Harmonic entropy: Difference between revisions

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Our derivation only analytically continues the entropy function for the "unnormalized" set of probabilities, which we previously wrote as <math>Q(j|c)</math>. For this definition to be philosophically perfect, we would want to analytically continue the entropy function for the normalized sense of probabilities, previously written as <math>P(j|c)</math>.

However, in practice, the "unnormalized entropy" appears to be an extremely good approximation to the normalized entropy for large values of <math>N</math>. The resulting curve has the same minima and maxima as HE, the same general shape, and for all intents and purposes looks exactly like HE, just shifted on the y-axis.

However, in practice, the "unnormalized entropy" appears to be an extremely good approximation to the normalized entropy for large values of <math>N</math>. The resulting curve has approximately the same minima and maxima as HE, the same general shape, and for all intents and purposes looks exactly like HE, just shifted on the y-axis.

Here are some examples for different values of <math>s</math>. All of these are Shannon HE (<math>a=1</math>), using <math>\sqrt{nd}</math> weights, with unreduced rationals (more on this below), with the bound that <math>nd < 1000000</math>, just with different values of <math>s</math>. All have been scaled so that the minimum entropy is 0, and the maximum entropy is 1:

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 Our derivation only analytically continues the entropy function for the "unnormalized" set of probabilities, which we previously wrote as <math>Q(j|c)</math>. For this definition to be philosophically perfect, we would want to analytically continue the entropy function for the normalized sense of probabilities, previously written as <math>P(j|c)</math>.
-However, in practice, the "unnormalized entropy" appears to be an extremely good approximation to the normalized entropy for large values of <math>N</math>. The resulting curve has the same minima and maxima as HE, the same general shape, and for all intents and purposes looks exactly like HE, just shifted on the y-axis.
+However, in practice, the "unnormalized entropy" appears to be an extremely good approximation to the normalized entropy for large values of <math>N</math>. The resulting curve has approximately the same minima and maxima as HE, the same general shape, and for all intents and purposes looks exactly like HE, just shifted on the y-axis.
 Here are some examples for different values of <math>s</math>. All of these are Shannon HE (<math>a=1</math>), using <math>\sqrt{nd}</math> weights, with unreduced rationals (more on this below), with the bound that <math>nd < 1000000</math>, just with different values of <math>s</math>. All have been scaled so that the minimum entropy is 0, and the maximum entropy is 1: