# Information Theory 2014

#### News:

30 Nov 2014: Philip added some more topics to the list of topics for the final presentations.
26 Nov 2014: The final exam will consist of student presentations. Check the list of topics here.
27 Oct 2014: Good reasons to put away your mobile phone during class
See here for the Spring 2014 edition 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.

## Intended Learning Outcomes

At the end of the course, you are able to
• Define Shannon entropy and Mutual Information and compute these quantities on examples.
• Work with joint discrete random variables (conditioning, Bayes' rule)
• Define basic discrete probability distributions (Bernoulli, Binomial, Geometric) and compute their expected value and variance
• State Jensen's inequality for convex and concave functions and use it in proofs
• Use entropy diagrams to read off and find new relations between entropic quantities
• Prove Shannon's theorem about perfectly secure encryption (e.g. by using an entropy diagram)
• Define typical and jointly typical sets, prove properties about their size and probability mass and know how they are used in source coding and channel coding
• Use Kraft's inequality e.g. to check if a prefix-free code exists for given codeword lengths
• Compute a d-ary Huffman code
• Describe how much a given source can be compressed and give a way to do it
• Prove properties about Arithmetic Codes
• Find the confusability graph of a given channel, and find channels for a given confusability graph
• Compute the independence number and (zero-error) Shannon capacity of a given confusability graph
• Compute the capacity and maximizing input distribution of a given channel
• Define basic channels (binary symmetric, erasure channel)
• State Shannon's noisy channel-coding theorem and and understand the key ingredients of the proof
• Grasp definitions of types of entropy different than Shannon entropy

## Course website

Updated information about the course can be found on http://homepages.cwi.nl/~schaffne/courses/inftheory/2014/

## 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.

The final exam will consist of student presentations about slightly more advanced topics connected to the course. The detailed procedure and list of topics can be found here.

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 2014

Date Content Homework
Mon, 27 Oct

Overview of the course, Probability Theory

Section 2.1 of [CF]

Slides #1

Wed, 29 Oct

Shannon Entropy, Jensen's inequality, Properties of Shannon entropy

Sections 3.1-3.3 of [CF]

Slides #2

Exercises #1
Mon, 3 Nov

Chain Rule, Mutual Information, Entropy Diagrams, Markov Chains

Sections 3.3, 3.4 of [CF], Sections 2.5, 2.8 of [CT]

Wed, 5 Nov

Data-Processing Inequality, Sufficient Statistics, Perfectly Secure Encryption: One-time pad

Sections 2.8-2.9 of [CT], Section 4 of [CF]

Exercises #2
Mon, 10 Nov

Perfectly Secure Encryption: Shannon's theorem. Fano's inequality. Data compression / Source coding

Section 4 of [CF], Sections 2.10 and 3.1 of [CT]

Insecurity of Key Reuse in OTP

Wed, 12 Nov

Data Compression: Asymptotic Equipartition Property, Typical set, Source-coding Theorem, high-probability set

Section 3.2+3.3 of [CT]

Slides #6

Homework #3
Mon, 17 Nov

Data Compression: symbol codes, properties, source-coding theorem reloaded, Kraft's inequality

Section 5 of [CF], Chapter 5 of [CT], Chapter 5 of [MacKay]

Slides #7

Wed, 19 Nov

Huffman codes and their optimality, Game of 20 Questions, Arithmetic Codes

Section 5.4+5.5 of [CF], Section 6.2 of [MacKay]

Online Game of 20 questions

Homework #4
Mon, 24 Nov

Advantages of Arithmetic Codes, Noisy-Channel Coding, Basic Definitions, Graph Theory

Pages 4+5 of this survey paper

Section 7.5 of [CT]

Wed, 26 Nov

Zero-error channel coding

Homework #5
Mon, 1 Dec

Noisy-channel coding: capacity, set-up and proof of converse

Sections 7.1, 7.4, 7.5, 7.9, 7.12 of [CT]

Wed, 3 Dec

Sections 7.6, 7.7 of [CT], Chapter 10 of [MacKay]

Slides #12

Homework #6
Mon, 8 Dec

Source-channel Separation, Error-correcting codes

Section 7.13 of [CT], Chapter 1 of [MacKay]

Wed, 10 Dec

Philip Schulz about Information Theory in Machine Translation

Homework #7
Wed, 17 Dec, 10:00-12:30
 10:00 - 10:30 Almudena: Uniqueness of the uncertainty measure Slides 10:30 - 11:00 Fangzhou: Kolmogorov Complexity I Slides 11:15 - 11:45 Jesus: Kolmogorov Complexity II Slides 11:45 - 12:15 Giulio: Gambling Slides

Location: CWI, ground floor of the new wing, room L0.17

Thu, 18 Dec, 13:00-15:00
 13:00 - 13:30 Arianna: Wald's Inequality Slides 13:30 - 14:00 Philip: Expectation-based syntactic comprehension Slides 14:15 - 14:45 Davide: Code Breaking Slides

## 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: