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

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).