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

• URL: http://arxiv.org/abs/2603.04479v1
• Date: Wed, 04 Mar 2026 17:16:49 GMT
• Title: Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective
• Authors: Nicolò Bonacorsi, Matteo Bordoni,
• Abstract summary: Empirically, $(n)$ is a skewed and heavily overdispersed count with pronounced arithmetic heterogeneity.<n>We develop two complementary models.<n>A mechanisticgenerative approximation based on the odd-block decomposition yields a calibrated approximation via a Dirichlet-multinomial update.<n>On held-out data, the NB2-GLM achieves substantially higher predictive likelihood than the odd-block generators.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 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 $τ(n)$.

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