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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 in- equalities 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.