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The information content of an outcome of a variable is $h(x_i)=-\log_2 (p(x_i))$. I am interested in using this concept to provide more insight on the differences in the distributions of two independent variables.

I am aware of entropy and relative entropy for comparing the expected value of these variables, and I'm using these as needed. However, I am looking for a metric that captures the shape of the distributions that would be graphically captured by a histogram. Is there any reason that I can't take the sum of information content values for the various outcomes of a variable? Effectively, calculating: $$ h(X)=\sum(-log_2 (p(x_i))$$

These summed information content values give a measure of the shape of the distribution.

I've tried looking for examples of people using this metric, but have come up empty. I'm not sure what it would be called -- presumably just the information content of the variable. I can't see any mathematical reason why this can't be done, but I thought someone here might be able to fill me in if there's something I'm missing.

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  • $\begingroup$ Perhaps you'd be interested in KL-divergence: en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence $\endgroup$ – rubikscube09 Apr 23 '19 at 15:08
  • $\begingroup$ Yes, thanks. I'm using that for comparison of some of the variables. However, a couple of the variables I'm looking at have different domains, or alphabets, so they aren't defined over the same probability space. The comparison of information content of each variable, while not probabilistically comparable, would seem to provide a measure of the shape of the distributions. $\endgroup$ – Don29826 Apr 23 '19 at 16:26
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You'd need to justify why you believe that that value is useful, and why it gives

a measure of the shape of the distribution.

Anyway, if the alphabet size is $n$ we have

$$\begin{align} \sum_x -\log p(x) &= \sum_x \log (1/p(x))\\ &= \sum_x (\log \frac{1/n}{p(x)} + \log(n)) \\ &= n \log n + n\sum_x \frac{1}{n}\log \frac{1/n}{p(x)} \\ &= n \left(\log n + KL(u || p(x)\right) \end{align} $$

hence the value depends directly on the KL divergence (relative entropy) between an uniform and the given distribution.

Furthermore, for a nice finite entropy distribution with infinite values, like the geometric ($p(x_i) = 2^{-i}$), the value is infinite - which is not very nice.

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Maybe the Shannon entropy, or weighted average of the information content of outcomes of a probability distribution, would be useful. It measures the amount of uncertainty associated with a distribution, so it counts as a proxy for its "shape". When well-defined, the uniform distribution has the highest entropy, while the Dirac distribution has an entropy of zero, i.e. the lowest possible.

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