  Stein's method (Stein, 1973; 1981) is a powerful tool for statistical
applications, and has had a significant impact in machine learning. Stein's
lemma plays an essential role in Stein's method. Previous applications of
Stein's lemma either required strong technical assumptions or were limited to
Gaussian distributions with restricted covariance structures. In this work, we
extend Stein's lemma to exponential-family mixture distributions including
Gaussian distributions with full covariance structures. Our generalization
enables us to establish a connection between Stein's lemma and the
reparamterization trick to derive gradients of expectations of a large class of
functions under weak assumptions. Using this connection, we can derive many new
reparameterizable gradient-identities that goes beyond the reach of existing
works. For example, we give gradient identities when expectation is taken with
respect to Student's t-distribution, skew Gaussian, exponentially modified
Gaussian, and normal inverse Gaussian.
