{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:15:38Z","timestamp":1760242538207,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2017,10,6]],"date-time":"2017-10-06T00:00:00Z","timestamp":1507248000000},"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>Suppose we have a pair of information channels,      \u03ba 1  ,  \u03ba 2     , with a common input. The Blackwell order is a partial order over channels that compares     \u03ba 1     and     \u03ba 2     by the maximal expected utility an agent can obtain when decisions are based on the channel outputs. Equivalently,     \u03ba 1     is said to be Blackwell-inferior to     \u03ba 2     if and only if     \u03ba 1     can be constructed by garbling the output of     \u03ba 2    . A related partial order stipulates that     \u03ba 2     is more capable than     \u03ba 1     if the mutual information between the input and output is larger for     \u03ba 2     than for     \u03ba 1     for any distribution over inputs. A Blackwell-inferior channel is necessarily less capable. However, examples are known where     \u03ba 1     is less capable than     \u03ba 2     but not Blackwell-inferior. We show that this may even happen when     \u03ba 1     is constructed by coarse-graining the inputs of     \u03ba 2    . Such a coarse-graining is a special kind of \u201cpre-garbling\u201d of the channel inputs. This example directly establishes that the expected value of the shared utility function for the coarse-grained channel is larger than it is for the non-coarse-grained channel. This contradicts the intuition that coarse-graining can only destroy information and lead to inferior channels. We also discuss our results in the context of information decompositions.<\/jats:p>","DOI":"10.3390\/e19100527","type":"journal-article","created":{"date-parts":[[2017,10,6]],"date-time":"2017-10-06T10:56:34Z","timestamp":1507287394000},"page":"527","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Coarse-Graining and the Blackwell Order"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6411-7775","authenticated-orcid":false,"given":"Johannes","family":"Rauh","sequence":"first","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}]},{"given":"Pradeep","family":"Banerjee","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3987-1271","authenticated-orcid":false,"given":"Eckehard","family":"Olbrich","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}]},{"given":"J\u00fcrgen","family":"Jost","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}]},{"given":"Nils","family":"Bertschinger","sequence":"additional","affiliation":[{"name":"Frankfurt Institute for Advanced Studies, 60438 Frankfurt, Germany"}]},{"given":"David","family":"Wolpert","sequence":"additional","affiliation":[{"name":"Santa Fe Institute, Santa Fe, NM 87501, USA"},{"name":"Massachusetts Institute of Technology, Cambridge, MA 02139, USA"}]}],"member":"1968","published-online":{"date-parts":[[2017,10,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1214\/aoms\/1177729032","article-title":"Equivalent Comparisons of Experiments","volume":"24","author":"Blackwell","year":"1953","journal-title":"Ann. Math. Stat."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Torgersen, E. (1991). Comparison of Statistical Experiments, Cambridge University Press.","DOI":"10.1017\/CBO9780511666353"},{"key":"ref_3","first-page":"127","article-title":"Comparison of Experiments\u2014A Short Review","volume":"30","year":"1996","journal-title":"Stat. Probab. Game Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1109\/TIT.1973.1054980","article-title":"Random coding theorem for broadcast channels with degraded components","volume":"19","author":"Bergmans","year":"1973","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","unstructured":"Cohen, J., Kemperman, J., and Zb\u0103ganu, G. (1998). Comparisons of Stochastic Matrices with Applications in Information Theory, Statistics, Economics, and Population Sciences, Birkh\u00e4user."},{"key":"ref_6","first-page":"411","article-title":"Comparison of two noisy channels","volume":"Volume 16","author":"Marton","year":"1975","journal-title":"Topics in Information Theory"},{"key":"ref_7","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_8","doi-asserted-by":"crossref","unstructured":"Rauh, J., Banerjee, P.K., Olbrich, E., Jost, J., and Bertschinger, N. (arXiv, 2017). On extractable shared information, arXiv.","DOI":"10.3390\/e19070328"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Raginsky, M. (August, January 31). Shannon meets Blackwell and Le Cam: Channels, codes, and statistical experiments. Proceedings of the 2011 IEEE International Symposium on Information Theory Proceedings, St. Petersburg, Russia.","DOI":"10.1109\/ISIT.2011.6033729"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/10\/527\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:46:40Z","timestamp":1760208400000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/10\/527"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,6]]},"references-count":9,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2017,10]]}},"alternative-id":["e19100527"],"URL":"https:\/\/doi.org\/10.3390\/e19100527","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2017,10,6]]}}}