{"id":10959,"date":"2026-03-06T07:02:29","date_gmt":"2026-03-06T07:02:29","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2026\/03\/06\/2603-04479\/"},"modified":"2026-03-06T07:02:29","modified_gmt":"2026-03-06T07:02:29","slug":"2603-04479","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2026\/03\/06\/2603-04479\/","title":{"rendered":"Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective"},"content":{"rendered":"

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

\n
<\/BR>
\n
\n \t

\n
<\/BR><\/p>\n

arXiv:2603.04479v1 Announce Type: new
\nAbstract: We study the Collatz total stopping time $tau(n)$ over $nle 10^7$ from a probabilistic machine learning viewpoint. Empirically, $tau(n)$ 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 $tau(n)$ 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 $mbmod 8$ markedly improves the generator’s distributional fit, indicating that low-order modular structure is a key driver of heterogeneity in $tau(n)$.<\/div>\n

\t

\n
<\/BR>
\n Nicol`o Bonacorsi, Matteo Bordoni
\n \t

\n
<\/BR>
\nGo to original source<\/a>
\n \t

\n
<\/BR><\/p>\n","protected":false},"excerpt":{"rendered":"

Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective arXiv:2603.04479v1 Announce Type: new Abstract: We study the Collatz total stopping time $tau(n)$ over $nle 10^7$ from a probabilistic machine learning viewpoint. Empirically, $tau(n)$ is a skewed and heavily overdispersed count with pronounced arithmetic heterogeneity. We develop two complementary models. First, a Bayesian hierarchical […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,113,420,190,181,112,191],"tags":[3105,4887,4886],"class_list":["post-10959","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-math-pr","category-math-st","category-stat-ap","category-stat-ml","category-stat-th","tag-block","tag-odd","tag-tau"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/10959"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=10959"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/10959\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=10959"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=10959"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=10959"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}