Dependency networks (Heckerman et al., 2000) provide a flexible framework for modeling complex systems with many variables by combining independently learned local conditional distributions through pseudo-Gibbs sampling. Despite their computational advantages over Bayesian and Markov networks, the theoretical foundations of dependency networks remain incomplete, primarily because their model distributions -- defined as stationary distributions of pseudo-Gibbs sampling -- lack closed-form expressions. This paper develops an information-geometric analysis of pseudo-Gibbs sampling, interpreting each sampling step as an m-projection onto a full conditional manifold. Building on this interpretation, we introduce the full conditional divergence and derive an upper bound that characterizes the location of the stationary distribution in the space of probability distributions. We then reformulate both structure and parameter learning as optimization problems that decompose into independent subproblems for each node, and prove that the learned model distribution converges to the true underlying distribution as the number of training samples grows to infinity. Experiments confirm that the proposed upper bound is tight in practice.
Figures & tables
Figure 1: Node i .
Figure 2: m-projection onto an e-flat manifold.
Figure 3: Movement of pt in genuine Gibbs sampling with sequential scan.
Figure 4: Movement of pt in pseudo-Gibbs sampling with sequential scan.
Figure 5: Movement of pt in pseudo-Gibbs sampling with random scan.
This paper introduces General Proximal Flow Networks (GPFNs), a generalization of Bayesian Flow Networks that broadens the class of admissible belief-update operators. In Bayesian Flow Networks, each update step is a Bayesian posterior update, which is equivalent to a proximal step with respect to the Kullback-Leibler divergence. GPFNs replace this fixed choice with an arbitrary divergence or distance function, such as the Wasserstein distance, yielding a unified proximal-operator framework for iterative generative modeling. The corresponding training and sampling procedures are derived, establishing a formal link to proximal optimization and recovering the standard BFN update as a special case. Empirical evaluations confirm that adapting the divergence to the underlying data geometry yields measurable improvements in generation quality, highlighting the practical benefits of this broader framework.
Probabilistic inference in high-dimensional Bayesian networks is difficult because exact manipulation of the joint distribution scales exponentially with network size. We propose a decomposition framework based on directed convex subgraphs and introduce a minimal d-decomposition tree. Together, they provide a principled alternative to classical junction-tree constructions. The proposed framework represents the joint distribution by lower-dimensional sub-models that can be learned and stored separately. This decomposition reduces computational cost and naturally enables parallel computation. Based on a minimal d-decomposition tree, we further develop two parallel algorithms for parameter estimation and probabilistic inference. Experiments show that the proposed method substantially improves computational efficiency over junction-tree methods while maintaining inference accuracy, especially for low-dimensional queries.
Pei Heng, Xinyi Hu, Yi Sun
Institute of Statistics and Data Science, Xinjiang University of Finance and Economics, Urumqi, China
We study the task of learning the structure of a d-sparse Gaussian graphical model on n variables from a single trajectory of Glauber dynamics. Beyond algorithmic considerations, many applications present temporally correlated observations rather than i.i.d.\ samples. In the classical i.i.d.\ setting, under comparably general sparsity and minimum edge-strength assumptions, sublinear-in-n sample guarantees are known, but achieving them in polynomial-time remains open. Motivated in part by this gap, we give a polynomial-time algorithm that recovers the conditional-independence graph from a single Glauber trajectory, with a trajectory-length guarantee that does not depend on the mixing time. Technically, our algorithm has three components. First, we estimate the conditional variances and rescale the trajectory to reduce to the unit-diagonal case, without changing the underlying graph. Second, we design a local edge test that extracts adjacency information from short update windows by isolating pairwise influence. Third, we aggregate these local statistics using a robust median-based estimator, and prove accuracy despite temporal dependence arising from a single trajectory.