# Information Theoretic Methods in Statistics

### Wed 4:00-7:00 PM, Spring 2007 (3 Credits)

Information Theory has been primarily motivated by problems in telecommunication systems, e.g., minimizing the description length of random processes or maximizing the number of distinguishable signals in the presence of noise. This perspective - optimization - has led to a number of insights that contribute to the body of knowledge in Statistics and Probability Theory. This course will discuss some applications of information theoretic methods in statistics.

The course will begin with a very brief introduction to information theory. Notions of entropy, mutual information, and Kullback-Leibler divergence will be reviewed. Their significance in data compression and error correcting codes will be brought out via the source- and channel-coding theorems.

Three application areas will form the bulk of the course:

• Information Geometry: The notion of I-projections will be introduced. Examples will be drawn from problems encountered in maximum entropy and maximum likelihood estimation, the EM algorithm, etc. Projection of a probability mass function onto a linear family of probability mass functions, and iterative algorithms for finding the minimum divergence between two convex sets of probability mass functions will be studied.
• Large Deviations: Error exponents, Sanov's theorem and related results will be presented from an information theoretic viewpoint. The role of I-divergence will be highlighted, and error bounds for selected estimation problems will be presented as examples.
• Redundancy and Data Compression: Lossy and lossless compression of data will be investigated. In particular, Rissanen's Minimum Description Length (MDL) principle, and other model based techniques for universal lossless data compression will be studied, and connections with statistical modelling will be explored.

Participants are expected to be familiar with probability and statistics at the level of an advanced undergraduate or core graduate course. Exposure to information theory (ECE 520.447) will be helpful but not essential.

### Lecture Notes

Instructor: Sanjeev Khudanpur
Office: Barton 221, 410-516-7024
email: MyLastName at jhu dot edu