Exponential Shannon Entropy and Subadditivity in Portfolio Selection

Summary

The document raises a question about using exponential differential Shannon entropy as an uncertainty measure in portfolio selection. It asks whether the measure of a sum of random variables is bounded by the sum of their individual measures, and compares that proposed property with the entropy power inequality for independent variables.

It also asks whether a similar subadditivity result holds for exponential Rényi entropy at particular parameter values. The document supplies no proof, answer, or empirical evidence; it frames mathematical questions for further investigation. Its relevance to portfolio research is the choice and behavior of entropy-based measures, but the stated claims should not be treated as established results from this text alone.

Key ideas

• The document questions whether exponential differential Shannon entropy is subadditive under addition of random variables.
• It contrasts the proposed inequality with the entropy power inequality for independent variables.
• It asks whether exponential Rényi entropy has a related property for any parameter values.
• The text poses open questions and provides no proof or resolution.

Tags

# Is the exponential Shannon entropy sub-additive?


# Is the exponential Shannon entropy sub-additive?












In a recent paper of Salazar et al. (2014), The Diversification Delta: A Different Perspective, forthcoming in the Journal of Portfolio Management , the authors propose to use the exponential Shannon entropy as an uncertainty measure in portfolio selection and make the following claim:

If we define $$H(X)=−\int_xf(x)\text{ln}f(x)dx$$ to be the differential Shannon entropy, then we have that for random variables $X$ and $Y$ (pp.10): $$\text{exp}(H(X+Y))\leqslant \text{exp}(H(X))+\text{exp}(H(X))$$ After thorough research I haven't been able to find any proof of this sub-additive property which makes me wonder: is this property really correct? And would you have any proof of it? Moreover is seems that this sub-additive property is at odds with the entropy power inequality that says that for independent $X$ and $Y$: $$\text{exp}(2H(X+Y))⩾\text{exp}(2H(X))+\text{exp}(2H(X))$$

Also, I am interested in knowing how this property would generalize to Rényi entropy defined as $$H_\alpha(X)=\frac{1}{1-\alpha}\text{ln}∫_x(f(x))^αdx$$ Is $\text{exp}(H_\alpha(X))$ also sub-additive for some values of $\alpha$?

I would really appreciate any insights you might have on this problem.

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