# How is Efron–Stein inequality useful

From Wikipedia

The Efron–Stein inequality (or influence inequality, or MG bound on variance) bounds the variance of a general function.

Suppose that $X_1 \dots X_n, X_1' \dots X_n'$ are independent with $X_i'$ and $X_i$ having the same distribution for all $i$.

Let $X = (X_1,\dots , X_n), X^{(i)} = (X_1, \dots , X_{i-1}, X_i',X_{i+1}, \dots , X_n)$. Then $$\mathrm{Var}(f(X)) \leq \frac{1}{2} \sum_{i=1}^{n} E[(f(X)-f(X^{(i)}))^2].$$

I wonder how this equality can be useful? It is supposed to provide an upper bound on $\mathrm{Var}(f(X))$, but I don't see $E[(f(X)-f(X^{(i)}))^2]$ in this bound is easier to compute than $\mathrm{Var}(f(X))$.

Thanks!

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