Generalized Hierarchical Bayesian Segmentation with Irregular Designs, Multi-Sequence Hierarchies, and Grouped/Latent-Group Designs

Bayesian change-point and segmentation models provide uncertainty-aware piecewise-constant representations of ordered data, but exact inference is often limited to narrow likelihood classes, single sequences, or index-uniform designs. We present \texttt{BayesBreak}, a modular offline Bayesian segmentation framework that separates local block scoring from global inference: each candidate block supplies a marginal likelihood and any needed moment numerators, while a dynamic program combines these scores to compute posteriors over segment counts, boundaries, and latent signals. For weighted exponential-family likelihoods with conjugate priors, block evidences and posterior moments are available in closed form from cumulative sufficient statistics, enabling exact sum-product inference for $p(y\mid k)$, $p(k\mid y)$, boundary marginals, and Bayes regression curves. We distinguish these summaries from the \emph{joint} MAP segmentation, recovered by a separate max-sum recursion. BayesBreak supports design-aware partition priors for irregular observations, exact pooling across replicates with shared boundaries, and latent-template mixtures with exact EM updates. For non-conjugate GLM blocks, the same DP layer can use deterministic local approximations such as Laplace, variational methods, EP, or quadrature. We prove a posterior-odds stability bound: uniform per-block log-evidence error $\varepsilon$ perturbs $k$-odds and boundary-odds by at most $(k+k')\varepsilon$ and $2k\varepsilon$. Validation includes synthetic recovery, calibration, and scaling experiments, plus four real-data illustrations: well-log geology, array-CGH copy number, equity-return volatility, and CpG-atlas methylation.