Information Theory

Information theory is a mathematical field that studies the quantification, storage, and communication of digital information. Pioneered by Claude Shannon, it establishes the fundamental limits on how much data can be compressed without loss (data compression) and transmitted reliably over a noisy channel (channel capacity). The central concept is entropy, which measures the average level of uncertainty or "surprise" inherent in a variable's possible outcomes, thereby quantifying the amount of information contained in a message. This foundational theory provides the theoretical underpinning for numerous applications in computer science, including data compression algorithms, error-correcting codes, cryptography, and even concepts within machine learning.

1. Foundational Concepts of Information
   1. Historical Context and Development
      1. Early Communication Theory
         1. Telegraphy and Early Electrical Communication
         2. Statistical Approaches to Communication Problems
         3. Hartley's Contributions to Information Measurement
      2. Claude Shannon's Revolutionary Work
         1. "A Mathematical Theory of Communication" (1948)
         2. Key Insights and Breakthroughs
         3. Impact on Engineering and Computer Science
      3. Post-Shannon Developments
         1. Expansion into Network Information Theory
         2. Applications in Computer Science
         3. Modern Information-Theoretic Applications
   2. Information as a Quantifiable Concept
      1. The Notion of Surprise and Uncertainty
         1. Relationship Between Probability and Information
         2. Rare Events and High Information Content
         3. Predictable Events and Low Information Content
      2. Units of Information Measurement
         1. The Bit (Binary Digit)
         2. The Nat (Natural Unit)
         3. The Hartley (Decimal Unit)
         4. Conversion Between Units
      3. Logarithmic Nature of Information
         1. Why Logarithms Are Used
         2. Choice of Logarithm Base
         3. Additivity Property of Information
         4. Self-Information of an Event
   3. Mathematical Prerequisites
      1. Probability Theory Fundamentals
         1. Sample Spaces and Events
         2. Probability Axioms
         3. Conditional Probability
         4. Bayes' Theorem
      2. Random Variables
         1. Discrete Random Variables
         2. Continuous Random Variables
         3. Mixed Random Variables
      3. Probability Distributions
         1. Probability Mass Functions
         2. Probability Density Functions
         3. Cumulative Distribution Functions
      4. Expectation and Moments
         1. Expected Value
         2. Variance and Standard Deviation
         3. Higher-Order Moments
         4. Moment Generating Functions
      5. Joint Distributions
         1. Joint Probability Mass Functions
         2. Joint Probability Density Functions
         3. Marginal Distributions
         4. Conditional Distributions
      6. Independence and Dependence
         1. Statistical Independence
         2. Conditional Independence
         3. Correlation and Dependence Measures