{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T07:42:40Z","timestamp":1781163760435,"version":"3.54.1"},"reference-count":14,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,4,11]],"date-time":"2018-04-11T00:00:00Z","timestamp":1523404800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Makkeh, Theis, and Vicente found that Cone Programming model is the most robust to compute the Bertschinger et al. partial information decomposition (BROJA PID) measure. We developed a production-quality robust software that computes the BROJA PID measure based on the Cone Programming model. In this paper, we prove the important property of strong duality for the Cone Program and prove an equivalence between the Cone Program and the original Convex problem. Then, we describe in detail our software, explain how to use it, and perform some experiments comparing it to other estimators. Finally, we show that the software can be extended to compute some quantities of a trivaraite PID measure.<\/jats:p>","DOI":"10.3390\/e20040271","type":"journal-article","created":{"date-parts":[[2018,4,11]],"date-time":"2018-04-11T12:16:50Z","timestamp":1523449010000},"page":"271","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["BROJA-2PID: A Robust Estimator for Bivariate Partial Information Decomposition"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3581-8262","authenticated-orcid":false,"given":"Abdullah","family":"Makkeh","sequence":"first","affiliation":[{"name":"Institute of Computer Science, University of Tartu, \u00dclikooli 17, 51014 Tartu, Estonia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dirk","family":"Theis","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, University of Tartu, \u00dclikooli 17, 51014 Tartu, Estonia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Raul","family":"Vicente","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, University of Tartu, \u00dclikooli 17, 51014 Tartu, Estonia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,4,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2161","DOI":"10.3390\/e16042161","article-title":"Quantifying unique information","volume":"16","author":"Bertschinger","year":"2014","journal-title":"Entropy"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Griffith, V., and Koch, C. (2014). Quantifying synergistic mutual information. Guided Self-Organization: Inception, Springer.","DOI":"10.1007\/978-3-642-53734-9_6"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"012130","DOI":"10.1103\/PhysRevE.87.012130","article-title":"Bivariate measure of redundant information","volume":"87","author":"Harder","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_4","unstructured":"Williams, P.L., and Beer, R.D. (arXiv, 2010). Nonnegative decomposition of multivariate information, arXiv."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Makkeh, A., Theis, D.O., and Vicente, R. (2017). Bivariate Partial Information Decomposition: The Optimization Perspective. Entropy, 19.","DOI":"10.3390\/e19100530"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Boyd, S., and Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press.","DOI":"10.1017\/CBO9780511804441"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Banerjee, P.K., Rauh, J., and Mont\u00fafar, G. (arXiv, 2017). Computing the Unique Information, arXiv.","DOI":"10.1109\/ISIT.2018.8437757"},{"key":"ref_8","unstructured":"Chicharro, D. (arXiv, 2017). Quantifying multivariate redundancy with maximum entropy decompositions of mutual information, arXiv."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Domahidi, A., Chu, E., and Boyd, S. (2013, January 17\u201319). ECOS: An SOCP solver for embedded systems. Proceedings of the European Control Conference (ECC), Zurich, Switzerland.","DOI":"10.23919\/ECC.2013.6669541"},{"key":"ref_10","unstructured":"O\u2019Donoghue, B., Chu, E., Parikh, N., and Boyd, S. (2017, November 26). SCS: Splitting Conic Solver, Version 1.2.7, 2016. Available online: https:\/\/github.com\/cvxgrp\/scs."},{"key":"ref_11","unstructured":"Luenberger, D.G., and Ye, Y. (1984). Linear and Nonlinear Programming, Springer."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"G\u00e4rtner, B., and Matousek, J. (2012). Approximation Algorithms and Semidefinite Programming, Springer.","DOI":"10.1007\/978-3-642-22015-9"},{"key":"ref_13","unstructured":"Chares, R. (2009). Cones and Interior-Point Algorithms for Structured Convex Optimization Involving Powers Andexponentials. [Ph.D. Thesis, UCL-Universit\u00e9 Catholique de Louvain]."},{"key":"ref_14","unstructured":"Makkeh, A. (2018). Applications of Optimization in Some Complex Systems. [Ph.D. Thesis, University of Tartu]. forthcoming."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/4\/271\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:00:23Z","timestamp":1760194823000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/4\/271"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,11]]},"references-count":14,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,4]]}},"alternative-id":["e20040271"],"URL":"https:\/\/doi.org\/10.3390\/e20040271","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,4,11]]}}}