bounded difference for variance

[[concept]]

Topics

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Corollary (Bounded Difference for Variance)

Let $x=(x_1,\dots ,x_N)$ have independent entries and $f:{\Sigma }_1\times \cdots \times {\Sigma }_N\to \mathbb{R}$ be a function on the product measure space. Then if ${\Delta }_{i}f:={\sup }_{\begin{array}{c}{x}_1\in {\Sigma }_1\\ x_2\in {\Sigma }_2\\ ⋮\\ x_N\in {\Sigma }_N\\ y_i\in {\Sigma }_i\end{array}}\{f(x_1,\dots ,x_{i-1},x_i,x_{i+1},\dots ,x_N)-f(x_1,..,x_{i-1},y_i,x_{i+1},\dots x_N)\}\ge 0$

By Efron-Stein, we get $\text{Var}(f(x))\le \frac{1}{4}{\sum }_{i=1}^N({\Delta }_{i}f{)}^2$

References

References

See Also

Mentions

Mentions

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