Han's inequality for relative entropy

If $P={⨂}_{i=1}^n{P}_i$ is a product measure, $Q$ is any probability measure, and $X^{(i)}$ omits coordinate $i$, then

For a random vector $X_{1:n}$,This is equivalent to Han's inequality for relative entropy after expansion against a product reference measure.

The strong formfollows by iterating entropy submodularity.