Coarse-graining and the Blackwell Order

Johannes Rauh, Pradeep Kumar Banerjee, Eckehard Olbrich, Jürgen Jost, Nils Bertschinger, and David Wolpert

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Submission date: 10. Nov. 2017
Pages: 14
published in: Entropy, 19 (2017) 10, art-no. 527 
DOI number (of the published article): 10.3390/e19100527
Bibtex
MSC-Numbers: 62B15, 94A15, 94A17
Keywords and phrases: Channel preorders, Blackwell order, degradation order, garbling, more capable, coarse-graining
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Link to arXiv: See the arXiv entry of this preprint.

Abstract:
Suppose we have a pair of information channels, κ₁,κ₂, with a common input. The Blackwell order is a partial order over channels that compares κ₁ and κ₂ by the maximal expected utility an agent can obtain when decisions are based on the channel outputs. Equivalently, κ₁ is said to be Blackwell-inferior to κ₂ if and only if κ₁ can be constructed by garbling the output of κ₂. A related partial order stipulates that κ₂ is more capable than κ₁ if the mutual information between the input and output is larger for κ₂ than for κ₁ for any distribution over inputs. A Blackwell-inferior channel is necessarily less capable. However, examples are known where κ₁ is less capable than κ₂ but not Blackwell-inferior. We show that this may even happen when κ₁ is constructed by coarse-graining the inputs of κ₂. 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.