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The more improbable the message, the less "compressible" it is, and the more bandwidth it requires. This is Shannon's point: the essence is its improbability.

William Poundstone
Essence, Messages, Shannon
William Poundstone (2010). “Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street”, p.57, Macmillan
The more improbable the message, the less compressible it is, and the more bandwidth it requires. This is Shannon's point: the essence is its improbability.