tensorization of PI

[[concept]]

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Lemma ("Tensorization" of PI)

Suppose ${\mu }_1,\dots ,{\mu }_N$ are probability measures over $\mathbb{R}$ each with PI $C$. Then the product measure $\mu =\underset{\text{Law}(X_1,\dots ,X_N)}{\underbrace{{\mu }_1\otimes \cdots \otimes {\mu }_N}}$ Also has PI $C$.

Proof

Using the Efron-Stein theorem, we see

\text{Var}_{X\sim \mu}(f(x)) &\leq \sum_{i=1}^N \mathbb{E}_{\text{all except }i}\left[ \overbrace{\text{Var}_{X_{i}\sim \mu_{i}}(f(x)) }^{PI \text{ of }\mu_{i}}\right] \\ &\leq \sum_{i=1}^N \mathbb{E}_{\text{all except }i}\left[C\mathbb{E}_{X_{i}\sim \mu_{i}}\left[(\partial_{i} f(x))^2\right] \right] \\ &\leq C \mathbb{E}_{}\left[\lvert \lvert \nabla f(x) \rvert \rvert ^2\right] \end{align}$$

References

References

See Also

Mentions

Mentions

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