Four equivalent forms of the Fourier entropy-influence conjecture

Let $f$ be a Boolean function with spectral distribution $\mu(S)=\hat f(S)^2$, and let $\gamma(T)=\sum_{U\supseteq T}\mu(U)$ be the generalized influence of a set $T$. The Fourier entropy splits as $H(f)=A+B$ with $A=\mathbb{E}_\mu\log_2(1/\gamma(S))$ and $B=\mathbb{E}_\mu\log_2(\gamma(S)/\mu(S))\ge 0$. Since derivatives of Boolean functions take values in a lattice, $A\le H(f)/2+I(f)-\mathrm{Var}(f)$, and so the Fourier entropy-influence conjecture is equivalent to $B\le C\,I(f)$. We give three further equivalent forms: a bound on the cross-entropy of the spectral distribution against the noisy generalized influences $\|T_{\sqrt t}\,\partial_T f\|_2^2$ at a fixed $t<1$; a bound on the integral over $t\in[\tfrac12,1]$ of the expected number of elements a noise-tilted posterior adds to the spectral sample; and a bound of $O(1)$ bits on the log-loss of predicting a uniformly chosen element of the size-biased spectral sample from the others, which also has a variational form over all predictors. Form (c) gives the unconditional bound $H(f)\le 4I(f)-2\mathrm{Var}(f)+2M(f)$, where $M(f)$ is the expected number of such added elements at $t=1$.