Stein's density approach for discrete distributions and information inequalities

We inscribe Stein's density approach for discrete distributions in a new, flexible framework, hereby extending and unifying a large portion of the relevant literature. We use this to derive a Stein identity whose power we illustrate by obtaining a wide variety of so-called inequalities between probability metrics and information functionals. Whenever competitor inequalities are available in the literature, the constants in ours are better. We also argue that our inequalities are local versions of the famous Pinsker inequality.