A Flat Connection: The Pooling Factor and the Geometry of Centring in Hierarchical MCMC

Statistics > Computation [Submitted on 17 Jun 2026] Title:A Flat Connection: The Pooling Factor and the Geometry of Centring in Hierarchical MCMC View PDF HTML (experimental)Abstract:Standard MCMC diagnostics ($\hat{R}$, effective sample size, divergence counts) detect whether a chain has mixed, but not why it has not. We ask whether the centring/non-centring obstruction in hierarchical models has a geometric cause beyond the metric. The joint parameter space is a fiber bundle (hyperparameters the base, group-level parameters the fibers), and the Fisher information metric induces an Ehresmann connection $A = -G_{FF}^{-1}G_{BF}$; the natural hypothesis is that the obstruction is its curvature, felt by the sampler as holonomy. We prove this false. The connection is flat for any smooth hierarchical posterior, not only the Gaussian case, because its horizontal leaves are the level sets of the fiber score $\partial_\alpha \log p$: there is no geometric obstruction above the metric. What remains is statistical, not geometric, and the flat connection identifies it as a single quantity: the conditional dependence of fiber on base, governed per group by the prior fraction $\pi_j$, the classical pooling factor. From it the framework recovers the established picture, that prior-dominated groups mix slowly and that the optimal per-group non-centring weight follows in closed form, and a simulation study separates this base-fiber coupling from the funnel, a distinct base-space pathology, by their opposite dependence on the hierarchical variance. A direct attribution test confirms that NUTS does not transport the fiber: the chain-level footprint is excess conditional autocorrelation in prior-dominated groups, exactly as $\pi_j$ predicts. Genuine, even rotational, curvature does appear, but only for connections built from a sampler's working metric (a fixed mass matrix), where holonomy re-enters as an algorithmic rather than geometric phenomenon. The prior-fraction diagnostic is distributed as the R package fibr, with the geometric methods as accompanying reproduction code. Current browse context: stat.CO References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.