Coarse-Graining and the Blackwell Order

Entropy

Nils Bertschinger¹  Eckehard Olbrich²  Pradeep Kr. Banerjee²  Johannes Rauh²  Jürgen Jost²  David Wolpert³ 

[1] Frankfurt Institute for Advanced Studies, 60438 Frankfurt, Germany;Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany;Santa Fe Institute, Santa Fe, NM 87501, USA;

关键词: Channel preorders;    Blackwell order;    degradation order;    garbling;    more capable;    coarse-graining;   

DOI  :  10.3390/e19100527

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown