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.

Statement

Suppose X is a normally distributed random variable with expectation μ and variance σ^2. Further suppose g is a differentiable function for which the two expectations E(g(X) (X − μ)) and 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

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

Multidimensional

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

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

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

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

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

Gradient descent

Stein's lemma can be used to stochastically estimate gradient:$${\displaystyle \nabla E_{\epsilon \sim {\mathcal {N}}(0,I)}{\bigl (}g(x+\Sigma ^{1/2}\epsilon ){\bigr )}=\Sigma ^{-1/2}E_{\epsilon \sim {\mathcal {N}}(0,I)}{\bigl (}g(x+\Sigma ^{1/2}\epsilon )\epsilon {\bigr )}\approx \Sigma ^{-1/2}{\frac {1}{N}}\sum _{i=1}^{N}g(x+\Sigma ^{1/2}\epsilon _{i})\epsilon _{i}}$$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]

Proof

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:

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

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

Generalizations

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]

See also

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

References

1. Ingersoll, J., Theory of Financial Decision Making, Rowman and Littlefield, 1987: 13-14.
2. Csiszár, Imre; Körner, János (2011). Information Theory: Coding Theorems for Discrete Memoryless Systems. Cambridge University Press. p. 14. ISBN 9781139499989.
3. Thomas M. Cover, Joy A. Thomas (2006). Elements of Information Theory. John Wiley & Sons, New York. ISBN 9781118585771.
4. Liu, Qiang; Wang, Dilin (2019-09-09). "Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm". arXiv:1608.04471 [stat.ML].
5. Liu, Yang; Ramachandran, Prajit; Liu, Qiang; Peng, Jian (2017-04-07). "Stein Variational Policy Gradient". arXiv:1704.02399 [cs.LG].
6. Cellier, Dominique; Fourdrinier, Dominique; Robert, Christian (1989). "Robust shrinkage estimators of the location parameter for elliptically symmetric distributions". Journal of Multivariate Analysis. 29 (1): 39–52. doi:10.1016/0047-259X(89)90075-4.
7. Hamada, Mahmoud; Valdez, Emiliano A. (2008). "CAPM and option pricing with elliptically contoured distributions". The Journal of Risk & Insurance. 75 (2): 387–409. CiteSeerX 10.1.1.573.4715. doi:10.1111/j.1539-6975.2008.00265.x.
8. Landsman, Zinoviy; Nešlehová, Johanna (2008). "Stein's Lemma for elliptical random vectors". Journal of Multivariate Analysis. 99 (5): 912––927. doi:10.1016/j.jmva.2007.05.006.