Stein's Lemma for the Reparameterization Trick with Exponential Family Mixtures

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.