Functorial semantics and diagrammatic languages for statistical theories and models
Abstract
Building data-driven models has become an essential activity across academia, industry, and government, including in information and cybersecurity. Although the library of statistical methods and algorithms is ever growing, the logical foundations of statistics have hardly evolved since Wald systematized statistical decision theory in the early twentieth century. Recent research in data science has neglected foundational aspects. Yet the discipline faces serious challenges to do with the replicability, generalizability, and trustworthiness of data-driven models. Ultimately, such problems limit our ability to learn from observation and experiment, and to make safe and reliable decisions in the face of uncertainty. These challenges will not be overcome simply by creating more powerful and efficient statistical methods; rather, they concern the logic of science and the ways in which models are connected to data and to each other. To create a reliable science of data, as intended by the phrase data science, we must revisit its foundations and build new mathematical and computational tools that reflect its logical structure. The issues facing data science stem from a fundamental mismatch between practice and theory- science is supposed to proceed by steady accumulation and integration of empirical research, yet existing statistical theory is almost exclusively concerned with the analysis of a single model. Thus, we seek a mathematical framework for statistics that can specify not just one model in isolation but the hierarchy of models involved in connecting theory to empirical data, and that can aid in navigating the network of possible models. Our framework will be based on modern mathematical tools from categorical logic and the synthetic theory of probability.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 29, 2024
- Source ID
- FA95502310133
Entities
People
- Evan Patterson
Organizations
- Air Force Office of Scientific Research
- United States Air Force