Stein's Method

• Liu, Qiang, Jason Lee, and Michael Jordan. "A kernelized Stein discrepancy for goodness-of-fit tests." International Conference on Machine Learning. 2016.
• Ley, Christophe, Gesine Reinert, and Yvik Swan. "Stein’s method for comparison of univariate distributions." Probability Surveys 14 (2017): 1-52.

1. Definition

Definition. Assume that $\mathcal{X}$ is a subset of $\mathbb{R}^{d}$ and $p(x)$ a continuous differentiable (also called smooth) density whose support is $\mathcal{X}$. The (Stein) score function of $p$ is defined as

$$\boldsymbol{s}_{p}=\nabla_{x} \log p(x)=\frac{\nabla_{x} p(x)}{p(x)}$$

Definition. We say that a function $f : \mathcal{X} \rightarrow \mathbb{R}$ is in the Stein class of $p$ if $f$ is smooth and satisfies

$$\int_{x \in \mathcal{X}} \nabla_{x}(f(x) p(x)) \, \mathrm{d} x=0$$

Notice that the RBF kernel $k\left(x, x^{\prime}\right)= \exp \left(-\frac{1}{2 h^{2}}\left\|x-x^{\prime}\right\|_{2}^{2}\right)$ is in the Stein class for smooth densities supported on $\mathcal{X}=\mathbb{R}^{d}$.

Definition. The Stein’s operator of $p$ is a linear operator acting on the Stein class of $p$, defined as

$$\mathcal{A}_{p} \, f(x)=s_{p}(x) \, f(x)+\nabla_{x} \, f(x)$$

Applying $\mathcal{A}_{p}$ on a vector-valued $\boldsymbol{f}(x)$ results a $d \times d^\prime$ matrix-valued function,

$$\mathcal{A}_{p} \, \boldsymbol{f}(x)=\boldsymbol{s}_{p}(x) \, \boldsymbol{f}(x)^{\top} + \nabla_{x} \, \boldsymbol{f}(x)$$

2. Stein's Identity

Theorem. Assume $p(x)$ is a smooth density supported on $\mathcal{X}$, then

$$\mathbb{E}_{p}\left[\mathcal{A}_{p} \, \boldsymbol{f}(x)\right]=\mathbb{E}_{p}\left[\boldsymbol{s}_{p}(x) \, \boldsymbol{f}(x)^{\top}+ \nabla \, \boldsymbol{f}(x)\right]=0$$

for any $\boldsymbol{f}$ that is in the Stein class of $p$.

Another useful variate of Stein's Identity is as follows.

Theorem. Assume $p(x)$ and $q(x)$ are smooth densities supported on $\mathcal{X}$ and $\boldsymbol{f}(x)$ in the Stein class of $p$ , we have

$$\mathbb{E}_{p}\left[\mathcal{A}_{q} \, \boldsymbol{f}(x)\right] = \mathbb{E}_{p}\left[\left(\boldsymbol{s}_{q}(x) - \boldsymbol{s}_{p}(x)\right) \, \boldsymbol{f}(x)^{\top}\right]$$