Update 2017-07-03: Corrected equation for associative definition, thank you /u/Syrak. This may not be the first time someone recognized this, but I have recently discovered some interesting and useful properties of the entropy function and now share them. First a definition: Entropy — H(p 1 , p 2 , ..., p n ) — is a function that quantifies surprise in selecting an object from a set where the probability of selecting each object is given: {p 1 , p 2 , ..., p n }. This has utility in communications, information theory and other fields of math. H b (p 1 , p 2 , ..., p n ) = Σ(i..n)-p i log b (p i ) where b is normally 2, to express entropy in bits. Other definitions of H() use expected values and random variables. As an analog to the definition above, I will discuss entropy of a set of frequencies. p i = f i / Σ f i. Entropy defined without bits: A definition that doesn't use bits is: H(p 1 , p 2 , ..., p n ) = Π(i.....
