Appendix for “A Kernelized Stein Discrepancy for Goodness-of-fit Tests”

Because k(·, x) is in the Stein class of p for any x , we can show that ∇ x k(·, x) is also in the Stein class, since x ∇ x (p(x)∇ x k(x, x))dx = ∇ x x ∇ x (p(x)k(x, x))dx = 0, and hence v(·, x) is also in the Stein class; apply Lemma 2.3 on v(·, x) with fixed x gives S(p, q) = E x,x ∼p [(s q (x) − s p (x)) v(x, x))] = E x,x ∼p [s q (x) v(x, x) + trace(∇ x v(x, x))] The result then follows by noting that ∇ x v(x, x) = ∇ x k(x, x)s q (x) + ∇ x x k(x, x).