{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T06:46:57Z","timestamp":1772779617861,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T00:00:00Z","timestamp":1772582400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2014\/50851-0"],"award-info":[{"award-number":["2014\/50851-0"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2018\/19150-6"],"award-info":[{"award-number":["2018\/19150-6"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2021\/12220-1"],"award-info":[{"award-number":["2021\/12220-1"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"National Council for Scientific and Technological Development","doi-asserted-by":"publisher","award":["465755\/2014-3"],"award-info":[{"award-number":["465755\/2014-3"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Learning dependence graphs from multivariate continuous data is challenging when marginal distributions are heterogeneous, since likelihood-based nonparametric scores can be sensitive to smoothing choices and can confound marginal irregularities, including non-identifiability, with dependence. This work studies structure learning in the class of decomposable (chordal) Markov random fields, where junction tree factorizations enable tractable inference and local score updates. Our first contribution is a theoretical result showing that, under decomposability, mutual information can be expressed as a difference of clique\/separator copula entropies, yielding a dependence-only decomposition aligned with the clique\/separator structure. Building on this identity, we define an information-theoretic objective for decomposable graphs with a complexity penalty that preserves clique\/separator additivity, and we derive closed-form local score differences for chordality-preserving single-edge insertions and deletions. To make the score computable from data, we instantiate clique\/separator copula entropies using pseudo-observations and a probit-transformed kernel density estimator with predictive log score evaluation to mitigate boundary effects on the unit hypercube. The resulting nonparametric greedy procedure improves edge recovery accuracy on synthetic chordal benchmarks compared with a likelihood-driven nonparametric baseline, and it produces interpretable dependence summaries on an airway epithelial gene expression dataset. Concretely, this paper contributes (1) a decomposable mutual information identity via clique\/separator copula entropies, (2) a copula information score with an additive complexity penalty for decomposable graphs, (3) a closed-form local score, enabling greedy chordal add or delete search, (4) a practical nonparametric copula entropy estimation pipeline, and (5) empirical gains on synthetic and real data.<\/jats:p>","DOI":"10.3390\/e28030293","type":"journal-article","created":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T16:17:31Z","timestamp":1772641051000},"page":"293","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Learning Continuous Decomposable Models Using Mutual Information and Statistical Copulas"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8629-1870","authenticated-orcid":false,"given":"Luiz","family":"Desu\u00f3 Neto","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, S\u00e3o Paulo State University (UNESP), Guaratinguet\u00e1 12516-410, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3624-7924","authenticated-orcid":false,"given":"Henrique","family":"de Oliveira Caetano","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of S\u00e3o Paulo (USP), S\u00e3o Carlos 13566-590, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7683-4843","authenticated-orcid":false,"given":"Matheus","family":"de Souza Sant\u2019Anna Fogliatto","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of S\u00e3o Paulo (USP), S\u00e3o Carlos 13566-590, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0137-6678","authenticated-orcid":false,"given":"Carlos","family":"Dias Maciel","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, S\u00e3o Paulo State University (UNESP), Guaratinguet\u00e1 12516-410, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v114.i12","article-title":"Benchpress: A Versatile Platform for Structure Learning in Causal and Probabilistic Graphical Models","volume":"114","author":"Rios","year":"2025","journal-title":"J. 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