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Article

Coarse-Graining and the Blackwell Order

1
Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
2
Frankfurt Institute for Advanced Studies, 60438 Frankfurt, Germany
3
Santa Fe Institute, Santa Fe, NM 87501, USA
4
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
Author to whom correspondence should be addressed.
Entropy 2017, 19(10), 527; https://doi.org/10.3390/e19100527
Received: 15 June 2017 / Revised: 13 September 2017 / Accepted: 20 September 2017 / Published: 6 October 2017
Suppose we have a pair of information channels, κ 1 , κ 2 , with a common input. The Blackwell order is a partial order over channels that compares κ 1 and κ 2 by the maximal expected utility an agent can obtain when decisions are based on the channel outputs. Equivalently, κ 1 is said to be Blackwell-inferior to κ 2 if and only if κ 1 can be constructed by garbling the output of κ 2 . A related partial order stipulates that κ 2 is more capable than κ 1 if the mutual information between the input and output is larger for κ 2 than for κ 1 for any distribution over inputs. A Blackwell-inferior channel is necessarily less capable. However, examples are known where κ 1 is less capable than κ 2 but not Blackwell-inferior. We show that this may even happen when κ 1 is constructed by coarse-graining the inputs of κ 2 . Such a coarse-graining is a special kind of “pre-garbling” 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. View Full-Text
Keywords: Channel preorders; Blackwell order; degradation order; garbling; more capable; coarse-graining Channel preorders; Blackwell order; degradation order; garbling; more capable; coarse-graining
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MDPI and ACS Style

Rauh, J.; Banerjee, P.K.; Olbrich, E.; Jost, J.; Bertschinger, N.; Wolpert, D. Coarse-Graining and the Blackwell Order. Entropy 2017, 19, 527. https://doi.org/10.3390/e19100527

AMA Style

Rauh J, Banerjee PK, Olbrich E, Jost J, Bertschinger N, Wolpert D. Coarse-Graining and the Blackwell Order. Entropy. 2017; 19(10):527. https://doi.org/10.3390/e19100527

Chicago/Turabian Style

Rauh, Johannes, Pradeep K. Banerjee, Eckehard Olbrich, Jürgen Jost, Nils Bertschinger, and David Wolpert. 2017. "Coarse-Graining and the Blackwell Order" Entropy 19, no. 10: 527. https://doi.org/10.3390/e19100527

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