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.