# Information Theory 2015

#### News:

9 Dec 2015: added 2014 exam
24 Nov 2015: updated Exercise #4
24 Oct 2015: updated course outline
25 September 2015: updated required previous knowledge
5 March 2015: created new page
27 Oct 2014: Good reasons to put away your mobile phone during class
See here for the Spring 2014 and Fall 2014 editions of this course.

## Content of the course

Information theory was developed by Claude E. Shannon in the 1950s to investigate the fundamental limits on signal-processing operations such as compressing data and on reliably storing and communicating data. These tasks have turned out to be fundamental for all of computer science.

In this course, we quickly review the basics of (discrete) probability theory and introduce concepts such as (conditional) Shannon entropy, mutual information and Renyi entropy. Then, we prove Shannon's theorems about data compression and channel coding. In the course, we also cover some aspects of information-theoretic security and show applications of information theory to the area of machine learning.

## Requirements

Students are required to know the (theory) contents of the course Basic Probability, Computing and Statistics in the Master of Logic (no programming will be required for this course). Study the script and the theory homework exercises.

## Intended Learning Outcomes

At the end of the course, you will be able to solve problems of the following kinds:

(Probability)

• compute the probability of an event using the most common discrete probability distributions (Bernoulli, binomial, and geometric);

• compute inverse probabilities using Bayes' rule;

• compute the means and variances of commonly used probability distributions;

• compute the means and variances of sums or products of a random variables with known distributions;

• bound the probability of an extreme event using inequalities such as the Markov bound, Chebyshev's inequality, or Hoeffding's inequality.

(Entropy and related concepts)

• compute the entropy of a random variable;

• compute the mutual information between two random variables;

• use entropy diagrams to reason about the relative size of the entropies, conditional entropies, and mutual information of two or three random variables;

• use Jensen's inequality to bound the mean of a random variable defined in terms of convex or concave function of another random variable.

(Data compression)

• construct a d-ary Huffman code for a random variable.

• use Kraft's inequality to check whether a prefix-free code can be constructed to fit certain codeword lengths;

• bound the possible rate of lossless compression of output from a given source using Shannon's source coding theorem;

• define a typical set and reason about its size, probability, and elements;

• compute the Shannon-Fano-Elias codeword for a sample from a stochastic process.

• compute the entropy rate of a Markov process.

(Noisy-channel coding)

• construct a probability model of a communication channel given a verbal description;

• compute the channel capacity of a channel;

• use Shannon's channel coding theorem to bound the achievable rate of reliable communication over a channel;

• use Bayes' rule to decode corrupted messages sent using an error-correcting code;

• evaluate the rate and reliability of such codes;

• define the jointly typical sets of a source and channel, and use such sets to decode outputs from the channel;

• draw the confusability graph of a given channel, and describe the channel depicted by a given confusability graph;

• compute the independence number and the zero-error capacity of a confusability graph;

## Course website

Updated information about the course can be found on http://homepages.cwi.nl/~schaffne/courses/inftheory/2015/

## Study Material

The material will be presented in black-boards lectures. The following are good references:

## Lectures and Exercise sessions (2 x 45min each)

please check Datanose for the definite times and locations.

This is a 6 ECTS course, which comes to roughly 20 hours of work per week.

There will be homework exercises every week to be handed in one week later. The answers should be in English. Feel free to use LaTeX, here is a template to get you started, but readable handwritten solutions are fine, too. Cooperation while solving the exercises is allowed and encouraged, but everyone has to hand in their own solution set in their own words.

There will be a final written exam. The exam is open-book, meaning that you can bring the study material [CF], [CT], [MacKay] mentioned above as well as any notes you made, but no electronic devices are allowed. Here is the written exam from Spring 2014.

The final grade for the course consists by 1/2 of the average homework grade (ignoring the worst grade) and 1/2 of the grade obtained at the final exam.

## Course schedule Fall 2015

The following table provides an overview of the course contents as currently planned:

Day

Contents

[CF]

[CT]

[MacKay]

Exercises

1

Wed 28 Oct

Introduction, Probability Theory, Entropy

Slides #1

2.1

exercise sheet #1

Fri 30 Oct

Preparation Homework: Understand Section 2.2 of [CF], solve Exercise 9 of Sheet #1

Jensen's inequality, properties of the entropy, mutual information and dependence

Slides #2

3.1–3.3

2.1, 2.2, 2.6

2

Wed 4 Nov

Preparation Homework: Figure out how to construct a Huffman code

Data Compression: symbol codes, Kraft's inequality, source-coding theorem (symbol-code version), Huffman codes

Slides #3

5.1, 5.2

5

5, L4

exercise sheet #2

Fri 6 Nov

Asymptotic Equipartition Theorem (AEP), the source coding theorem (asymptotic version)

Slides #4

3

4.4

3

Wed 11 Nov

Preparation Homework: Play the online game of 20 questions

d-ary Huffman codes, 20 questions, optimality and disadvantages of Huffman codes, codes as probability distributions, arithmetic codes

Slides #5   Mathias's slides on arithmetic coding

5.9, 13.3

6, L5

exercise sheet #3

Fri 13 Nov

Stochastic Processes, Entropy rates

4

4

Wed 18 Nov

Ergodicity

exercise sheet #4 (updated)

Fri 21 Nov

Entropy diagrams, Markov chains, data-processing inequality, perfectly secure encryption

Slides #8

Insecurity of Key Reuse in OTP

3.3–3.4, 4

2.5, 2.8, 7.11

8

5

Wed 25 Nov

Noisy channels, error-correcting codes, Hamming code

Slides #9

7.11

1, L1

exercise sheet #5

Fri 27 Nov

Preparation Homework: look up and remember the definitions of the following concepts: graph, vertex, edge, independent set, independence number. What is the independence number of ?

Zero-error channel coding, Shannon capacity of a graph

Slides #10

7.8

6

Wed 2 Dec

Fano's inequality, Channel capacity, Shannon's channel coding theorem: intuition and proof of converse

Slides #10

2.10, 7.9–7.10

exercise sheet #6

Fri 4 Dec

Shannon's channel coding theorem

Slides #12

7.1–7.7

9, 10

7

Wed 9 Dec

Source-channel separation, Quantum Information Theory

The cocktail-party effect

Slides #13

7.13

exercise sheet #7 (slightly updated)

Fri 11 Dec

Quantum Cryptography

Serge Fehr's lecture notes on Quantum Cryptography

Photos from the exercise class: The derivative of the binary entropy equals the binary logit function. Matrix notation for channels; an example channel: picture 1, picture 2, and picture 3. Strong products, and a part of the noisy 5-typewriter's confusability graph. Symmetry and concavity: explanation, example, and illustration.

## Life after "Information Theory"

If you got hooked on the world of entropies, you have several options after the course to pursue the topics of information theory and cryptography: