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MiS Preprint

Coarse-graining and the Blackwell Order

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


Suppose we have a pair of information channels, $\kappa_{1},\kappa_{2}$, with a common input. The Blackwell order is a partial order over channels that compares $\kappa_{1}$ and $\kappa_{2}$ by the maximal expected utility an agent can obtain when decisions are based on the channel outputs. Equivalently, $\kappa_{1}$ is said to be Blackwell-inferior to $\kappa_{2}$ if and only if $\kappa_{1}$ can be constructed by garbling the output of $\kappa_{2}$. A related partial order stipulates that $\kappa_{2}$ is more capable than $\kappa_{1}$ if the mutual information between the input and output is larger for $\kappa_{2}$ than for $\kappa_{1}$ for any distribution over inputs. A Blackwell-inferior channel is necessarily less capable. However, examples are known where $\kappa_{1}$ is less capable than $\kappa_{2}$ but not Blackwell-inferior. We show that this may even happen when $\kappa_{1}$ is constructed by coarse-graining the inputs of $\kappa_{2}$. 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.

MSC Codes:
62B15, 94A15, 94A17
Channel preorders, Blackwell order, degradation order, garbling, more capable, coarse-graining

Related publications

2017 Journal Open Access
Johannes Rauh, Pradeep Kumar Banerjee, Eckehard Olbrich, Jürgen Jost, Nils Bertschinger and David Wolpert

Coarse-graining and the Blackwell order

In: Entropy, 19 (2017) 10, p. 527