Date
Publisher
arXiv
The computational revolution is advancing at an unprecedented pace. The combination of proof-assistant technologies and generative AI tools has recently enabled the solution of complex problems in pure mathematics at a scale that seemed unattainable only a few years ago. However, these technologies have not yet become standard tools in the development of statistical theory. In this paper, we do not present new theoretical results. Instead, we discuss five case studies involving classical problems in statistics and describe how they can be analyzed using a machine proof-checking. Our goal is not to propose a definitive workflow, but to stimulate reflection on how these technologies may transform theoretical research and advanced statistical education. We focus on two main aspects. First, statistical theory often compresses substantial mathematical content into expressions such as "under the usual regularity conditions". Formalization in a machine-verifiable language forces each assumption to be explicit, reveal hidden dependencies, and provide a deeper understanding of the formalized objects. Second, we argue that the statistical community could benefit from a collaborative effort to build repositories of formalized axioms, definitions, and theorems, supporting more precise and reliable theoretical developments. Finally, we discuss the role of these tools in graduate education. Just as high-level programming languages revolutionized empirical research by enabling rapid experimentation and prototyping, machine-assisted formalization may introduce a new paradigm for the development, verification, and communication of statistical theory.
Study at a glance
- Question: How does formalizing proofs with computers and AI tools change statistical theory and advanced education?
- Setting and data: Five classical theoretical statistical problems formalized in Lean 4; no empirical datasets analyzed.
- Method: Formal mathematical verification and case-study gap analysis using a proof assistant and generative AI models.
Why it matters
Proof assistants and generative AI are transforming pure mathematics but remain rare in statistical theory. Understanding how machines check proofs can clarify compressed technical assumptions and reshape advanced statistical research and education.
Key findings
- Machine formalization identified 32 mathematical conditions across five statistical cases, linking each condition to specific proof steps that informal textbook citations usually compress.
- A formal proof revealed that the pointwise strong law of large numbers fails to justify evaluating averages at random intermediate points without extra uniform conditions.
- Proof assistants verify logical derivations from stated premises but cannot check whether a formal mathematical statement correctly represents the intended claim.
Summary drafted with AI from the full paper and may contain errors. See the full paper for details.
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