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

arXiv: https://arxiv.org/abs/2603.04479 · HTML: https://mlsanity.com/paper/2603.04479/

## Key facts

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- First posted on arXiv 4 Mar 2026.

## Abstract (arXiv)

We study the Collatz total stopping time τ(n) over n≤ 107 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=2K(m)m' with m' odd and K(m)=v2(3m+1)≥ 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 mbmod 8 markedly improves the generator's distributional fit, indicating that low-order modular structure is a key driver of heterogeneity in τ(n).