# ECE/TCOM-5583: Information Theory and Statistical Inference

There has been a strong resurgence of AI in recent years. An important core technology of AI is statistical learning, which aims to automatically “program” machines with data. While the idea can date back to the 50's of the last century, the plethora of data and inexpensive computational power allow the techniques to thrive and penetrate into every aspect of our daily lives — customer behavior prediction, financial market prediction, fully automatic surveillance, self-driving vehicles, autonomous robots, and beyond.

Information theory was first introduced and developed by the great communications engineer, Claude Shannon in the 50's of the last century. The theory was introduced in an attempt to explain the principle behind point-to-point communication and data storing. However, the technique has been incorporated into statistical learning and has inspire many of the underlying principles. In this graduate course, we would try to explore the exciting area of statistical learning from the perspectives of information theorists. It facilitates students to have a deeper understanding of the omnipresent field of statistical learning and to appreciate the wide-spread significance of information theory.

The course will start by providing an overview of information theory and statistical learning. We will then aid students to establish a solid foundation on core information theory principles including information measures, AEP, source and channel coding theory. We will then introduce common and yet powerful statistical techniques such as Bayesian learning, decision forests, and belief propagation algorithms and discuss how these modern statistical learning techniques are connected to information theory. The main reference text is a book by Professor Mackay — Information Theory, Inference, and Learning Algorithms but we will also borrow heavily from materials available online. Other most important reference texts are

• Learning from data by Abu-Mostafa, Magdon-Ismail, and Lin

• Element of Information Theory by Cover and Thomas

• Pattern Recognition and Machine Learning by Bishop

• Machine Learning: A Probabilistic Perspective by Kevin P. Murphy

### Office Hours

There are no “regular” office hours. But you are welcome to come catch me anytime or contact me through emails.

### Course Syllabus (Tentative)

• Probability review

• Maximum likelihood estimator, MAP, Bayesian estimator

• Graphical models and message passing algorithms

• Lossless source coding theory, Huffmann coding, and introduction to Arithmetic coding

• Asymptotic equipartition property (AEP), typicality and joint typicality

• Entropy, conditional entropy, mutual information, and their properties

• Channel coding theory, capacity, and Fano’s inequality Continuous random variables, differential entropy, Gaussian source, and Gaussian channel

• Error correcting codes, linear codes, and introduction to low-density parity check code

• Methods of type, large deviation theory, maximum entropy principle

N.B. You will expect to expose to some Python and Matlab. You won't become an expert on these things after this class. But it is good to get your hands dirty and play with them early.

### Projects

Homework: 30%.

"Mid-term": 30%. Closed book test, and no Internet and cell-phones, but you can bring calculator and two pieces of letter-size cheat sheet

Final Project: 40%.

• A: $$\sim$$ 80 and above

• B: $$\sim$$ between 60 and 80

• C: $$\sim$$ between 40 and 60

• D: $$\sim$$ between 20 and 40

• F: Below 20

### Calendar

 Topics Materials 8/24 Overview of IT, probability overview, univariate normal distribution Slides from Berkeley CS188, Slides from Wash U CSE 473 8/29 Hermitian matrices, covariance matrices, independence vs conditional independence slides, gaussian.pdf 9/5 ML, MAP, Bayesian inference, marginalizing multivariate normal distribution slides 9/12 Principal component analysis (PCA), conditioning of multivariate normal distribution, product of multivariate normal distribution slides 9/19 Mixture of gaussian, conjugate prior, Bernoulli distribution, binomial distribution, Beta distribution slides 9/26 Multinormial distribution, Dirichlet distribution, Poisson distribution slides 10/3 Constraint optimization, Lagrange multiplier, Karush-Kuhn-Tucker condition, overview of Source Coding Theorem, Kraft's inequality, converse proof of Source Coding Theorem slides 10/10 Shannon-Fano-Elias codes, forward proof of Source Coding Theorem, different entropy, Jensen's inequality slides 10/17 Joint entropy, conditional entropy, Kullback-Leibler divergence, mutual information, Shannon's perfect secrecy slides 10/24 Decison trees, random forests, law of large number, asymptoic equipartion (AEP), typical sequences slides, Random Forest notebook by Jason Sanchez 10/31 Joint typical sequences, packing and covering lemmas, channel capacity, Channel Coding Theorem slides 11/7 Converse proof of Channel Coding Theorem, rate-distortion problem, rate-distortion theorem slides 11/14 Review 11/21 Proof of rate-distortion theorem, Mid-term slides 11/28 Method of type, large deviation theory slides 12/5 Bayesian networks, factor graphs, belief propagation algorithm slides
• All lecture slides in one pdf