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.

本文将 Stein 的引理扩展到指数族混合分布，建立了 Stein 的引理和重新参数化技巧之间的联系，从而得到了许多新的可重新参数化的梯度等式，包括在学生 t 分布，偏斜高斯分布，指数修改高斯分布和正态逆高斯分布下的期望梯度