  HWI inequalities are interpolation inequalities relating entropy, Fisher
information and optimal transport distances. We adapt an argument of Y. Wu for
proving the Gaussian HWI inequality via a coupling argument to the discrete
setting, establishing new interpolation inequalities for the discrete hypercube
and the discrete torus. In particular, we obtain an improvement of the modified
logarithmic Sobolev inequality for the discrete hypercube of Bobkov and Tetali.
