Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective

Researchers Nicolò Bonacorsi and Matteo Bordoni have published a novel paper, 'Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective,' applying advanced statistical techniques to one of mathematics' most famous unsolved problems. The study analyzes the total stopping time—the number of steps to reach 1—for integers up to 10 million, treating it as a skewed and overdispersed count variable. The core announcement is the development of two complementary probabilistic models to predict and understand this erratic behavior, moving beyond pure number theory into data-driven inference.

The first model is a Bayesian hierarchical Negative Binomial regression (NB2-GLM) that predicts stopping time using simple covariates like log(n) and n mod 8, providing full posterior uncertainty. The second is a mechanistic generative model based on randomizing the 'odd-block' lengths in the Collatz sequence, calibrated with a Dirichlet-multinomial update. A key finding is that the regression model achieved substantially higher predictive likelihood on held-out data. Crucially, conditioning the block-length distribution on the residue class modulo 8 significantly improved the generator's fit, revealing that low-order modular arithmetic is a primary driver of the stopping time's notorious heterogeneity. This work demonstrates how modern probabilistic ML can shed new light on classic, hard computational problems.