[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82829-en":3,"doc-seo-82829-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82829,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","A Differentiable Covariance Calculus for Linear Gaussian Bayesian Networks","Linear Gaussian Bayesian networks, equivalent to linear Gaussian structural equation models, are characterized by a joint covariance obtained via a differentiable K-recursion from local edge and innovation parameters. Existing inference and estimation are often derived separately per task and topology. This work unifies them by building a differentiable covariance calculus where each task reduces to block-matrix primitives on a single covariance chart, enabling end-to-end gradients for arbitrary vector-valued DAGs via automatic differentiation. The calculus supports conditioning, conditional-independence via mutual information, maximum-likelihood with hidden nodes, and Slepian–Bangs Fisher information with identifiability and Cramér–Rao reliability. Validation uses a linear Gaussian state-space model and a skipconnected extension.","A Differentiable Covariance Calculus for Linear Gaussian Bayesian Networks  \nTadashi Wadayama  \nDepartment of Computer Science, Nagoya Institute of Technology, Nagoya 466-8555, Japan  \n[Email: wadayama@nitech.ac.jp](Email: wadayama@nitech.ac.jp)  \narXiv :2607 .04578v 1 [ cs .IT] 6 Jul 2026  \nAbstract—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 Cramr–Rao reliability it induces. It is validated on a linear Gaussian state-space model and a skipconnected (non-chain) extension against the Kalman recursions, d-separation, and the Cramr–Rao bound.  \nKeywords—Linear Gaussian Bayesian networks, structural equation models, covariance recursion, automatic differentiation, conditional independence, Fisher information, identifiability, Cram r– Rao bound, forward sampling.  \nI. INTRODUCTION  \nLinear Gaussian models on directed acyclic graphs (DAGs) recur across statistics, control, and communications. They are linear structural equation models (SEMs) and Gaussian graphical models in statistics [7], [3], linear state-space models in control and signal processing, and cascaded linear channels with additive Gaussian noise in communications, as in MIMO transceivers, multi-hop amplify-and-forward relays, and cooperative sensor arrays. In each case a node carries a Gaussian variable, an edge applies a linear map, and an independent Gaussian innovation enters at every non-root node.  \nIn the applications that motivate this work, the nodes are vectors and the edges are matrices: an antenna array, a latent feature vector, or a dynamical state at each node, and a MIMO channel or processing matrix on each edge. The joint distribution of such a network is zero-mean Gaussian and is therefore determined entirely by its covariance, and essentially every quantity of interest (marginals and conditionals, mutual information, likelihoods, and Fisher information) is a function of sub-blocks of that covariance. The covariance is, in turn, a function of the local conditional parameters: the edge matrices and the innovation covariances. Mapping the local parameters  \nto the global covariance is thus the computational hub on which all downstream analysis rests.  \nThis local-to-global covariance map is classical; its evaluation as a differentiable, inverse-free forward operator, together with its relation to the classical Gaussian-network, path-analysis, and state-space literature, is developed in the companion paper [1] . We take that covariance backend as given: the object of this paper is the inference and estimation framework built on it, not the covariance recursion itself.  \nWhat is missing is not another formula but an organizing one. Taken individually, each downstream operation (conditioning by a Schur complement, a log-determinant mutual information, a node-wise regression for maximum likelihood) is elementary; assembling all of them, over a","cbCaij3Zg9oHHKvT","https://ap.wps.com/l/cbCaij3Zg9oHHKvT","pdf",667568,4,1,11,"English","en",105,"# Introduction\n## Problem motivation and gap\n## Differentiable covariance chart and K-recursion backend\n## Contributions: unified inference and estimation","[{\"question\":\"What is the central idea of this work on linear Gaussian Bayesian networks?\",\"answer\":\"It treats the local-to-global covariance mapping as a differentiable covariance chart (via the K-recursion) and builds a unified inference and estimation calculus on top of it.\"},{\"question\":\"How does the method compute gradients across tasks and topologies?\",\"answer\":\"By using reverse-mode automatic differentiation on a smooth computation graph composed of matrix products, sums, and transposes, producing gradients with respect to all edge parameters in a single backward sweep.\"},{\"question\":\"Which inference and estimation capabilities are covered by the covariance calculus?\",\"answer\":\"Conditioning via Schur complements, mutual-information-based conditional-independence testing, maximum-likelihood estimation with hidden nodes, and Fisher information for local identifiability and Cramér–Rao reliability.\"}]",1784183249,28,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-differentiable-covariance-calculus-for-linear-gaussian-bayesian-networks","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/a-differentiable-covariance-calculus-for-linear-gaussian-bayesian-networks/82829/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the central idea of this work on linear Gaussian Bayesian networks?","Question",{"text":75,"@type":76},"It treats the local-to-global covariance mapping as a differentiable covariance chart (via the K-recursion) and builds a unified inference and estimation calculus on top of it.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method compute gradients across tasks and topologies?",{"text":80,"@type":76},"By using reverse-mode automatic differentiation on a smooth computation graph composed of matrix products, sums, and transposes, producing gradients with respect to all edge parameters in a single backward sweep.",{"name":82,"@type":73,"acceptedAnswer":83},"Which inference and estimation capabilities are covered by the covariance calculus?",{"text":84,"@type":76},"Conditioning via Schur complements, mutual-information-based conditional-independence testing, maximum-likelihood estimation with hidden nodes, and Fisher information for local identifiability and 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