Volume 6 (2010) Article 4 pp. 81-84 [Note]
Decision Trees and Influence: an Inductive Proof of the OSSS Inequality
$\DeclareMathOperator{\zo}{\{0,1\}} %bit set \newcommand{\oo}{\{-1,1\}} %bit set \DeclareMathOperator*{\Var}{Var} \DeclareMathOperator{\Inf}{Inf}$
We give a simple proof of the OSSS inequality (O’Donnell, Saks, Schramm, Servedio, FOCS 2005). The inequality states that for any decision tree $T$ calculating a Boolean function $f:\zo^n\rightarrow \oo$, we have $\Var[f] \leq \sum_i \delta_i(T)\Inf_i(f)$, where $\delta_i(T)$ is the probability that the input variable $x_i$ is read by $T$ and $\Inf_i(f)$ is the influence of the $i$th variable on $f$.