Markov’s Calculus of Probabilities: The Foundation of Modern AI

2026-07-11 · Source: Valeriy’s Substack · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Intermediate, medium

Summary

Andrei Andreyevich Markov's 1900 work, "The Calculus of Probabilities," established the mathematical foundations for modern artificial intelligence, predating his famous Markov chains by six years. This foundational text, now available in English, codified the logic used by autoregressive language models to predict tokens. Markov, a key figure in the Russian School of Probability, sought to transform probability into a rigorous science, defining events with precision and demonstrating how observer information impacts independence. He developed generating functions to scale calculations for repeated trials, encoding discrete probability states as polynomial coefficients. For infinite trials, he derived a version of Laplace's theorem, using Stirling's approximation to prove Bernoulli's theorem, the law of large numbers. This proof, demonstrating that variance shrinks with increased samples, is crucial for neural networks to learn, allowing billions of weights to converge on stable predictions despite high-dimensional language complexity.

Key takeaway

For AI Scientists and Machine Learning Engineers developing or training large models, understanding Markov's foundational probability calculus is crucial. Your models' ability to learn and make stable predictions relies directly on the law of large numbers, ensuring chaotic individual token deviations average out at scale. Revisit these core principles to deepen your grasp of why massive datasets are indispensable for robust neural network convergence and reliable autoregressive predictions.

Key insights

Markov's 1900 probability calculus provides the rigorous mathematical bedrock for modern AI's statistical learning and prediction.

Principles

• Independence is observer-dependent, not an inherent physical property.
• Rigorous formal proof is essential for robust statistical models.
• Millions of random events converge into predictable certainty.

Method

Markov encoded discrete probability states as polynomial coefficients using generating functions for repeated trials, and used continuous approximations (Laplace's theorem, Stirling's approximation) for infinite trials.

In practice

• Use generating functions to scale discrete probability calculations.
• Apply the law of large numbers for neural network weight convergence.
• Consider observer's information when defining variable independence.

Topics

• Probability Theory
• Markov Chains
• Autoregressive Models
• Law of Large Numbers
• Generating Functions
• AI Foundations

Best for: AI Scientist, Machine Learning Engineer, AI Student

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Editorial summary, takeaway, and curation by AIssential. Original article published by Valeriy’s Substack.