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If one is given a joint probability distribution over a finite set of discrete random variables then I guess there a notion of $\alpha-$Renyi entropy defined for it as $S_\alpha (X_1,..,X_n) = \frac{1}{1-\alpha} \log ( \sum_{a_1,..,a_n} p_{X_1,..,X_n}^\alpha (X_1 = a_1,..,X_n = a_n ) )$

  • Can the sum in the above is read as taking the trace of the $\alpha$ power of some density matrix?

  • Lets say the joint probability distribution is given as a probabilistic graphical model. Now given such a graph one can split it into sub-graphs and look at the joint distributions of each of these subgraphs. So one presumably has the $S_\alpha$ defined for each of these subgraphs. Do we have any relation between $S_\alpha (graph)$ and $\sum_{subgraphs} S_\alpha (subgraphs)$? (consider that the decomposition could be into disjoint parts or not)

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The answer to your first question is "yes, but such a matrix may not necessarily have an operational significance, until guessing strategies come in." A good starting point may be the paper Revisiting Conditional Renyi Entropies ... which seems to have a good discussion of the non uniqueness of conditional Renyi entropies, and is publicly accessible.

If you are in addition imposing a graphical probability model, you will need to develop this kind of theory further, in the light of your goals.

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  • $\begingroup$ Why do you say these are "conditional Renyi entropies"? Also could you describe what is the density matrix you have in mind and what are these guessing strategies? $\endgroup$ – Anirbit Mar 31 '16 at 3:32

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