[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86429-en":3,"doc-seo-86429-105":29,"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":11,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},86429,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Geometric Causal Models","Scientists seek causal conclusions from structured data that violates i.i.d. assumptions, including spatial, network, and molecular measurements. This work introduces geometric causal models (GCMs), a symmetry-based framework for causal inference with dependent observations. By expressing invariances via group theory, the document establishes when causal mechanisms can be identified and estimated. It uses ergodic theory for amenable groups and combines geometric deep learning with scalable Bayesian inference, recovering i.i.d. and do-calculus in special cases and enabling new DNA-symmetric models.","arXiv :2607 .05153v2 [ stat .ML] 11 Jul 2026  \nGeometric Causal Models  \nEli N. Weinstein∗, David M. Blei†  \nJuly 14, 2026  \nAbstract  \nScientists often seek to draw causal inferences from structured data that is not independently and identically distributed, such as spatial data, network data, or molecular data. We develop geometric causal models (GCMs), a framework for causal inference from dependent data that exploits underlying symmetries of the data generating process. For example, in spatial data, we consider processes that are symmetric under translations, or in graph data, symmetric under permutations of the nodes. We show how symmetries, formalized with group theory, can enable causal identification and estimation. We deploy ergodic theory for amenable groups to establish identification, and combine geometric deep learning with scalable Bayesian inference for estimation. We recover i.i.d. causal models and do-calculus when the data is a sequence and the symmetry is permutation equivariance, and find novel types of causal models when we use alternate structures and symmetries. As an example, we construct a causal model that satisfies the symmetries of DNA. This GCM enables new estimators for the effects of genetic variation, combining deep functional genomics models to describe outcomes and DNA language models to describe propensities. We illustrate on semisynthetic data.  \n1 Introduction  \nIn classical causal inference, each unit exists in isolation: its treatment and outcome is independent of all other units. But consider a dataset about trees and temperature measured at different locations in a city. We expect spatial dependence: planting trees will affect temperature in the surrounding area. Similarly in a community: talking to one person can affect the decisions of their friends. Orin a genome: a mutation at one locus affects gene expression nearby along the chromosome.  \nA central challenge in causal inference is to develop methods to account for such dependence among units. Many methods have been proposed, including techniques tailored to spatial, temporal or network data, as well as techniques for handling less structured forms of interference and dependence [Giffin et al., 2021, 2023, Papadogeorgou et al. , 2019, Papadogeorgou and Samanta, 2023, Gilbert et al., 2024, Wang et al., 2025, Rischard et al., 2018, Peters et al., 2013, Bojinov and Shephard, 2019, Christiansen et al., 2022, Papadogeorgou et al., 2022, Ogburn et al., 2022, Sridharet al. , 2022, Cristali and Veitch, 2022, Hudgens and Halloran, 2008, Guo et al. , 2025, Agarwal et al. , 2021, Sävje et al., 2021, Liang and Recht, 2023, Røysland et al., 2024, Didelez, 2008] .  \nWe study non-i.i.d. causal inference through the unified lens of symmetry. Our basic idea is that even when units are not independent, their underlying causal mechanisms may still be symmetric, in the sense that they are preserved under certain transformations. For instance, while temperature might not be i.i.d. across locations in a city, the same physical forces are at play everywhere. So the causal mechanisms driving urban heat are translation invariant, symmetric under a shift in location. We formalize the problem using the tools of group theory. We propose geometric causal models (GCMs), causal models with mechanisms that are symmetric under a group of transformations. For  \n∗ Department of Chemistry, Technical University of Denmark, Kgs. Lyngby, [DK.](DK. enawe@dtu.dk)[ enawe@dtu.dk](DK. enawe@dtu.dk)  \n†Departments of Statistics and Computer Science, Columbia University, New York, NY, USA. david .blei@ [columbia.edu](columbia.edu)  \nspatial data, this can be the group of translations in space; for network data, this can be the group of permutations of the network’s nodes. We show that as long as causal mechanisms obey some symmetry, causal inference is possible, even in the face of complex dependence among units.  \nTo create GCMs, we generalize the axioms of structural causal ","cbCaisyhXMRMjiSU","https://ap.wps.com/l/cbCaisyhXMRMjiSU","pdf",1367604,1,43,"English","en",105,"# Introduction\n## Non-i.i.d. causal inference through symmetry\n## Geometric causal models and identification","[{\"question\":\"What problem do geometric causal models (GCMs) address?\",\"answer\":\"They address causal inference when data units are dependent and not independently and identically distributed, such as in spatial, network, or molecular settings.\"},{\"question\":\"How do GCMs enable causal identification and estimation?\",\"answer\":\"They formalize assumptions about symmetry of the data-generating causal mechanisms using group theory, then use ergodic-theoretic tools for amenable groups to support identification and estimation.\"},{\"question\":\"What is the genomics application of GCMs in this document?\",\"answer\":\"It constructs a causal model whose symmetries match DNA, enabling new estimators for effects of genetic variation by combining models for functional genomics outcomes with DNA language models for propensities, illustrated on semisynthetic 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problem do geometric causal models (GCMs) address?","Question",{"text":75,"@type":76},"They address causal inference when data units are dependent and not independently and identically distributed, such as in spatial, network, or molecular settings.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do GCMs enable causal identification and estimation?",{"text":80,"@type":76},"They formalize assumptions about symmetry of the data-generating causal mechanisms using group theory, then use ergodic-theoretic tools for amenable groups to support identification and estimation.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the genomics application of GCMs in this document?",{"text":84,"@type":76},"It constructs a causal model whose symmetries match DNA, enabling new estimators for effects of genetic variation by combining models for functional genomics outcomes with DNA language models for propensities, illustrated on semisynthetic 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