  Pinsker's inequality states that the relative entropy $d_{\mathrm{KL}}(X, Y)$
between two random variables $X$ and $Y$ dominates the square of the total
variation distance $d_{\mathrm{TV}}(X,Y)$ between $X$ and $Y$. In this paper we
introduce generalized Fisher information distances $\mathcal{J}(X, Y)$ between
discrete distributions $X$ and $Y$ and prove that these also dominate the
square of the total variation distance. To this end we introduce a general
discrete Stein operator for which we prove a useful covariance identity. We
illustrate our approach with several examples. Whenever competitor inequalities
are available in the literature, the constants in ours are at least as good,
and, in several cases, better.
