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

TL;DR

This paper models the Collatz total stopping time using Bayesian hierarchical regression and a mechanistic probabilistic generator, revealing modular structure influences its variability and improving prediction accuracy.

Contribution

Introduces a Bayesian hierarchical Negative Binomial model and a mechanistic stochastic generator for Collatz stopping times, highlighting the role of modular arithmetic in heterogeneity.

Findings

The NB2-GLM outperforms the mechanistic generator in predictive likelihood.

Conditioning on residue classes improves the generator's fit.

Modular structure significantly influences stopping time variability.

Abstract

We study the Collatz total stopping time $\tau(n)$ over $n\le 10^7$ from a probabilistic machine learning viewpoint. Empirically, $\tau(n)$ is a skewed and heavily overdispersed count with pronounced arithmetic heterogeneity. We develop two complementary models. First, a Bayesian hierarchical Negative Binomial regression (NB2-GLM) predicts $\tau(n)$ from simple covariates ($\log n$ and residue class $n \bmod 8$), quantifying uncertainty via posterior and posterior predictive distributions. Second, we propose a mechanistic generative approximation based on the odd-block decomposition: for odd $m$, write $3m+1=2^{K(m)}m'$ with $m'$ odd and $K(m)=v_2(3m+1)\ge 1$; randomizing these block lengths yields a stochastic approximation calibrated via a Dirichlet-multinomial update. On held-out data, the NB2-GLM achieves substantially higher predictive likelihood than the odd-block generators. Conditioning the block-length distribution on $m\bmod 8$ markedly improves the generator's distributional fit, indicating that low-order modular structure is a key driver of heterogeneity in $\tau(n)$.