lec-crypto/entropy.tex

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\title{Entropy as a measure of information}
\author{Erik Neller}
\date{\today}

\begin{document}
\maketitle
\section{What is entropy?}
Across disciplines, entropy is a measure of uncertainty or randomness.
Originating in classical thermodynamics,
over time it has been applied in different sciences such as chemistry and information theory.
%As the informal concept of entropy gains popularity, its specific meaning can feel far-fetched and ambiguous.
The name 'entropy' was first coined by german physicist \textit{Rudolf Clausius} in 1865
while postulating the second law of thermodynamics, one of 3(4)
laws of thermodynamics based on universal observation regarding heat and energy conversion.

Specifically, the second law states that not all thermal energy can be converted into work in a cyclic process.
Or, in other words, that the entropy of an isolated system cannot decrease,
as they always tend toward a state of thermodynamic equilibrium where entropy is highest for a given internal energy.
Another result of this observation is the irreversibility of natural processes, also referred to as the \textit{arrow of time}.
Even though the first law (conservation of energy) allows for a cup falling off a table and breaking
as well as the reverse process of reassembling itself and jumping back onto the table,
the second law only allows the former and denies the latter,
requiring the state with higher entropy to occur later in time.

Only 10 years later, in 1875, \textit{Ludwig Boltzmann} and \textit{Willard Gibbs} derived the formal definition
that is still in use in information theory today.
\begin{equation}
S = -k_B \sum_i p_i \ln(p_i)
\end{equation}
It gives statistical meaning to the macroscopic phenomenon of classical thermodynamics 
by defining the entropy $S$ of a macrostate as 
the result of probabilities $p_i$ of all its constituting micro states.
$k_B$ refers to the Boltzmann constant, which he himself did not determine but is part of todays SI system.

\section{Shannon's axioms}
\textit{Claude Shannon} adapted the concept of entropy to information theory.
In an era of advancing communication technologies, the question he addressed was of increasing importance:
How can messages be encoded and transmitted efficiently?
He proposed 3(4) axioms a measure of information would have to comply with:
\begin{enumerate}
    \item $I(1) = 0$: events that always occur do not communicate information.
    \item $ I'(p) \leq 0$ is monotonically decreasing in p: an increase in the probability of an event
        decreases the information from an observed event, and vice versa.
    \item $I(p_1 \cdot p_2) = I(p_1) + I(p_2)$: the information learned from independent events
        is the sum of the information learned from each event.
    \item $I(p)$ is a twice continuously differentiable function of p.
\end{enumerate}

As a measure, Shannon's formula uses the \textit{Bit}, quantifying the efficiency of codes
and media for transmission and storage.
In information theory, entropy can be understood as the expected information of a message.
\begin{equation}
    H = E(I) = - \sum_i p_i \log_2(p_i)
    \label{eq:entropy-information}
\end{equation}
This leaves $ I =log(1/p_i) = - log_2(p_i)$, implying that an unexpected message (low probability) carries
more information than one with higher probability.
Intuitively, we can imagine David A. Johnston, a volcanologist reporting day after day that there is no
activity on Mount St. Helens. After a while, we grow to expect this message because it is statistically very likely
that tomorrows message will be the same. When some day we get the message 'Vancouver! This is it!' it carries a lot of information
not only semantically (because it announces the eruption of a volcano) but statistically because it was very unlikely
given the transmission history.

However, uncertainty (entropy) in this situation would be relatively low.
Because we attach high surprise only to the unlikely message of an eruption, the significantly more likely message
carries less information - we already expected it before it arrived.

Putting the axioms and our intuitive understanding of information and uncertainty together,
we can see the logarithmic decay of information transported by a message as its probability increases in \autoref{fig:graph-information},
as well as the entropy for a 2-event source given by solving \autoref{eq:entropy-information} for $i=2$, resulting in
$-p * \log_2(p) - (1-p) * \log_2(1-p) $.

\begin{figure}[H]
\begin{minipage}{.5\textwidth}
\begin{tikzpicture}
  \begin{axis}[
    domain=0:1,
    samples=100,
    axis lines=middle,
    xlabel={$p$},
      ylabel={Information [bits]},
    xmin=0, xmax=1,
    ymin=0, ymax=6.1,
    grid=both,
    width=8cm,
    height=6cm,
    every axis x label/.style={at={(current axis.right of origin)}, anchor=west},
    every axis y label/.style={at={(current axis.above origin)}, anchor=south},
  ]
    \addplot[thick, blue] {-log2(x)};
  \end{axis}
\end{tikzpicture}
\caption{Information contained in a message depending on its probability $p$}
\label{fig:graph-information}
\end{minipage}
\begin{minipage}{.5\textwidth}
\begin{tikzpicture}
  \begin{axis}[
    domain=0:1,
    samples=100,
    axis lines=middle,
    xlabel={$p$},
    ylabel={Entropy [bits]},
    xmin=0, xmax=1,
    ymin=0, ymax=1.1,
    xtick={0,0.25,0.5,0.75,1},
    grid=both,
    width=8cm,
    height=6cm,
    every axis x label/.style={at={(current axis.right of origin)}, anchor=west},
    every axis y label/.style={at={(current axis.above origin)}, anchor=south},
  ]
    \addplot[thick, blue] {-x * log2(x) - (1-x) * log2(1-x)};
  \end{axis}
\end{tikzpicture}
\caption{Entropy of an event source with two possible events, depending on their probabilities $(p, 1-p)$}
\label{fig:graph-entropy}
\end{minipage}
\end{figure}

The base 2 is chosen for the logarithm as our computers rely on a system of the same base, but theoretically
arbitrary bases can be used as they are proportional according to $\log_a b = \frac{\log_c b}{\log_c a} $.

Further, the $\log_2$ can be intuitively understood for an event source with $2^n$ possible outcomes -
using standard binary coding, we can easily see that a message has to contain $\log_2(2^n) = n$ Bits
in order to be able to encode all possible outcomes.
For numbers where $a \neq 2^n$ such as $a=10$, it is easy to see that there exists a number $a^k = 2^n$
which defines a message size that can encode the outcomes of $k$ event sources with $a$ outcomes each,
leaving the required Bits per event source at $\log_2(a^k) \div k = \log_2(a)$.

%- bedingte Entropie
%- Redundanz
%- Quellentropie
\section{Applications}
\subsection{Decision Trees}
A decision tree is a supervised learning approach commonly used in machine learning.
The goal is to create an algorithm, i.e a series of questions to pose to new data (input variables)
in order to predict the target variable, a class label.
Graphically, each question can be visualized as a node in a tree, splitting the dataset into two or more groups.
This process is applied to the source set and then its resulting sets in a process called \textit{recursive partitioning}.
Once a leaf is reached, the class of the input has been successfully determined.

In order to build the shallowest possible trees, we want to use input variables that minimize uncertainty.
While other measures for the best choice such as the \textit{Gini coefficient} exist,
entropy is a popular measure used in decision trees.

Using what we learned about entropy, we want the maximum decrease in entropy of our target variable,
as explained in~\autoref{ex:decisiontree}.
\begin{figure}[H]
\centering
\begin{minipage}{.3\textwidth}
\begin{tabular}{c|c|c}
    & hot & cold \\
    \hline
    rain &4  &5 \\
    \hline
    no rain & 3 & 2 \\
\end{tabular}
\end{minipage}
\begin{minipage}{.6\textwidth}
When choosing rain as a target variable, the entropy prior to partitioning is $H_{prior} = H(\frac{9}{14},\frac{5}{14})$, 
after partitioning by temperature (hot/cold)$H_{hot}= H(\frac{4}{7}, \frac{3}{7})$
and $H_{cold}= H(\frac{5}{7}, \frac{2}{7})$ remain.
This leaves us with an expected entropy of
$p_{hot} * H_{hot} + p_{cold} * H_{cold} $ .
The \textbf{information gain} can then be calculated as the difference of entropy proor and post partitioning.
Since $H_{prior}$ is constant in this equation, it is sufficient to minimize post-partitioning $E[H]$.
\end{minipage}
\caption{Example of information gain in decision trees}
\label{ex:decisiontree}
\end{figure}

Advantages of decision trees over other machine learning approaches include low computation cost and
interpretability, making it a popular choice for many applications.
However, drawbacks include overfitting and poor robustness, where minimal alterations to training data
can lead to a change in tree structure.

\subsection{Cross-Entropy}
When dealing with two distributions, the \textit{cross-entropy}, also called Kullback-Leibler divergence
between a true distribution $p$
and an estimated distribution $q$ is defined as:
\begin{equation}
    H(p, q) = -\sum_x p(x) \log_2 q(x)
\end{equation}
The \textit{Kullback–Leibler divergence} measures how much information is lost when $q$
is used to approximate $p$:
\begin{equation}
    D_{KL}(p \| q) = H(p, q) - H(p)
\end{equation}
In machine learning, this term appears in many loss/cost functions — notably in classification problems 
(cross-entropy loss) and in probabilistic models such as Variational Autoencoders (VAEs).
There, the true and predicted label are used as the true and estimated distribution, respectively.
In a supervised training example, the cross entropy loss degenerates to $-\log(p_{pred i})$ as the
true label vector is assumed to be the unit vector $e_i$ (one-hot).

\subsection{Coding}
The concept of entropy also plays a crucial role in the design and evaluation of codes used for data compression and transmission.
In this context, \textit{coding} refers to the representation of symbols or messages
from a source using a finite set of codewords.
Each codeword is typically composed of a sequence of bits,
and the design goal is to minimize the average length of these codewords while maintaining unique decodability.

According to Shannon's source coding theorem, the theoretical lower bound for the average codeword length of a source
is given by its entropy $H$.
In other words, no lossless coding scheme can achieve an average length smaller than the source entropy when expressed in bits.
Codes that approach this bound are called \textit{efficient} or \textit{entropy-optimal}.
A familiar example of such a scheme is \textit{Huffman coding},
which assigns shorter codewords to more probable symbols and longer ones to less probable symbols,
resulting in a prefix-free code with minimal expected length.

Beyond compression, coding is essential for reliable communication over imperfect channels.
In real-world systems, transmitted bits are often corrupted by noise, requiring mechanisms to detect and correct errors.
One simple but powerful concept to quantify the robustness of a code is the \textit{Hamming distance}.
The Hamming distance between two codewords is defined as the number of bit positions in which they differ.
For example, the codewords $10110$ and $11100$ have a Hamming distance of 2.

A code with a minimum Hamming distance $d_{min}$ can detect up to $d_{min}-1$ errors
and correct up to $\lfloor (d_{min}-1)/2 \rfloor$ errors.
This insight forms the basis of error-correcting codes such as Hamming codes,
which add redundant bits to data in a structured way that enables the receiver to both identify and correct single-bit errors.

Thus, the efficiency and reliability of communication systems are governed by a trade-off:
higher redundancy (lower efficiency) provides greater error correction capability,
while minimal redundancy maximizes data throughput but reduces error resilience.

%Coding of a source of an information and communication channel
% https://www.youtube.com/watch?v=ErfnhcEV1O8
% relation to hamming distance and efficient codes

\subsection{Noisy communication channels}
The noisy channel coding theorem was stated by \textit{Claude Shannon} in 1948, but first rigorous proof was
provided in 1954 by Amiel Feinstein.
One of the important issues Shannon tackled with his 'Mathematical theory of commmunication'
was the insufficient means of transporting discrete data through a noisy channel that were more efficient than
the telegram - or, how to communicate reliably over an unreliable channel.
The means of error correction until then had been limited to very basic means.

\begin{figure}[H]
\begin{tikzpicture}
    \def\boxw{2.5cm}
    \def\n{5}
    \pgfmathsetmacro{\gap}{(\textwidth - \n*\boxw)/(\n-1)}
    % Draw the boxes
    \node (A) at (0, 0) [draw, text width=\boxw, align=center] {Information Source};
    \node (B) at (\boxw + \gap, 0) [draw, text width=\boxw, align=center] {Transmitter};
    \node (C) at ({2*(\boxw + \gap)}, 0) [draw, text width=\boxw, align=center] {Channel};
    \node (N) at ({2*(\boxw + \gap)}, -1) [draw, text width=\boxw, align=center] {Noise};
    \node (D) at ({3*(\boxw + \gap)}, 0) [draw, text width=\boxw, align=center] {Receiver};
    \node (E) at ({4*(\boxw + \gap)}, 0) [draw, text width=\boxw, align=center] {Destination};
    
    % Draw arrows between the boxes
    \draw[->] (A) -- (B);
    \draw[->] (B) -- (C);
    \draw[->] (C) -- (D);
    \draw[->] (D) -- (E);
    \draw[->] (N) -- (C);
\end{tikzpicture}
\caption{Model of a noisy communication channel}
\label{fig:noisy-channel}
\end{figure}
First, analogue connections like the first telephone lines, bypassed the issue altogether and relied
on the communicating parties and their brains' ability to filter human voices from the noise that was inevitably transmitted
along with the intended signal.
After some development, the telegraph in its final form used morse code, a series of long and short clicks, that,
together with letter and word gaps, would encode text messages.
Even though the long-short coding might appear similar to todays binary coding, the means of error correction were lacking.
For a long time, it relied on simply repeating the message multiple times, which is highly inefficient.
The destination would then have to determine the most likely intended message by performing a majority vote.
One might also propose simply increasing transmitting power, thereby decreasing the error rate of the associated channel.
However, the noisy channel coding theorem provides us with a more elegant solution.
It is of foundational importance to information theory, stating that given a noisy channel with capacity $C$
and information transmitted at rate $R$, there exists an $R<C$ so the error rate at the receiver can be
arbitrarily small.



\paragraph{Channel capacity and mutual information}
For any discrete memoryless channel, we can describe its behavior with a conditional probability distribution
$p(y|x)$ — the probability that symbol $y$ is received given symbol $x$ was sent.
The \textit{mutual information} between the transmitted and received signals measures how much information, on average, passes through the channel:
\begin{equation}
    I(X;Y) = \sum_{x,y} p(x, y) \log_2 \frac{p(x, y)}{p(x)p(y)} = H(Y) - H(Y|X)
\end{equation}
The \textit{channel capacity} $C$ is then defined as the maximum achievable mutual information across all possible input distributions:
\begin{equation}
    C = \max_{p(x)} I(X;Y)
\end{equation}
It represents the highest rate (in bits per symbol) at which information can be transmitted with arbitrarily small error,
given optimal encoding and decoding schemes.

\paragraph{Binary symmetric channel (BSC)}
The binary symmetric channel is one example of such discrete memoryless channels, where each transmitted bit
has a probability $p$ of being flipped during transmission and a probability $(1-p)$ of being received correctly.

\begin{figure}[H]
\begin{tikzpicture}
    \def\boxw{2.5cm}
    \def\n{4}
    \pgfmathsetmacro{\gap}{(\textwidth - \n*\boxw)/(\n-1)}
    \node (S) at (0,0) [draw, align=center, text width=\boxw] {Transmitter};
    \node (S0) at (\boxw + \gap,1) [draw, circle] {0};
    \node (S1) at (\boxw + \gap,-1) [draw, circle] {1};
    \node (D0) at ({2*(\boxw + \gap)},1) [draw, circle] {0};
    \node (D1) at ({2*(\boxw + \gap)},-1) [draw, circle] {1};
    \node (D) at ({3*(\boxw + \gap)},0) [draw, align=center, text width=\boxw] {Receiver};

    \draw[->] (S) -- (S0);
    \draw[->] (S) -- (S1);

    \draw[->,dashed] (S0) -- (D0) node[midway, above] {$1-p$};
    \draw[->,dashed] (S0) -- (D1) node[pos=0.8, above] {$p$};
    \draw[->,dashed] (S1) -- (D0) node[pos= 0.2, above] {$p$};
    \draw[->,dashed] (S1) -- (D1) node[midway, below] {$1-p$};

    \draw[->] (D0) -- (D);
    \draw[->] (D1) -- (D);
\end{tikzpicture}
\caption{Binary symmetric channel with crossover probability $p$}
\label{fig:binary-channel}
\end{figure}

The capacity of the binary symmetric channel is given by:
\begin{equation}
    C = 1 - H_2(p)
\end{equation}
where $H_2(p) = -p \log_2(p) - (1-p)\log_2(1-p)$ is the binary entropy function.
As $p$ increases, uncertainty grows and channel capacity declines.
When $p = 0.5$, output bits are completely random and no information can be transmitted ($C = 0$).
As already shown in \autoref{fig:graph-entropy}, an error rate over $p > 0.5$ is equivalent to $ 1-p < 0.5$,
though not relevant in practice.

Shannon’s theorem is not constructive as it does not provide an explicit method for constructing such efficient codes,
but it guarantees their existence.
In practice, structured codes such as Hamming and Reed–Solomon codes are employed to approach channel capacity.

\printbibliography

\end{document}