# Introduction to Information Theory (Fall 2021)

**Program:** Bachelor Mathematics, Informatics, AI, Physics & Astronomy (WI, WInf, IN, KI, NS)

**Lecturer:** Michael Walter

**Teaching assistants:** Yachen Liu, Misho Yanakiev

**Schedule:** Nov-Dec 2021 (period 2)

**Further information:** DataNose , Canvas

See **here** for the latest edition of this course.

The Canvas course page is the primary source for all course material. This page is offered only for your convenience.

## Course description

This course will give an introduction to *information theory* – the mathematical theory of information.
Ever since its inception, information theory has had a profound impact on society.
It underpins important technological developments, from reliable memories to mobile phone standards, and its versatile mathematical toolbox has found use in computer science, machine learning, physics, and even pure mathematics.

Starting from probability theory, we will discuss how to mathematically model *information sources* and *communication channels*, how to optimally *compress* information, and how to design *error-correcting codes* that allow us to reliably communicate over noisy communication channels. We will also see how techniques used in information theory can be applied more generally to *make predictions* from noisy data.

## Recommended prior knowledge

We will discuss rigorous definitions and proofs, hence some “mathematical maturity” will be assumed. We will use *basic probability theory* throughout the course (and will remind you of the most important facts). In addition, part of the homework will require *programming* in Python.

## Lecture notes and video recordings

Video recordings are available on Canvas.

- Lecture 1 (Nov 1, 2021): Welcome, Introduction to Information Theory
- Lecture 2 (Nov 3, 2021): Probability Theory Refresher
- Lecture 3 (Nov 8, 2021): Numerical Random Variables, Convexity and Concavity, Entropy
- Lecture 4 (Nov 10, 2021): Symbol Codes: Lossless Compression, Huffman Algorithm
- Lecture 5 (Nov 15, 2021): Block Codes: Shannon’s Source Coding Theorem, its Proof, and Variations
- Lecture 6 (Nov 17, 2021): Stream Codes: Lempel-Ziv Algorithm
- Lecture 7 (Nov 22, 2021): Stream Codes: Arithmetic Coding
- Lecture 8 (Nov 24, 2021): Joint Entropies & Communication over Noisy Channels
- Lecture 9 (Nov 29, 2021): Shannon’s Noisy Coding Theorem
- Lecture 10 (Dec 1, 2021): Proof of the Noisy Coding Theorem
- Lecture 11 (Dec 6, 2021): Proof of the Converse, Shannon’s Theory vs Practice
- Lecture 12 (Dec 8, 2021): Reed-Solomon Codes
- Lecture 13 (Dec 13, 2021): Message Passing for Decoding and Inference, Outlook
- Student Presentations (Dec 15, 2021)

Our main reference (which is optional, since handwritten lecture notes and video recordings will be provided) is David J. C. MacKay’s beautiful textbook “Information Theory, Inference, and Learning Algorithms”, Cambridge (2003). It is available online 📖 and contains much more material than what we will be able to discuss in class.

## Homework

- Problem set 1 (due Nov 7, 2021)
- Problem set 2 (due Nov 14, 2021)
- Problem set 3 (due Nov 21, 2021)
- Problem set 4 (due Nov 28, 2021)
- Problem set 5 (due Dec 5, 2021)
- Problem set 6 (due Dec 12, 2021)

## Practice Problems

In addition, we will offer many practice problems (one set for each exercise class) so that you can test your understanding as the course progresses.

## Learning objectives

At the end of the course, you will be able to:

- mathematically model memoryless and Markov information sources using probability theory
- mathematically model memoryless information channels using probability theory
- compute basic properties of probability distributions, such as entropies and essential bit contents, using mathematical techniques such as Jensen’s inequality
- define typical sets and estimate their size and probability
- prove and apply Shannon’s source coding theorem
- prove and apply the Kraft inequality for uniquely decodable (UD) codes, its converse for prefix codes, and use it to bound the average length for optimal UD or prefix codes
- design optimal symbol codes using Huffman’s coding algorithm
- analyze and apply the Lempel-Ziv algorithm for stream coding
- analyze and apply arithmetic coding for different probabilistic models
- compute basic properties of joint probability distributions, such as conditional entropies and mutual informations
- understand basic properties of block codes, such as their rate and various probabilities of error
- prove and apply Shannon’s noisy channel coding theorem
- encode Reed-Solomon codes and decode them for erasure errors
- understand maximum likelihood approach to codeword and bitwise decoding
- design and apply message passing algorithms for encoding and inference problems
- independently read about a topic in information theory and present it to your peers

We will check off these objectives as we move along in the course.

## Format

Lectures, exercises classes, self-study, and a short presentation.

## Assessment

The final grade will be determined by the following calculation:

*60% exam grade + 30% homework grade + 10% presentation grade*

The same rule applies for the re-sit exam.

There will be one *homework* problem set per week (in total 6), posted on the course homepage by Monday.
You must submit your completed homework *on Canvas* before Monday the week after.
We will ask you to collaborate and submit *in groups of 3-4 students*.
The solutions will be discussed in the Wednesday exercise class (among other things).
Assignments will be accepted late only if you have extenuating circumstances (such as sickness or family emergency) and provided you confirm with the lecturer before the deadline.
Your problem set with the lowest score will be ignored (this includes any problem set you did not submit).

Instead of having a midterm exam, we would like you to read about a topic in information theory and give a *short presentation* to your peers.
You will present as a group of 3-4 students; everyone should speak for a few minutes.
We will give you many suggestions for topics (on Canvas) but you are free to pick your own topic (but please confirm it with us).

Everything that we discussed in class and on the homework is in principle in scope for the exam.
We recommend that you revisit the homework problems and the exercises as a preparation. You are allowed to bring one self-prepared *“cheat sheet”* to the exam (A4 paper, hand-written, you can use both sides).