Coarse-Graining and the Blackwell Order

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.

Generalised Measures of Multivariate Information Content

“…More specifically, since this paper considers information contents, this redundancy lattice is actually same as the specificity lattice from pointwise partial information decomposition [ 27 , 28 ]. This observation connects the work presented in this paper to the existing body of theoretical literature on information decomposition [ 23 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 ], and its applications [ 63 , 64 , 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 , 79 , 80 , 81 , 82 , 83 , 84 , 85 , 86 ]. (For a brief summary of this literature, see [ 24 ].)…”

On Extractable Shared Information

“…However, one can check that the channel S → X 1 is not a garbling of the channel S → X 2 . This example is discussed in more detail in Rauh et al [13].…”

Information Decomposition of Target Effects from Multi-Source Interactions: Perspectives on Previous, Current and Future Work

“…This is achieved via a construction which translates any measure of shared information into one that satisfies this property. The authors then explore the properties of this measure, and show for example that it is not compatible with a Blackwell interpretation of unique information (see their other contribution, [ 32 ], discussed in Section 2.2 ).…”

“…Rauh et al [ 32 ] examine the decision-theoretic Blackwell partial order, which ranks information channels (with a common input) according to the utility that can be obtained when decisions are made on the channel outputs. The authors present the unexpected result that a coarse-graining of one channel output may actually result in improved utility.…”