[Paper Review] The algebra and machine representation of statistical models

Evan Patterson|arXiv (Cornell University)|Jun 16, 2020

Statistics Education and Methodologies179 references4 citations

TL;DR

This dissertation introduces a categorical algebraic framework to formalize statistical models as mathematical structures akin to logical theories, enabling machine representation through category theory. It unifies statistical inference with computational workflows via a software system that captures semantic meaning across Python and R, advancing reproducible and composable data science through formal, language-agnostic modeling of scientific knowledge.

ABSTRACT

As the twin movements of open science and open source bring an ever greater share of the scientific process into the digital realm, new opportunities arise for the meta-scientific study of science itself, including of data science and statistics. Future science will likely see machines play an active role in processing, organizing, and perhaps even creating scientific knowledge. To make this possible, large engineering efforts must be undertaken to transform scientific artifacts into useful computational resources, and conceptual advances must be made in the organization of scientific theories, models, experiments, and data. This dissertation takes steps toward digitizing and systematizing two major artifacts of data science, statistical models and data analyses. Using tools from algebra, particularly categorical logic, a precise analogy is drawn between models in statistics and logic, enabling statistical models to be seen as models of theories, in the logical sense. Statistical theories, being algebraic structures, are amenable to machine representation and are equipped with morphisms that formalize the relations between different statistical methods. Turning from mathematics to engineering, a software system for creating machine representations of data analyses, in the form of Python or R programs, is designed and implemented. The representations aim to capture the semantics of data analyses, independent of the programming language and libraries in which they are implemented.

Motivation & Objective

• . To establish a rigorous algebraic foundation for statistical models using categorical logic.
• . To formalize statistical theories as algebraic structures with morphisms that represent methodological relationships.
• . To design and implement a software system that generates machine-readable, semantics-preserving representations of data analyses in Python and R.
• . To bridge the gap between statistical methodology and scientific theory by modeling the hierarchy of scientific knowledge.
• . To enable machine-assisted reasoning, verification, and composition of data science workflows through formal, computable representations.

Proposed method

• . Uses category theory—particularly colored PROPs and monoidal categories—to model statistical theories and their morphisms.
• . Defines statistical models as algebras over theories, with parameterized families of distributions forming the core structure.
• . Introduces a formal ontology for data science workflows, encoding semantics independent of programming language or library.
• . Employs program analysis techniques to extract semantic meaning from Python and R code, transforming it into structured, machine-processable representations.
• . Constructs a higher-categorical structure for statistical theories, where composition and product operations are non-strict, suggesting future work in higher-dimensional formalism.
• . Applies the framework to unify experimental design, model construction, and inference within a single formal system grounded in category theory.

Experimental results

Research questions

• RQ1 . How can statistical models be systematically represented as algebraic structures within a formal category-theoretic framework?
• RQ2 . What is the role of morphisms between statistical models in formalizing relationships between different statistical methods?
• RQ3 . How can data analysis workflows in Python and R be transformed into machine-readable, semantics-preserving representations independent of implementation details?
• RQ4 . In what way can statistical theories be embedded within a hierarchy of scientific models to support generalizability and theory propagation?
• RQ5 . How might formal, computable representations of statistical models and data analyses improve reproducibility, verification, and automation in data science?

Key findings

• . The dissertation successfully formalizes statistical models as algebras over colored PROPs, providing a precise categorical framework for statistical inference.
• . It demonstrates that morphisms between statistical models can represent methodological transformations, such as model reduction or parameterization changes.
• . A working software system is implemented that translates Python and R data analysis code into formal, semantic representations that preserve meaning across languages.
• . The framework enables the representation of experimental designs and models of experiments as part of a unified, formal knowledge structure.
• . The work reveals a fundamental mismatch between the current statistical paradigm—centered on null hypothesis testing—and the actual structure of scientific knowledge, advocating for a more holistic, hierarchical modeling approach.
• . The proposed system supports the long-term vision of machine-assisted scientific reasoning by enabling formal, composable, and verifiable data science workflows.