Inequalities for entropies of sets of subsets of random variables

Han´s inequality on the entropy rates of subsets of random variables is a classic result in information theory, which often finds its application in multiuser information theoretic problems. In this note, we generalize Han´s inequality to allow common components among the random variables, or, in an equivalent manner, to replace the simple random variables in Han´s inequality by subsets of random variables. This additional ingredient significantly complicates the matter and the form of the resultant inequalities are rather different from the original Han´s inequality. Our proof only relies on the sub-modularity property of the entropy function and the super-modularity property of the conditional entropy function. This new set of inequalities also provides a new link between Han´s inequality and the n-way sub-modularity inequality.