Stein's lemma

Stein's lemma, named in honor of Charles Stein, is a theorem of probability theory that is of interest primarily because of its applications to statistical inference — in particular, to James–Stein estimation and empirical Bayes methods — and its applications to portfolio choice theory.^[1] The theorem gives a formula for the covariance of one random variable with the value of a function of another, when the two random variables are jointly normally distributed.

Note that the name "Stein's lemma" is also commonly used^[2] to refer to a different result in the area of statistical hypothesis testing, which connects the error exponents in hypothesis testing with the Kullback–Leibler divergence. This result is also known as the Chernoff–Stein lemma^[3] and is not related to the lemma discussed in this article.

Suppose X is a normally distributed random variable with expectation μ and variance σ². Further suppose g is a differentiable function for which the two expectations ${\displaystyle \operatorname {E} (g(X)(X-\mu ))}$ and ${\displaystyle \operatorname {E} (g'(X))}$ both exist. (The existence of the expectation of any random variable is equivalent to the finiteness of the expectation of its absolute value.) Then

${\displaystyle \operatorname {E} {\bigl (}g(X)(X-\mu ){\bigr )}=\sigma ^{2}\operatorname {E} {\bigl (}g'(X){\bigr )}.}$

In general, suppose X and Y are jointly normally distributed. Then

${\displaystyle \operatorname {Cov} (g(X),Y)=\operatorname {Cov} (X,Y)\operatorname {E} (g'(X)).}$

For a general multivariate Gaussian random vector ${\displaystyle (X_{1},...,X_{n})\sim {\mathcal {N}}(\mu ,\Sigma )}$ it follows that

${\displaystyle \operatorname {E} {\bigl (}g(X)(X-\mu ){\bigr )}=\Sigma \cdot E{\bigl (}\nabla g(X){\bigr )}.}$

Similarly, when ${\displaystyle \mu =0}$, $${\displaystyle \operatorname {E} [\partial _{i}g(X)]=\operatorname {E} [g(X)(\Sigma ^{-1}X)_{i}],\quad \operatorname {E} [\partial _{i}\partial _{j}g(X)]=\operatorname {E} [g(X)((\Sigma ^{-1}X)_{i}(\Sigma ^{-1}X)_{j}-\Sigma _{ij}^{-1})]}$$

Stein's lemma can be used to stochastically estimate gradient:$${\displaystyle \nabla \operatorname {E} _{\epsilon \sim {\mathcal {N}}(0,I)}{\bigl (}g(x+\Sigma ^{1/2}\epsilon ){\bigr )}=\Sigma ^{-1/2}\operatorname {E} _{\epsilon \sim {\mathcal {N}}(0,I)}{\bigl (}g(x+\Sigma ^{1/2}\epsilon )\epsilon {\bigr )}\approx \Sigma ^{-1/2}{\frac {1}{N}}\sum _{i=1}^{N}g(x+\Sigma ^{1/2}\epsilon _{i})\epsilon _{i}}$$where ${\displaystyle \epsilon _{1},\dots ,\epsilon _{N}}$ are IID samples from the standard normal distribution ${\displaystyle {\mathcal {N}}(0,I)}$. This form has applications in Stein variational gradient descent^[4] and Stein variational policy gradient.^[5]

The univariate probability density function for the univariate normal distribution with expectation 0 and variance 1 is

${\displaystyle \varphi (x)={1 \over {\sqrt {2\pi }}}e^{-x^{2}/2}}$

Since ${\displaystyle \int x\exp(-x^{2}/2)\,dx=-\exp(-x^{2}/2)}$ we get from integration by parts:

${\displaystyle \operatorname {E} [g(X)X]={\frac {1}{\sqrt {2\pi }}}\int g(x)x\exp(-x^{2}/2)\,dx={\frac {1}{\sqrt {2\pi }}}\int g'(x)\exp(-x^{2}/2)\,dx=\operatorname {E} [g'(X)]}$.

The case of general variance ${\displaystyle \sigma ^{2}}$ follows by substitution.

Isserlis' theorem is equivalently stated as$${\displaystyle \operatorname {E} (X_{1}f(X_{1},\ldots ,X_{n}))=\sum _{i=1}^{n}\operatorname {Cov} (X_{1},X_{i})\operatorname {E} (\partial _{X_{i}}f(X_{1},\ldots ,X_{n})).}$$where ${\displaystyle (X_{1},\dots X_{n})}$ is a zero-mean multivariate normal random vector.

Suppose X is in an exponential family, that is, X has the density

${\displaystyle f_{\eta }(x)=\exp(\eta 'T(x)-\Psi (\eta ))h(x).}$

Suppose this density has support ${\displaystyle (a,b)}$ where ${\displaystyle a,b}$ could be ${\displaystyle -\infty ,\infty }$ and as ${\displaystyle x\rightarrow a{\text{ or }}b}$, ${\displaystyle \exp(\eta 'T(x))h(x)g(x)\rightarrow 0}$ where ${\displaystyle g}$ is any differentiable function such that ${\displaystyle E|g'(X)|<\infty }$ or ${\displaystyle \exp(\eta 'T(x))h(x)\rightarrow 0}$ if ${\displaystyle a,b}$ finite. Then

${\displaystyle E\left[\left({\frac {h'(X)}{h(X)}}+\sum \eta _{i}T_{i}'(X)\right)\cdot g(X)\right]=-E[g'(X)].}$

The derivation is same as the special case, namely, integration by parts.

If we only know ${\displaystyle X}$ has support ${\displaystyle \mathbb {R} }$, then it could be the case that ${\displaystyle E|g(X)|<\infty {\text{ and }}E|g'(X)|<\infty }$ but ${\displaystyle \lim _{x\rightarrow \infty }f_{\eta }(x)g(x)\not =0}$. To see this, simply put ${\displaystyle g(x)=1}$ and ${\displaystyle f_{\eta }(x)}$ with infinitely spikes towards infinity but still integrable. One such example could be adapted from ${\displaystyle f(x)={\begin{cases}1&x\in [n,n+2^{-n})\\0&{\text{otherwise}}\end{cases}}}$ so that ${\displaystyle f}$ is smooth.

Extensions to elliptically-contoured distributions also exist.^[6][7][8]

• Stein's method
• Taylor expansions for the moments of functions of random variables
• Stein discrepancy