Random Variables, Conditional Independence and Categories of Abstract Sample Spaces (2503.02477v2)

Abstract: 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 been successfully axiomatized, and widely generalized, using the notion of Markov category, the categorical semantics of the random variable picture remain less clear. Simpson's probability sheaves are a recent approach, in which probabilistic concepts like random variables are allowed vary over a site of sample spaces. Simpson has identified rich structure on these sites, most notably an abstract notion of conditional independence, and given examples ranging from probability over databases to nominal sets. We aim bring this development together with the generality and abstraction of Markov categories: We show that for any suitable Markov category, a category of sample spaces can be defined which satisfies Simpson's axioms, and that a theory of probability sheaves can be developed purely synthetically in this setting. We recover Simpson's examples in a uniform fashion from well-known Markov categories, and consider further generalizations.

• The paper presents a categorical framework unifying random variable and probability channel perspectives on probability theory using Markov categories and abstract sample spaces.
• It defines categories of abstract sample spaces within any Markov category, satisfying conditional independence axioms, and develops a synthetic theory of probability sheaves.
• This framework clarifies stochastic computations and has implications for statistical modeling, probabilistic programming, and practical applications like Bayesian inversion.

Overview of "Random Variables, Conditional Independence and Categories of Abstract Sample Spaces"

The paper "Random Variables, Conditional Independence and Categories of Abstract Sample Spaces" by Dario Stein presents a formal framework unifying two conceptualizations of probability theory: the traditional perspective emphasizing random variables and the alternative approach centered on probability channels, or Markov kernels. The former approach considers random variables as functions on a sample space, while the latter focuses on transitions between probabilistic distributions. Stein's work explores the categorical semantics of these frameworks, aiming to bridge the gap between Simpson's notion of probability sheaves over varying sample spaces and the rigorously axiomatized channel-based picture using Markov categories.

Reconciling Random Variables and Probability Channels

The paper establishes a categorical structure where concepts related to random variables can be consistently synthesized within the framework of Markov categories. The author demonstrates that it is possible to construct categories of sample spaces adhering to Simpson's axioms of conditional independence, which he articulated as "independent pullbacks." This integration allows probability sheaves to be defined synthetically without relying on specifics of individual models, thereby maintaining both generality and abstraction.

Major Contributions

Several key contributions underscore the significance of this research:

1. Unified Framework for Sample Spaces:
   • For any Markov category $C$, Stein defines a category of sample spaces $S(C)$ satisfying Simpson's axioms, establishing a robust foundation for categorical probability theory encompassing both distribution-based and random variable-based representations.
2. Probability Sheaves and Conditional Independence:
   • The paper develops a synthetic theory of probability sheaves over these sample spaces, expanding Simpson's model-specifically rooted examples into a broader, more universal framework through synthetic constructions.
3. Recuperation and Generalization of Known Models:
   • Known examples from Markov categories such as FinStoch and BorelStoch are shown to naturally exemplify the new constructs, but the work also extends to less commonly synthesized domains—such as Gaussian probability and fresh name generation—illustrating their utility under categorical probability paradigms.
4. Use of Markov Categories:
   • Markov categories encapsulate nondeterministic processes, enabling stochastic, measure-theoretic, and potentially computational interpretations under a common categorical umbrella, enriching the language to compose and analyze random, or probabilistic, structures.

Theoretical and Practical Implications

The implications of Stein’s work on theoretical aspects of probability theory are profound. By employing a categorical perspective, the work clarifies the relationship between stochastic computations expressed via random variables and those through probability channels. This clearer understanding could impact fields such as statistical modeling and probabilistic programming, where conditional independence and random variable semantics are pivotal.

Practically, realizing categorical probability in frameworks that are as convenient and expressive as possible creates new pathways for implementing robust probabilistic reasoning in areas such as machine learning, data analysis, and computer science. For instance, a solid theory of Bayesian inversion emerges in this setting as a fundamental operation, which is central to updating and conditioning in learning models.

Future Directions

In conclusion, the research sets the stage for further advancements in categorical probability theory. Subsequent efforts might focus on exploring the logical underpinnings of atomic sheaf toposes, developing a comprehensive treatment of expectation and conditional expectation in this categorical setup, and extending applications to domains beyond those immediately covered by the paper—like conditional independence structures relating to probabilistic separation logics and nominal techniques.

Overall, Stein’s integration of Markov categories with Simpson’s probability sheaves promises to catalyze continuing inquiry into the categorical structures that underlie complex probabilistic systems, offering mathematicians and theoretical computer scientists alike a well-structured lens through which to examine and expand upon existing foundations.