv2024

Deriving the Giry algebras on standard Borel spaces using $\mathbb{R}_{\infty}$-generalized points

The Giry monad on the category of measurable spaces restricts to the full subcategory of standard Borel spaces, $\mathbf{Std}$, which we show is amenable to analysis. $\mathbf{Std}$ contains the space $\mathbb{R}_{\infty}$ which is the one-point compactification of the real numbers. By viewing probability measures $P \in \mathcal{G}(A)$ as functionals operating on measurable functions $A \rightarrow \mathbb{R}_{\infty}$, and taking the restriction of those functionals to operate on affine measurable functions we show that $A \cong Hom_{\mathbb{R}_{\infty}^{\mathbb{R}_{\infty}}}(\mathbb{R}_{\infty}^A|,\mathbb{R}_{\infty})$ for all object $A$ lying in the subcategory $\mathbf{Std}_{Cvx}$ of $\mathbf{Std}$. The objects of $\mathbf{Std}_{Cvx}$ are standard spaces with a convex space structure which satisfies the generic ``fullness property''. The morphisms of the category $\mathbf{Std}_{Cvx}$ are affine measurable functions. The isomorphism is equivalent to the statement that the full subcategory of $\mathbf{Std}_{Cvx}$ consisting of the single object $\mathbb{R}_{\infty}$ is codense in $\mathbf{Std}_{Cvx}$ which allows us to easily construct the $\mathcal{G}$-algebras of objects in $\mathbf{Std}_{Cvx}$. This permits an adjoint factorization of the Giry monad as the composite of $\mathbf{Std} \xrightarrow{\hat{\mathcal{G}}} \mathbf{Std}_{Cvx}$, which is the Giry monad functor viewed as a functor into $\mathbf{Std}_{Cvx}$, and the partial forgetful functor $\mathbf{Std}_{Cvx} \xrightarrow{\mathcal{U}_{Cvx}} \mathbf{Std}$ which forgets the convex space structure. We prove that the category $\mathbf{Std}_{Cvx}$ is the category of algebras of the $\mathcal{G}$-monad.

math.CT↗

arXiv2022

Giry algebras for standard measurable spaces

The notion of "super convex spaces" generalizes the idea of convex spaces by replacing finite affine sums with countable affine sums. Using this notion permits a very elegant approach for analysis of the Giry monad on standard measurable spaces and identifying the $\mathcal{G}$-algebras for that monad. We use Isbell duality and restrict the adjunction $\mathbf{Spec} \dashv \mathcal{O}$ to a proper subcategory of super convex spaces and separated standard measurable spaces.

math.CT↗

arXiv2022

The factorization of the Giry monad

We construct a factorization of the Giry monad through the category of convex spaces, and show that, provided that no measurable cardinals exist, probability measures can be viewed as natural transformations. Using the adjunction of this factorization, we then show the category of Giry algebras is equivalent to the category of convex measurable spaces where the $σ$-algebra structure associated with a convex space satisfies an elementary property.

math.CT↗

arXiv2022

Characterizing Giry-algebras as coseparable super convex spaces

We investigate the Eilenberg-Moore algebras for the Giry monad defined on the category of measurable spaces using super convex spaces. The category of super convex spaces has a subcategory consisting of the one point extension of the real line, and the truncated Yoneda embedding arising from the full subcategory with that one object is full, although it is not faithful. By restricting to those super convex spaces which are coseparable by the one point extension of the real line, the truncated Yoneda embedding is full and faithful. This permits the construction of a barycenter map used to factorize the Giry monad, and obtain an equivalence of categories.

math.CT↗

arXiv2022

The existence and utility of Giry algebras in probability theory

Giry algebras are barycenters maps, which are coequalizers of contractible coequalizer pairs (like any algebras), and their existence, in general, requires the measurable space be coseparated by the discrete two point space, and the hypothesis that no measurable cardinals exist. Under that hypothesis, every measurable space which is coseparated has an algebra, and the category of Giry algebras provides a convenient setting for probability theory because it is a symmetric monoidal closed category with all limits and colimits, as well as having a seperator and coseperator. This is in stark contrast to the Kleisi category of the Giry monad, which is often used to model conditional probability, which has a seperator but not much else.

math.CT↗

arXiv2017

The equivalence between the categories of Giry-algebras and convex spaces

A duality between the category of convex spaces and measurable spaces arises from the existence of the unit interval, which is an object in both these categories. The full subcategory of the category of convex spaces, consisting of just the single object, the unit interval, is both a dense and codense subcategory in the category of convex spaces. Combined with the the symmetric monoidal closed category structure of the category, one obtains the double dualization monad into the unit interval, which sends a point to the evaluation map at that point. The restriction of the codomain of the unit of this monad to the weakly averaging affine functionals is an isomorphism. Moreover, every convex space has an associated measurable space, whose σ-algebra is generated by the Boolean subobjects of that convex space. The resulting σ- algebra of that measurable space makes it a separated measurable space. These properties are used to give a proof that the category of Giry algebras is equivalent to the category of convex spaces.

math.CT↗

arXiv2016

Bayesian Inference using the Symmetric Monoidal Closed Category Structure

Using the symmetric monoidal closed category structure of the category of measurable spaces, in conjunction with the Giry monad which we show is a strong monad, we analyze Bayesian inference maps and their construction in relation to the tensor product probability. This perspective permits the inference maps to be seen as a pullback construction.

math.CT↗

arXiv2015

Categorical Probability Theory

We present a categorical viewpoint of probability measures by showing that a probability measure can be viewed as a weakly averaging affine measurable functional taking values in the unit interval which preserves limits. The probability measures on a space are the elements of a submonad of a double dualization monad on the category of measurable spaces into the unit interval, and this monad is naturally isomorphic to the Giry monad. We show this submonad is the codensity monad of a functor from the category of convex spaces to the category of measurable spaces. A theorem proving the integral operator acting on the space of measurable functions and the space of probability measures on the domain space of those functions is given using the strong monad structure of the Giry monad.

math.CT↗

arXiv2013

Bayesian machine learning via category theory

From the Bayesian perspective, the category of conditional probabilities (a variant of the Kleisli category of the Giry monad, whose objects are measurable spaces and arrows are Markov kernels) gives a nice framework for conceptualization and analysis of many aspects of machine learning. Using categorical methods, we construct models for parametric and nonparametric Bayesian reasoning on function spaces, thus providing a basis for the supervised learning problem. In particular, stochastic processes are arrows to these function spaces which serve as prior probabilities. The resulting inference maps can often be analytically constructed in this symmetric monoidal weakly closed category. We also show how to view general stochastic processes using functor categories and demonstrate the Kalman filter as an archetype for the hidden Markov model.

math.CT↗

arXiv2013

A categorical foundation for Bayesian probability

Given two measurable spaces $H$ and $D$ with countably generated $σ$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $S: H \rightarrow D$, there is a corresponding inference map $I: D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $μ: 1 \rightarrow D$, a posterior probability $\widehat{P_H}= I \circ μ$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $I$ and the process repeats with each additional measurement. The main result uses an existence theorem for regular conditional probabilities by Faden, which holds in more generality than the setting of Polish spaces. This less stringent setting then allows for non-trivial decision rules (Eilenberg--Moore algebras) on finite (as well as non finite) spaces, and also provides for a common framework for decision theory and Bayesian probability.

math.CT↗

arXiv2012

Quantifiers as Adjoint in Probability

Using the Kleisi category of the Giry monad the deterministic existential and universal quantifiers are generalized to incorporate nondeterminism. These probabilistic quantifiers are quantified over the points of the category which are probability measures.

math.PR↗