Jeffreys centroids: A closed-form expression for positive histograms and a guaranteed tight approximation for frequency histograms Frank Nielsen [email protected]5793b870 Sony Computer Science Laboratories, Inc. April 2013 c 2013 Frank Nielsen, Sony Computer Science Laboratories, Inc. 1/25
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Jeffreys centroids:A closed-form expression for positive histograms
and a guaranteed tight approximation forfrequency histograms
Distance between two frequency histograms p and q:Kullback-Leibler divergence or relative entropy.
KL(p : q) = H×(p : q)− H(p),
H×(p : q) =d∑
i=1
pi log1
qi, cross− entropy
H(p) = H×(p : p) =
d∑
i=1
pi log1
pi,Shannon entropy.
→ expected extra number of bits per datum that must betransmitted when using the “wrong” distribution q instead of thetrue distribution p.p is hidden by nature (and hypothesized), q is estimated.
The Jeffreys positive centroid c = (c1, ..., cd ) of a set {h1, ..., hn}of n weighted positive histograms with d bins can be calculatedcomponent-wise exactly using the Lambert W analytic function:
c i =ai
W ( ai
g i e),
where ai =∑n
j=1 πjhij denotes the coordinate-wise arithmetic
weighted means and g i =∏n
j=1(hij )πj the coordinate-wise
geometric weighted means.
Lambert analytic function [2] W (x)eW (x) = x for x ≥ 0.
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