= Influence of a variable
{title2=$\operatorname{Inf}_i(f)$}
= Influence
{synonym}
For the $\{-1,1\}$-valued convention on the unbiased <Boolean hypercube>, the influence of coordinate $i$ is $\operatorname{Inf}_i(f)=\lVert D_i f\rVert_2^2$, where $D_i f$ is the <discrete derivative of a Boolean function>. It equals the <probability> that flipping coordinate $i$ changes $f$. For the $\{0,1\}$-valued convention, the same flip probability is instead $4\lVert D_i f\rVert_2^2$, since a nonzero discrete derivative has magnitude $1/2$.
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