Update.

[tex.git] / inftheory.tex

diff --git a/inftheory.tex b/inftheory.tex
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--- a/inftheory.tex
+++ b/inftheory.tex
@@ -1,5 +1,9 @@
 %% -*- mode: latex; mode: reftex; mode: flyspell; coding: utf-8; tex-command: "pdflatex.sh" -*-
 
 %% -*- mode: latex; mode: reftex; mode: flyspell; coding: utf-8; tex-command: "pdflatex.sh" -*-
 

+%% Any copyright is dedicated to the Public Domain.
+%% https://creativecommons.org/publicdomain/zero/1.0/
+%% Written by Francois Fleuret <francois@fleuret.org>
+

 \documentclass[10pt,a4paper,twoside]{article}
 \usepackage[paperheight=18cm,paperwidth=10cm,top=5mm,bottom=20mm,right=5mm,left=5mm]{geometry}
 %\usepackage[a4paper,top=2.5cm,bottom=2cm,left=2.5cm,right=2.5cm]{geometry}
 \documentclass[10pt,a4paper,twoside]{article}
 \usepackage[paperheight=18cm,paperwidth=10cm,top=5mm,bottom=20mm,right=5mm,left=5mm]{geometry}
 %\usepackage[a4paper,top=2.5cm,bottom=2cm,left=2.5cm,right=2.5cm]{geometry}


@@ -16,8 +20,6 @@
 \usetikzlibrary{tikzmark}
 \usetikzlibrary{decorations.pathmorphing}
 \usepackage[round]{natbib}
 \usetikzlibrary{tikzmark}
 \usetikzlibrary{decorations.pathmorphing}
 \usepackage[round]{natbib}

-%\usepackage{cmbright}
-%\usepackage{showframe}

 
 \usepackage{mleftright}
 
 
 \usepackage{mleftright}
 


@@ -36,68 +38,70 @@
 \def\argmin{\operatornamewithlimits{argmin}}
 \def\expect{\mathds{E}}
 
 \def\argmin{\operatornamewithlimits{argmin}}
 \def\expect{\mathds{E}}
 

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-\newcommand{\checknbdrafts}{
-\ifnum \thenbdrafts > 0
-\@latex@warning@no@line{*WARNING* The document contains \thenbdrafts \space draft note(s)}
-\fi}
-\newcommand{\todo}[1]{\addtocounter{nbdrafts}{1}{\color{red} #1}}
-\makeatother

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{document}
 
 Information Theory is awesome so here is a TL;DR about Shannon's entropy.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{document}
 
 Information Theory is awesome so here is a TL;DR about Shannon's entropy.
 

-This field is about quantifying the amount ``of information'' contained
-in a signal and how much can be transmitted under certain conditions.
+The field is originally about quantifying the amount of
+``information'' contained in a signal and how much can be transmitted
+under certain conditions.

 
 

-What makes it awesome IMO is that it is very intuitive, and like thermodynamics in Physics it give exact bounds about what is possible or not.
+What makes it awesome IMO is that it is very intuitive, and like
+thermodynamics in Physics, it gives exact bounds about what is possible
+or not.

 
 \section{Shannon's Entropy}
 
 
 \section{Shannon's Entropy}
 

-This is the key concept from which everything is defined.
+Shannon's entropy is the key concept from which everything is defined.

 
 

-Imagine that you have a distribution of probabilities p on a finite set of symbols and that you generate a stream of symbols by sampling them one after another independently with that distribution.
+Imagine that you have a distribution of probabilities $p$ on a finite
+set of symbols, and that you generate a stream of symbols by sampling
+them one after another independently with that distribution.

 
 

-To transmit that stream, for instance with bits over a communication line, you can design a coding that takes into account that the symbols are not all as probable, and decode on the other side.
+To transmit that stream, for instance with bits over a communication
+line, you can design a coding that takes into account that the symbols
+are not all as probable, and decode on the other side.

 
 For instance if $P('\!\!A')=1/2$, $P('\!\!B')=1/4$, and
 $P('\!\!C')=1/4$ you would transmit ``0'' for a ``A'' and ``10'' for a
 ``B'' and ``11'' for a ``C'', 1.5 bits on average.
 
 
 For instance if $P('\!\!A')=1/2$, $P('\!\!B')=1/4$, and
 $P('\!\!C')=1/4$ you would transmit ``0'' for a ``A'' and ``10'' for a
 ``B'' and ``11'' for a ``C'', 1.5 bits on average.
 

-If the symbol is always the same, you transmit nothing, if they are equiprobable you need $\log_2$(nb symbols) etc.
+If the symbol is always the same, you transmit nothing, if they are
+equiprobable you need $\log_2$(nb symbols) etc.

 
 

-Shannon's Entropy (in base 2) is the minimum number of bits you have to emit on average to transmit that stream.
+Shannon's Entropy (in base 2) is the minimum number of bits you have
+to emit on average per symbol to transmit that stream.

 
 

-It has a simple formula:
+It has a simple analytical form:

 %
 \[
  H(p) = - \sum_k p(k) \log_2 p(k)
 \]
 %
 %
 \[
  H(p) = - \sum_k p(k) \log_2 p(k)
 \]
 %

-where by convention $o \log_2 0 = 0$.
+where by convention $0 \log_2 0 = 0$.

 
 

-It is often seen as a measure of randomness since the more deterministic the distribution is, the less you have to emit.
+It is often seen as a measure of randomness since the more
+deterministic the distribution is, the less you have to emit.

 
 

-The codings above are "Huffman coding", which reaches the Entropy
-bound only for some distributions. The "Arithmetic coding" does it
-always.
+The examples above correspond to "Huffman coding", which reaches the
+Entropy bound only for some distributions. A more sophisticated scheme
+called "Arithmetic coding" does it always.

 
 From this perspective, many quantities have an intuitive
 value. Consider for instance sending pairs of symbols (X, Y).
 
 
 From this perspective, many quantities have an intuitive
 value. Consider for instance sending pairs of symbols (X, Y).
 

-If these two symbols are independent, you cannot do better than sending one and the other separately, hence
+If these two symbols are independent, you cannot do better than
+sending one and the other separately, hence

 %
 \[
 H(X, H) = H(X) + H(Y).
 \]
 
 %
 \[
 H(X, H) = H(X) + H(Y).
 \]
 

-However, imagine that the second symbol is a function of the first Y=f(X). You just have to send X since Y can be computed from it on the other side.
+However, imagine that the second symbol is a function of the first
+Y=f(X). You just have to send X since Y can be computed from it on the
+other side.

 
 Hence in that case
 %
 
 Hence in that case
 %


@@ -116,7 +120,7 @@ that quantifies the amount of information shared by the two variables.
 
 \section{Conditional Entropy}
 
 
 \section{Conditional Entropy}
 

-Okay given the visible interest for the topic, an addendum: Conditional entropy is the average of the entropy of the conditional distribution:
+Conditional entropy is the average of the entropy of the conditional distribution:

 %
 \begin{align*}
 &H(X \mid Y)\\
 %
 \begin{align*}
 &H(X \mid Y)\\


@@ -126,7 +130,9 @@ Okay given the visible interest for the topic, an addendum: Conditional entropy
 
 Intuitively it is the [minimum average] number of bits required to describe X given that Y is known.
 
 
 Intuitively it is the [minimum average] number of bits required to describe X given that Y is known.
 

-So in particular, if X and Y are independent 
+So in particular, if X and Y are independent, getting the value of $Y$
+does not help at all, so you still have to send all the bits for $X$,
+hence

 %
 \[
   H(X \mid Y)=H(X)
 %
 \[
   H(X \mid Y)=H(X)


@@ -135,10 +141,11 @@ So in particular, if X and Y are independent
 if X is a deterministic function of Y then
 %
 \[
 if X is a deterministic function of Y then
 %
 \[

-  H(X \mid Y)=0
+  H(X \mid Y)=0.

 \]
 
 \]
 

-And since if you send the bits for Y and then the bits to describe X given that X is known you have sent (X, Y), we have the chain rule:
+And if you send the bits for Y and then the bits to describe X given
+that Y, you have sent (X, Y). Hence we have the chain rule:

 %
 \[
 H(X, Y) = H(Y) + H(X \mid Y).
 %
 \[
 H(X, Y) = H(Y) + H(X \mid Y).


@@ -188,5 +195,8 @@ minus the entropy of p
 H(p) = -\sum_x p(x) \log p(x).
 \]
 
 H(p) = -\sum_x p(x) \log p(x).
 \]
 

-Notation horror: if $X$ and $Y$ are random variables $H(X, Y)$ is the entropy of their joint law, and if $p$ and $q$ are distributions, $H(p,q)$ is the cross-entropy between them.
+Notation horror: if $X$ and $Y$ are random variables $H(X, Y)$ is the
+entropy of their joint law, and if $p$ and $q$ are distributions,
+$H(p,q)$ is the cross-entropy between them.
+

 \end{document}
 \end{document}