Books
Books by Vadim Sokolov and coauthors. Each book is free to read as a PDF, and its source and code are on GitHub.
Bayes, Artificial Intelligence, and Deep Learning Foundations of Data Science
Nick Polson and Vadim Sokolov. 2026. Edition 1.
A textbook that develops Bayesian reasoning, statistical learning and deep learning as one probabilistic framework, from Bayes’ rule and decision theory to neural networks, transformers and AI agents, with code in R and Python.
On the Collatz Conjecture Martingales, Momentum and Continued Fractions
Nick Polson and Vadim Sokolov. 2026. Edition 1.
Why the Collatz (3x+1) conjecture resists proof: the map as a random walk, an optimizer and a Diophantine problem; proofs that fixed horizon monotone Lyapunov certificates and finite residue sieves cannot settle it and the cycle inequality cannot exclude all cycles; a momentum method from log2 3.
On Langlands Conjectures Probabilistic Methods
Nick Polson and Vadim Sokolov. 2026. Edition 1.
What probability can and cannot contribute to the Langlands programme: Sato Tate laws, random walks on trees and isogeny graphs, Thorin measures and generalized gamma convolutions as reformulations of the Riemann hypothesis, and Bayesian identification of Sato Tate groups, with code and data.
On Hilbert’s 6th Problem Probability, Mechanics and Bayesian Computation
Nick Polson and Vadim Sokolov. 2026. Edition 1.
Hilbert’s sixth problem from Kolmogorov’s axioms and the Boltzmann equation to the Navier Stokes equations, and the inverse passage: Bayesian estimation on the solution manifold of a differential equation, where the adjoint evolution of the data is a sufficient statistic for linear equations.
Computing the Riemann Hypothesis, Volume I Representations of Xi, Thorin Measures and the Reciprocal Xi Function
Nicholas G. Polson and Vadim Sokolov. 2026. Edition 2.
The theory: representations of Riemann’s Xi function, the Thorin measure whose positivity is equivalent to the Riemann Hypothesis, and the reciprocal Xi function as a generalised gamma convolution.
Computing the Riemann Hypothesis, Volume II Cauchy Deconvolution and Certified Computation
Nicholas G. Polson and Vadim Sokolov. 2026. Edition 2.
The computations: Riemann’s Xi function in ball arithmetic, the Thorin measure at finite resolution, Cauchy deconvolution, the de Bruijn Newman flow, the barrier method and the Thorin phase diagram.
P = RP A Monte Carlo Program for P versus NP
Nick Polson and Vadim Sokolov. 2026. Edition 1.
N queens counting as a rare event and a quantile integral, turned into a Monte Carlo program for P versus NP: NP = RP is shown equivalent to sampling queens completions, promise RP = promise P, which implies P = RP, is restated as a hitting set problem, with exact computations on the queens.
Clinical Trials in Practice From Foundations to AI Agents
Alexandra Sokolova and Vadim Sokolov. 2026. Edition 1.
Clinical trials as science and as regulated operations: design and statistics, the daily work of a Phase III study from site selection to the trial master file, regulatory submission, and what AI agents may and may not do in regulated research.
On Statistical Methods for Genetics Fisher, Ewens and the Bayesian Programme
Nicholas G. Polson. 2026. Edition 1.
Statistical theory of population genetics: an exact form of Fisher’s Fundamental Theorem, shrinkage estimates of the rate of adaptation, the Ewens sampling formula and its rigidity, rational conjugacy, and sufficiency for learned coalescent inference.
