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Math and science::INF ML AI

Entropy of an ensemble

The entropy of an ensemble, \( X = (x, A_x, P_x) \), is defined to be the average Shannon information content over all outcomes:

[\[ H(X) = \quad ? \] ]

Properties of entropy:

  • [\( H(X) \geq \; ? \) ] with equality iff [...].
  • Entropy is maximized if [something about the outcomes is true].
  • The Entropy is less than [some opperation applied to] the number of outcomes.

The last two points can be expressed as:

[\[ H(X) \leq \; ? \text{, with equality iff    ? } \]]

Proof on the back side.