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Entropy 2017, 19(10), 527; https://doi.org/10.3390/e19100527

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.
Received: 15 June 2017 / Revised: 13 September 2017 / Accepted: 20 September 2017 / Published: 6 October 2017
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Abstract

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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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Rauh, J.; Banerjee, P.K.; Olbrich, E.; Jost, J.; Bertschinger, N.; Wolpert, D. Coarse-Graining and the Blackwell Order. Entropy 2017, 19, 527.

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