Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective
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| Format: | Preprint |
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2026
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| _version_ | 1866912943100133376 |
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| author | Bonacorsi, Nicolò Bordoni, Matteo |
| author_facet | Bonacorsi, Nicolò Bordoni, Matteo |
| contents | 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 $τ(n)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_04479 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective Bonacorsi, Nicolò Bordoni, Matteo Machine Learning Probability Statistics Theory Applications 62F15(Primary), 60G40 62J12 65C05 68T05 (Secondary) G.3; I.2.6; I.6.8 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 $τ(n)$. |
| title | Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective |
| topic | Machine Learning Probability Statistics Theory Applications 62F15(Primary), 60G40 62J12 65C05 68T05 (Secondary) G.3; I.2.6; I.6.8 |
| url | https://arxiv.org/abs/2603.04479 |