Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective
We study the Collatz total stopping time $τ(n)$ over $n\le 10^7$ from a probabilistic machine learning viewpoint. Empirically, $τ(n)$ 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 $τ(n)$ 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 yield...
| Authors | Nicolò Bonacorsi, Matteo Bordoni |
|---|---|
| Primary category | stat.ML |
| Published | 4 March 2026 |
| Last updated at source | 4 March 2026 |
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