🤖 AI Summary
This work addresses the lack of a unified differentiable framework for inference and estimation in linear Gaussian Bayesian networks, which traditionally rely on ad hoc derivations tailored to specific graph structures. The authors propose a novel, unified differentiable covariance calculus centered on the joint covariance matrix, applicable to arbitrary directed acyclic graphs and structured parameterizations. Key tasks—including conditioning, d-separation testing, and maximum likelihood estimation with latent variables—are reformulated as linear algebraic operations on covariances, enabling gradient computation via a single backward pass. By integrating K-recursive covariance mappings, automatic differentiation, and information geometry, the framework naturally subsumes the Slepian–Bangs formula and the Cramér–Rao bound. Its correctness and consistency are validated on state-space models and their extensions with skip connections.
📝 Abstract
Linear Gaussian Bayesian networks, equivalently linear Gaussian structural equation models, recur across statistics, control, and communications; in the vector-valued setting that motivates this work, their nodes are vectors and their edges are matrices. Every quantity of interest is a function of sub-blocks of the joint covariance, which is itself a classical, differentiable map (the K-recursion) from the local edge and innovation parameters. Yet the resulting inference and estimation tasks are usually derived and implemented separately, per task and per topology. Taking this covariance chart as a single backend, we build on it a unified, differentiable covariance calculus in which each task reduces to a few linear-algebra primitives on the one covariance, and automatic differentiation returns every gradient in a single backward sweep, over arbitrary vector-valued directed acyclic graphs and parametrizations, including tied and structured ones. The calculus covers conditioning, conditional-independence testing through mutual information, maximum-likelihood estimation with hidden nodes, and the Slepian--Bangs Fisher information with the local identifiability and Cramér--Rao reliability it induces. It is validated on a linear Gaussian state-space model and a skip-connected (non-chain) extension against the Kalman recursions, d-separation, and the Cramér--Rao bound.