$\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)}
\newcommand{\b}[1]{\mathopen{}\left[ #1 \right]}
\newcommand{\set}[1]{\mathopen{}\left\{ #1 \right\}}
\newcommand{\abs}[1]{\mathopen{}\left\lvert #1 \right\rvert}
\newcommand{\floor}[1]{\mathopen{}\left\lfloor #1 \right\rfloor}
\newcommand{\ceil}[1]{\mathopen{}\left\lceil #1 \right\rceil}
\newcommand{\inner}[1]{\mathopen{}\left\langle #1 \right\rangle}
% .... (use phantom to force at least the standard height of double bars)
\newcommand{\norm}[1]{\mathopen{}\left\lVert #1 \vphantom{f} \right\rVert}
\newcommand{\frob}[1]{\norm{#1}_\mathrm{F}}
%% .. two-part
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\newcommand{\cond}[2]{ {\left.\incond{#1}{#2}\right.}}
\newcommand{\pco}[2]{\p{\incond{#1}{#2}}}
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\newcommand{\setco}[2]{\set{\incond{#1}{#2}}}
\newcommand{\at}[2]{ {\left.#1\right|_{#2}}}
\newcommand{\pat}[2]{\p{\at{#1}{#2}}}
\newcommand{\bat}[2]{\b{\at{#1}{#2}}}
% ..... (use phantom to force at least the standard height of double bar)
\newcommand{\oldpara}[2]{#1\vphantom{f} \mathop{}\middle\|\mathop{} #2}
%\newcommand{\para}[2]{#1\vphantom{f} \mathop{}\middle\|\mathop{} #2}
\newcommand{\para}[2]{\mathchoice{\begin{matrix}#1\\\hdashline#2\end{matrix}}{\begin{smallmatrix}#1\\\hdashline#2\end{smallmatrix}}{\begin{smallmatrix}#1\\\hdashline#2\end{smallmatrix}}{\begin{smallmatrix}#1\\\hdashline#2\end{smallmatrix}}}
\newcommand{\ppa}[2]{\p{\para{#1}{#2}}}
\newcommand{\bpa}[2]{\b{\para{#1}{#2}}}
%\newcommand{\bpaco}[4]{\bpa{\incond{#1}{#2}}{\incond{#3}{#4}}}
\newcommand{\bpaco}[4]{\bpa{\cond{#1}{#2}}{\cond{#3}{#4}}}
%
% Levels of closeness
\newcommand{\scirc}[1]{\sr{\circ}{#1}}
\newcommand{\sdot}[1]{\sr{.}{#1}}
\newcommand{\slog}[1]{\sr{\log}{#1}}
\newcommand{\createClosenessLevels}[7]{
\newcommand{#2}{\mathrel{(#1)}}
\newcommand{#3}{\mathrel{#1}}
\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}
\newcommand{\lq}{\preceq}
\newcommand{\lqq}{\mathrel{\lq\!\!\lq}}
\newcommand{\lqqq}{\mathrel{\lq\!\!\lq\!\!\lq}}
\newcommand{\gez}{\scirc\ge}
\newcommand{\gq}{\succeq}
\newcommand{\gqq}{\mathrel{\gq\!\!\gq}}
\newcommand{\gqqq}{\mathrel{\gq\!\!\gq\!\!\gq}}
\newcommand{\lz}{\scirc<}
\newcommand{\lt}{\prec}
\newcommand{\ltt}{\mathrel{\lt\!\!\lt}}
\newcommand{\lttt}{\mathrel{\lt\!\!\lt\!\!\lt}}
\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}
\newcommand{\lqd}{\sdot\lq}
\newcommand{\lqqd}{\sdot\lqq}
\newcommand{\lqqqd}{\sdot\lqqq}
\newcommand{\ged}{\sdot\ge}
\newcommand{\gqd}{\sdot\gq}
\newcommand{\gqqd}{\sdot\gqq}
\newcommand{\gqqqd}{\sdot\gqqq}
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\newcommand{\ltttd}{\sdot\lttt}
\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}
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%
% Miscellaneous
\newcommand{\LHS}{\mathrm{LHS}}
\newcommand{\RHS}{\mathrm{RHS}}
% .. operators
\DeclareMathOperator{\poly}{poly}
\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{}}
\newcommand{\df}[2]{ {\f{\d #1}{\d #2}}}
\newcommand{\ds}[2]{ {\s{\d #1}{\d #2}}}
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\newcommand{\partf}[2]{\f{\part #1}{\part #2}}
\newcommand{\parts}[2]{\s{\part #1}{\part #2}}
\newcommand{\grad}[1]{\mathop{\nabla\!_{#1}}}
% .. 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}
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\newcommand{\xor}{\oplus}
\newcommand{\XOR}{\bigoplus}
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%
% Linear algebra
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\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\Img}{Im}
\newcommand{\transp}{\mathsf{T}}
\renewcommand{\t}{^\transp}
% ... named tensors
\newcommand{\namedtensorstrut}{\vphantom{fg}} % milder than \mathstrut
\newcommand{\name}[1]{\mathsf{\namedtensorstrut #1}}
\newcommand{\nbin}[2]{\mathbin{\underset{\substack{#1}}{\namedtensorstrut #2}}}
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%
% Probability
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\let\Pr\undefined
\DeclareMathOperator*{\Pr}{Pr}
\DeclareMathOperator*{\G}{\mathbb{G}}
\DeclareMathOperator*{\Odds}{Od}
\DeclareMathOperator*{\E}{E}
\DeclareMathOperator*{\Var}{Var}
\DeclareMathOperator*{\Cov}{Cov}
\DeclareMathOperator*{\corr}{corr}
\DeclareMathOperator*{\median}{median}
\newcommand{\dTV}{d_{\mathrm{TV}}}
\newcommand{\dHel}{d_{\mathrm{Hel}}}
\newcommand{\dJS}{d_{\mathrm{JS}}}
% ... 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
\newcommand{\zo}{\set{0,1}}
\newcommand{\pmo}{\set{\pm 1}}
\newcommand{\zpmo}{\set{0,\pm 1}}
\newcommand{\harpoon}{\!\upharpoonright\!}
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\DeclareMathOperator{\Ind}{\mathrm{Ind}}
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\DeclareMathOperator{\T}{T}
\DeclareMathOperator{\sens}{\mathrm{s}}
\DeclareMathOperator{\bsens}{\mathrm{bs}}
\DeclareMathOperator{\fbsens}{\mathrm{fbs}}
\DeclareMathOperator{\Cert}{\mathrm{C}}
\DeclareMathOperator{\DT}{\mathrm{DT}}
\DeclareMathOperator{\CDT}{\mathrm{CDT}} % canonical
\DeclareMathOperator{\ECDT}{\mathrm{ECDT}}
\DeclareMathOperator{\CDTv}{\mathrm{CDT_{vars}}}
\DeclareMathOperator{\ECDTv}{\mathrm{ECDT_{vars}}}
\DeclareMathOperator{\CDTt}{\mathrm{CDT_{terms}}}
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\DeclareMathOperator{\CDTw}{\mathrm{CDT_{weighted}}}
\DeclareMathOperator{\ECDTw}{\mathrm{ECDT_{weighted}}}
\DeclareMathOperator{\AvgDT}{\mathrm{AvgDT}}
\DeclareMathOperator{\PDT}{\mathrm{PDT}} % partial decision tree
\DeclareMathOperator{\DTsize}{\mathrm{DT_{size}}}
\DeclareMathOperator{\W}{\mathbf{W}}
% .. functions (small caps sadly doesn't work)
\DeclareMathOperator{\Par}{\mathrm{Par}}
\DeclareMathOperator{\Maj}{\mathrm{Maj}}
\DeclareMathOperator{\HW}{\mathrm{HW}}
\DeclareMathOperator{\Thr}{\mathrm{Thr}}
\DeclareMathOperator{\Tribes}{\mathrm{Tribes}}
\DeclareMathOperator{\RotTribes}{\mathrm{RotTribes}}
\DeclareMathOperator{\CycleRun}{\mathrm{CycleRun}}
\DeclareMathOperator{\SAT}{\mathrm{SAT}}
\DeclareMathOperator{\UniqueSAT}{\mathrm{UniqueSAT}}
%
% Dynamic optimality
\newcommand{\OPT}{\mathsf{OPT}}
\newcommand{\Alt}{\mathsf{Alt}}
\newcommand{\Funnel}{\mathsf{Funnel}}
%
% 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}}
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\newcommand{\BK}{\boldsymbol{K}}
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\newcommand{\Bphi}{\boldsymbol{\phi}}
\newcommand{\Bfi}{\boldsymbol{\fi}}
\newcommand{\Bchi}{\boldsymbol{\chi}}
\newcommand{\Bpsi}{\boldsymbol{\psi}}
\newcommand{\Bomega}{\boldsymbol{\omega}}
% .. calligraphic
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% .. typewriter
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\newcommand{\TZ}{\mathtt{Z}}$
In the formula for ratio divergences
\[
D_f\ppa{Q}{P} \ce \E_{\Bx \sim P}\b{f\p{\f{Q(\Bx)}{P(\Bx)}}},
\]
since the function $f$ is convex, it can be lower bounded by the tangent at any point $t^*$:
\[
f(t) \ge f(t^*) - f'(t)(t-t^*)
\]
or equivalently, using the convex dual of $f$, it can be lower bounded by a tangent of any slope $a$:
\[
f(t) \ge at - f^*(a),
\]
with equality whenever $a = f'(t)$.
Plugging this into $D_f\ppa{Q}{P}$, we can even pick different values of $a$ depending on $\Bx$, which gives us a very general family of lower bounds on the ratio divergence:
\[
\al{
D_f\ppa{Q}{P}
&= \E_{\Bx \sim P}\b{f\p{\f{Q(\Bx)}{P(\Bx)}}}\\
&\ge \E_{\Bx \sim P}\b{a(x)\f{Q(\Bx)}{P(\Bx)} - f^*(a(\Bx))}\\
&= \E_{\Bx \sim Q}\b{a(\Bx)} - \E_{\Bx \sim P}\b{f^*(a(\Bx))}
}
\]
with equality when $a(x) = f'\p{\f{Q(x)}{P(x)}}$ for all $x$.
Rearranging, this becomes
\[
\E_{\Bx \sim Q}[a(\Bx)] \le \E_{\Bx \sim P}\b{f^*(a(\Bx))} + D_f\ppa{Q}{P}.
\]
That is, when $D_f\ppa{Q}{P}$ is not too large, it gives us a generic upper bound on expectations over $Q$ in terms of some other expectation over $P$!
Total variation distance
Things are particularly nice in the case of total variation distance:
\[
\dTV(P,Q) = \E_{\Bx \sim P}\b{\p{\f{Q(\Bx)}{P(\Bx)}-1}^+}.
\]
Indeed, $f(t) = (t-1)^+$ has the tangents
\[
f(t) \ge a(t-1)
\]
for any $a \in [0,1]$, though in general only two of them are interesting:
- $f(t) \ge 0$ for slope $a=0$,
- $f(t) \ge t-1$ for slope $a=1$.
Therefore we can write it as
\[
\dTV(P,Q) = \max_{a(\cdot) \in \zo} \p{\E_{\Bx \sim Q}[a(\Bx)] -\E_{\Bx \sim P}[a(\Bx)]},
\]
or rephrasing $a(\cdot)$ as the indicator of a set $A$,
\[
\dTV(P,Q) = \max_A \p{Q(A) - P(A)}.
\]
That is, the total variation distance is the maximum difference between the probabilities that $P$ and $Q$ give to some event.
For the information divergence $\D\ppa{Q}{P}$, we will in general be able to get a better bound if we consider all possible tilts of $f$ around $t=1$
\[
f(t) \ce t \log t + (r-1)(t-1).
\]
Indeed, we know that the inequality will be tight iff
\[
a = f'(t) = \log t + r \iff e^a \propto t,
\]
so $a(x)$ can be interpreted of as the logarithm of an (unnormalized) density function, with optimality (for some $r$) when $e^{a(x)} \propto \f{Q(x)}{P(x)}$ for all $x$.
The convex dual of $f$ at the slope $a = f'(t)$ is:
\[
\al{
f^*(a)
&= at - t \log t - (r-1)(t-1)\\
&= at - t(a - r) - (r-1)(t-1)\\
&= t+r-1\\
&= e^{a-r}+r-1,
}
\]
so
\[
\al{
\E_{\Bx \sim Q}\b{a(\Bx)}
&\le \E_{\Bx \sim P}\b{e^{a(\Bx)-r}+ r-1} + \D\ppa{Q}{P}\\
&= \f{\E_{\Bx \sim P}\b{e^{a(\Bx)}}}{e^r} + r-1 + \D\ppa{Q}{P},
}
\]
and optimizing over $r$ requires $e^r = \E_{\Bx \sim P}\b{e^{a(\Bx)}}$ (i.e. $e^r$ is the normalization constant of $e^a$ as a density), which gives
\[
\al{
\E_{\Bx \sim Q}\b{a(\Bx)}
&\le r + \D\ppa{Q}{P}\\
&= \log \E_{\Bx \sim P}\b{\exp a(\Bx)} + \D\ppa{Q}{P},
}
\]
or more simply, interpreting $a(\Bx)$ as a random variable $\Ba$,
\[
\E_{Q}[\Ba] \le \log \E_{P}[\exp \Ba] + \D\ppa{Q}{P}.
\]
#to-write generalize this optimization pattern (keeps popping up and almost magical, see lee--entropy-opt-1.pdf how magical it is that the density takes a simple form, completely unaffected by actual values)
Choosing the base
The above is actually true for any base of the logarithm and exponentiation, as long as $\D\ppa{Q}{P}$ is also expressed over that same base:
\[
\al{
\E_{Q}[\Ba]
&\le \log_b \E_{P}\b{b^{\Ba}} + \ub{\E_{Q}\b{\log_b \f{Q(\Bx)}{P(\Bx)}}}_\text{``$\D\ppa{Q}{P}$ in base $b$''}\\
&= \f{\ln \ob{\E_P\b{e^{s\Ba}}}^\text{moment-generating function} + \D\ppa{Q}{P}}{s},
}
\]
for $s \ce \ln b$.
That is, when $\D\ppa{Q}{P}$ is small, we can bound the expected value of $\Ba$ over $Q$ by its moment-generating function over $P$. The smaller $s$ is, the more $\D\ppa{Q}{P}$ gets amplified, so it is generally best to choose one of the largest possible values of $s$ for which the moment-generating function doesn’t blow up.
#to-write special case where $P$ is uniform
$\chi^2$-divergence
We get
\[
\p{\E_{Q}[\Ba] - \E_{P}[\Ba]}^2 \le \chi^2\ppa{Q}{P}\Var_{P}[\Ba],
\]
but I haven’t yet found an intuitive proof for this.
As concentration bounds
These variational representations are in a sense a generalization of concentration bounds. Instead of only bounding the probability that $\Ba$ exceeds a certain value, they bound $\E_Q[\Ba]$ over any distribution $Q$ that is not too far from the prior $P$.
Formally, if we want to bound $p \ce \Pr_{P}[\Ba \ge v]$ for some threshold $v$, we can let $Q$ be the conditional distribution $\at{P}{\cdot \ge v}$, which makes $D_f\ppa{Q}{P}$ a (decreasing) function of $p$, and write
\[
\al{
D_f \ppa{Q}{P}
&\ge \E_{Q}\b{\Ba} - \E_{P}\b{f^*(\Ba)}\\
&= \E_{P}\bco{\Ba}{\Ba\ge v} - \E_{P}\b{f^*(\Ba)}\\
&\ge v - \E_{P}\b{f^*(\Ba)},
}
\]
which then gives an upper bound on $p$.
We have
\[
\al{
\log \f1p
&= \D\ppa{Q}{P}\\
&\ge sv - \ln\E_{P}\b{e^{s\Ba}},
}
\]
that is, for $\Ba \sim P$,
\[
\Pr[\Ba \ge v] \le \f{\E\b{e^{s\Ba}}}{e^{sv}},
\]
which is the moment-generating function method (also known as Chernoff bound).
$\chi^2$-divergence
Let
\[
\al{
\mu &\ce \E_{P}[\Ba]\\
\sigma^2 &\ce \Var_{P}[\Ba]
}
\]
and let’s rewrite $v$ as $\mu + t\sigma$ for $t \ge 0$. Then we have
\[
\al{
\f1p - 1
&= \chi^2\ppa{Q}{P}\\
&\ge \f{\p{\E_{Q}[\Ba]-\E_{P}[\Ba]}^2}{\Var_{P}[\Ba]}\\
&\ge \f{\p{v-\mu}^2}{\sigma^2}\\
&= t^2,
}
\]
that is, for $\Ba \sim P$,
\[
\Pr[\Ba \ge \mu + t\sigma] \le \f{1}{1+t^2},
\]
which is Chebyshev’s inequality.
Total variation distance
This one requires a bit more adaptation (and because of this is perhaps more of a hairpull). We have
\[
\al{
1-p
&= \dTV(P,Q)\\
&\ge \E_{x \sim Q}[a(x)] - \E_{x \sim P}[a(x)],
}
\]
but this only holds if $a(x) \in [0,1]$. So we perform the change of variables
\[
a(x) \gets \f{\min(a(x), v)}{v},
\]
which means that for any $\Ba \ge 0$ we have
\[
\al{
1-p
&\ge \E_{Q}\b{\f{\min(\Ba, v)}v} - \E_{P}\b{\f{\min(\Ba, v)}v}\\
&= 1 - \f{\E_{P}\b{\min(\Ba, v)}}v\\
&\ge 1 - \f{\E_{P}\b{\Ba}}v,
}
\]
so for $\Ba \sim P$,
\[
\Pr[\Ba \ge v] \le \f{\E[\Ba]}{v},
\]
which is Markov’s inequality.
See also