$\require{mathtools}
%
%%% GENERIC MATH %%%
%
% Environments
\newcommand{\al}[1]{\begin{align}#1\end{align}} % need this for \tag{} to work
\renewcommand{\r}{\mathrm}
%
% Greek
\newcommand{\eps}{\epsilon}
\newcommand{\veps}{\varepsilon}
\newcommand{\Om}{\Omega}
\newcommand{\om}{\omega}
\newcommand{\Th}{\Theta}
\let\fi\phi % because it looks like an f
\let\phi\varphi % because it looks like a p
%
% Miscellaneous shortcuts
% .. over and under
\newcommand{\ss}[1]{_{\substack{#1}}}
\newcommand{\ob}{\overbrace}
\newcommand{\ub}{\underbrace}
\newcommand{\ol}{\overline}
\newcommand{\tld}{\widetilde}
\newcommand{\HAT}{\widehat}
\newcommand{\f}{\frac}
\newcommand{\s}[2]{#1 /\mathopen{}#2}
\newcommand{\rt}[1]{ {\sqrt{#1}}}
% .. relations
\newcommand{\sr}{\stackrel}
\newcommand{\sse}{\subseteq}
\newcommand{\ce}{\coloneqq}
\newcommand{\ec}{\eqqcolon}
\newcommand{\ap}{\approx}
\newcommand{\ls}{\lesssim}
\newcommand{\gs}{\gtrsim}
% .. miscer
\newcommand{\q}{\quad}
\newcommand{\qq}{\qquad}
\newcommand{\heart}{\heartsuit}
%
% Delimiters
% (I needed to create my own because the MathJax version of \DeclarePairedDelimiter doesn't have \mathopen{} and that messes up the spacing)
% .. one-part
\newcommand{\p}[1]{\mathopen{}\left( #1 \right)}
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% .... (use phantom to force at least the standard height of double bars)
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%% .. two-part
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% ..... (use phantom to force at least the standard height of double bar)
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%\newcommand{\bpaco}[4]{\bpa{\incond{#1}{#2}}{\incond{#3}{#4}}}
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%
% Levels of closeness
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\newcommand{#4}{\mathrel{#1\!\!#1}}
\newcommand{#5}{\mathrel{#1\!\!#1\!\!#1}}
\newcommand{#6}{\mathrel{(\sdot{#1})}}
\newcommand{#7}{\mathrel{(\slog{#1})}}
}
\let\lt\undefined
\let\gt\undefined
% .. vanilla versions (is it within a constant?)
\newcommand{\ez}{\scirc=}
\newcommand{\eq}{\simeq}
\newcommand{\eqq}{\mathrel{\eq\!\!\eq}}
\newcommand{\eqqq}{\mathrel{\eq\!\!\eq\!\!\eq}}
\newcommand{\lez}{\scirc\le}
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\newcommand{\gz}{\scirc>}
\newcommand{\gt}{\succ}
\newcommand{\gtt}{\mathrel{\gt\!\!\gt}}
\newcommand{\gttt}{\mathrel{\gt\!\!\gt\!\!\gt}}
% .. dotted versions (is it equal in the limit?)
\newcommand{\ed}{\sdot=}
\newcommand{\eqd}{\sdot\eq}
\newcommand{\eqqd}{\sdot\eqq}
\newcommand{\eqqqd}{\sdot\eqqq}
\newcommand{\led}{\sdot\le}
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\newcommand{\lqqd}{\sdot\lqq}
\newcommand{\lqqqd}{\sdot\lqqq}
\newcommand{\ged}{\sdot\ge}
\newcommand{\gqd}{\sdot\gq}
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\newcommand{\gqqqd}{\sdot\gqqq}
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\newcommand{\gd}{\sdot>}
\newcommand{\gtd}{\sdot\gt}
\newcommand{\gttd}{\sdot\gtt}
\newcommand{\gtttd}{\sdot\gttt}
% .. log versions (is it equal up to log?)
\newcommand{\elog}{\slog=}
\newcommand{\eqlog}{\slog\eq}
\newcommand{\eqqlog}{\slog\eqq}
\newcommand{\eqqqlog}{\slog\eqqq}
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%
% Miscellaneous
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\newcommand{\RHS}{\mathrm{RHS}}
% .. operators
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\DeclareMathOperator{\polylog}{polylog}
\DeclareMathOperator{\quasipoly}{quasipoly}
\DeclareMathOperator{\negl}{negl}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}
% .. functions
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\err}{err}
\DeclareMathOperator{\ReLU}{ReLU}
% .. analysis
\let\d\undefined
\newcommand{\d}{\operatorname{d}\mathopen{}}
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\newcommand{\partf}[2]{\f{\part #1}{\part #2}}
\newcommand{\parts}[2]{\s{\part #1}{\part #2}}
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% .. sets of numbers
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
%
%%% SPECIALIZED MATH %%%
%
% Logic
\renewcommand{\and}{\wedge}
\newcommand{\AND}{\bigwedge}
\newcommand{\or}{\vee}
\newcommand{\OR}{\bigvee}
\newcommand{\xor}{\oplus}
\newcommand{\XOR}{\bigoplus}
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%
% Linear algebra
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\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\Img}{Im}
\newcommand{\transp}{\mathsf{T}}
\renewcommand{\t}{^\transp}
% ... named tensors
\newcommand{\namedtensorstrut}{\vphantom{fg}} % milder than \mathstrut
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\newcommand{\nbin}[2]{\mathbin{\underset{\substack{#1}}{\namedtensorstrut #2}}}
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%
% Probability
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\let\Pr\undefined
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\DeclareMathOperator*{\E}{E}
\DeclareMathOperator*{\Var}{Var}
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\DeclareMathOperator*{\median}{median}
\newcommand{\dTV}{d_{\mathrm{TV}}}
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% ... information theory
\let\H\undefined
\DeclareMathOperator*{\H}{H}
\DeclareMathOperator*{\I}{I}
\DeclareMathOperator*{\D}{D}
%
%%% SPECIALIZED COMPUTER SCIENCE %%%
%
% Complexity classes
% .. classical
\newcommand{\Poly}{\mathsf{P}}
\newcommand{\NP}{\mathsf{NP}}
\newcommand{\PH}{\mathsf{PH}}
\newcommand{\PSPACE}{\mathsf{PSPACE}}
\renewcommand{\L}{\mathsf{L}}
% .. probabilistic
\newcommand{\formost}{\mathsf{Я}}
\newcommand{\RP}{\mathsf{RP}}
\newcommand{\BPP}{\mathsf{BPP}}
\newcommand{\MA}{\mathsf{MA}}
\newcommand{\AM}{\mathsf{AM}}
\newcommand{\IP}{\mathsf{IP}}
\newcommand{\RL}{\mathsf{RL}}
% .. circuits
\newcommand{\NC}{\mathsf{NC}}
\newcommand{\AC}{\mathsf{AC}}
\newcommand{\ACC}{\mathsf{ACC}}
\newcommand{\TC}{\mathsf{TC}}
\newcommand{\Ppoly}{\mathsf{P}/\poly}
\newcommand{\Lpoly}{\mathsf{L}/\poly}
% .. resources
\newcommand{\TIME}{\mathsf{TIME}}
\newcommand{\SPACE}{\mathsf{SPACE}}
\newcommand{\TISP}{\mathsf{TISP}}
\newcommand{\SIZE}{\mathsf{SIZE}}
% .. keywords
\newcommand{\co}{\mathsf{co}}
\newcommand{\Prom}{\mathsf{Promise}}
%
% Boolean analysis
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\DeclareMathOperator{\fbsens}{\mathrm{fbs}}
\DeclareMathOperator{\Cert}{\mathrm{C}}
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\DeclareMathOperator{\CDT}{\mathrm{CDT}} % canonical
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\DeclareMathOperator{\PDT}{\mathrm{PDT}} % partial decision tree
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\DeclareMathOperator{\W}{\mathbf{W}}
% .. functions (small caps sadly doesn't work)
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\DeclareMathOperator{\Maj}{\mathrm{Maj}}
\DeclareMathOperator{\HW}{\mathrm{HW}}
\DeclareMathOperator{\Thr}{\mathrm{Thr}}
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\DeclareMathOperator{\RotTribes}{\mathrm{RotTribes}}
\DeclareMathOperator{\CycleRun}{\mathrm{CycleRun}}
\DeclareMathOperator{\SAT}{\mathrm{SAT}}
\DeclareMathOperator{\UniqueSAT}{\mathrm{UniqueSAT}}
%
% Dynamic optimality
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%
% Alignment
\DeclareMathOperator{\Amp}{\mathrm{Amp}}
%
%%% TYPESETTING %%%
%
% In text
\renewcommand{\th}{^{\mathrm{th}}}
\newcommand{\degree}{^\circ}
%
% Fonts
% .. bold
\newcommand{\BA}{\boldsymbol{A}}
\newcommand{\BB}{\boldsymbol{B}}
\newcommand{\BC}{\boldsymbol{C}}
\newcommand{\BD}{\boldsymbol{D}}
\newcommand{\BE}{\boldsymbol{E}}
\newcommand{\BF}{\boldsymbol{F}}
\newcommand{\BG}{\boldsymbol{G}}
\newcommand{\BH}{\boldsymbol{H}}
\newcommand{\BI}{\boldsymbol{I}}
\newcommand{\BJ}{\boldsymbol{J}}
\newcommand{\BK}{\boldsymbol{K}}
\newcommand{\BL}{\boldsymbol{L}}
\newcommand{\BM}{\boldsymbol{M}}
\newcommand{\BN}{\boldsymbol{N}}
\newcommand{\BO}{\boldsymbol{O}}
\newcommand{\BP}{\boldsymbol{P}}
\newcommand{\BQ}{\boldsymbol{Q}}
\newcommand{\BR}{\boldsymbol{R}}
\newcommand{\BS}{\boldsymbol{S}}
\newcommand{\BT}{\boldsymbol{T}}
\newcommand{\BU}{\boldsymbol{U}}
\newcommand{\BV}{\boldsymbol{V}}
\newcommand{\BW}{\boldsymbol{W}}
\newcommand{\BX}{\boldsymbol{X}}
\newcommand{\BY}{\boldsymbol{Y}}
\newcommand{\BZ}{\boldsymbol{Z}}
\newcommand{\Ba}{\boldsymbol{a}}
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\newcommand{\Bomicron}{\boldsymbol{\omicron}}
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\newcommand{\Brho}{\boldsymbol{\rho}}
\newcommand{\Bsigma}{\boldsymbol{\sigma}}
\newcommand{\Btau}{\boldsymbol{\tau}}
\newcommand{\Bupsilon}{\boldsymbol{\upsilon}}
\newcommand{\Bphi}{\boldsymbol{\phi}}
\newcommand{\Bfi}{\boldsymbol{\fi}}
\newcommand{\Bchi}{\boldsymbol{\chi}}
\newcommand{\Bpsi}{\boldsymbol{\psi}}
\newcommand{\Bomega}{\boldsymbol{\omega}}
% .. calligraphic
\newcommand{\CA}{\mathcal{A}}
\newcommand{\CB}{\mathcal{B}}
\newcommand{\CC}{\mathcal{C}}
\newcommand{\CD}{\mathcal{D}}
\newcommand{\CE}{\mathcal{E}}
\newcommand{\CF}{\mathcal{F}}
\newcommand{\CG}{\mathcal{G}}
\newcommand{\CH}{\mathcal{H}}
\newcommand{\CI}{\mathcal{I}}
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\newcommand{\CN}{\mathcal{N}}
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\newcommand{\CP}{\mathcal{P}}
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\newcommand{\CS}{\mathcal{S}}
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\newcommand{\CU}{\mathcal{U}}
\newcommand{\CV}{\mathcal{V}}
\newcommand{\CW}{\mathcal{W}}
\newcommand{\CX}{\mathcal{X}}
\newcommand{\CY}{\mathcal{Y}}
\newcommand{\CZ}{\mathcal{Z}}
% .. typewriter
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\newcommand{\TU}{\mathtt{U}}
\newcommand{\TV}{\mathtt{V}}
\newcommand{\TW}{\mathtt{W}}
\newcommand{\TX}{\mathtt{X}}
\newcommand{\TY}{\mathtt{Y}}
\newcommand{\TZ}{\mathtt{Z}}$
Usual entropic inequalities
Fundamentally, shared information is visible in the whole $\BX$ but not in the individual parts $\BX_1, \ldots, \BX_n$, so that
- entropy is subadditive: $\H[\BX] \le \sum\H[\BX_i]$ (shared information overcounted on the right);
- entropy deficit is superadditive: $\D\b{\BX} \ge \sum \D\b{\BX_i}$ (shared information is only counted on the left).
More generally, Shearer’s lemma says that the same observation not just for individual coordinates, but any groupings such that
Using independence to reverse them
When the $\BX_i$ are independent, we can reverse these inequalities:
\[
\H[\BX] \ge \sum\H[\BX_i] \qquad \D\b{\BX} \le \sum\D\b{\BX_i}.
\]
And we even get inequalities for Information divergence and mutual information that can otherwise go both ways:
\[
\D{\BX'}{\BX} \geq \sum\D{\BX_i'}{\BX_i} \qquad \I[\BX;\BY] \ge \sum\I[\BX_i;\BY].
\]
Reversing them without independence
When the variables are not independent, we cannot in general hope for $\H[\BX]$ to be as big as the sum of the individual entropies $\H[\BX_i]$, or equivalently, for the individual deficits $\D\b{\BX_i}$ to be large if the overall deficit $\D\b{\BX}$ is large.
Conditional deficits
However, we can use the chain rule to show that the conditional deficits need to be large:
\[
\D\b{\BX} = \sum \D\bco{\BX_i}{\BX_{<i}} \le \D\bco{\BX_i}{\BX_{[n]\setminus \set{i}}}
\]
which is exactly the edge-isoperimetric inequality when $\BX$ is flat. In particular, at least one of the conditional deficits must be $\ge \frac{\D\b{\BX}}{n}$.
If $\BX$ is uniform over a subset of $\zo^n$, then we can identify it with the boolean function $f:\zo^n\to\zo$ such that $\BX$ is uniform over $f^{-1}(1)$. When $\D\b{\BX} \ge 1$ (i.e. $\E[f] \le 1/2$), this means that
\[
\D\bco{\BX_i}{\BX_{[n]\setminus \set{i}}} = \Pr[f(\Bx^{\xor i}) = 0 \mid f(\Bx)=1] =\frac{\Inf_i[f]}{2\E[f]} = \Theta\p{\frac{\Inf_i[f]}{\Var[f]}},
\]
so the Kahn–Kalai–Linial theorem shows that at least one of the conditional deficits is significantly bigger than $\frac{\D\b{\BX}}{n}$ (at least when $\D\b{\BX} \ll \log n$):
\[
\max_i \D\bco{\BX_i}{\BX_{[n]\setminus \set{i}}}
\ge \Omega\p{\frac{\log n}{n}}.
\]
Combining the chain rule and Kahn–Kalai–Linial, we get the
\[
\max_i \D\bco{\BX_i}{\BX_{[n]\setminus \set{i}}}
\ge \Omega\p{\frac{\max(\log n, \D\b{\BX})}{n}},
\]
which (perhaps surprisingly) is tight for every value of $\D\b{\BX}$, as exemplified by the Tribes DNF.
When $\D\b\BX \le 1$, we instead have $\D\bco{\BX_i}{\BX_{[n]\setminus \set{i}}} = \Theta(\Inf_i[f])$ and $\Var[f] = \Theta(\D\b\BX)$, and therefore
\[
\max_i \D\bco{\BX_i}{\BX_{[n]\setminus \set{i}}} \ge \D\b\BX \cdot\Omega\p{\frac{\max\p{\log n, \log\frac1{\D\b\BX}}}{n}}
\]
TODO: turn this into an actual #figure
Log-concave distributions
TODO: For distributions, “log-concavity” is a soft notion of independence which implies entropic inequalities, see Log-concave polynomials and distributions (not yet covered in that note).