Information Theory is a mathematical field that studies the quantification, storage, and communication of information. It provides a framework for measuring the amount of information contained within a message and how efficiently it can be transmitted through a communication channel, primarily using concepts from probability theory.
The foundation of Information Theory was laid by Claude Shannon in the 1940s, and it has since become a fundamental pillar for modern communication systems, data compression, and cryptography.
Key Points of Information Theory
The central concept of entropy
Binary Memoryless Source (BMS)
Entropy
Information Entropy is a concept that measures the uncertainty in information. It originates from entropy in physics, which describes the disorder of a system.
In Information Theory, entropy quantifies the uncertainty or randomness of information content. The higher the entropy, the greater the amount of information contained in a message and the stronger its ability to reduce uncertainty.
It uses mathematical definitions used to describe the average uncertainty of all the information that an information source can generate.
Suppose we have a random variable X with multiple possible outcomes, each with a probability of occurrence p(x). The information entropy, H(X), is defined as:
H(X) = -Σ p(x) log2 p(x)
Here, log2 denotes the logarithm base 2, and the unit of entropy is bits. The sum of the entropy values of each outcome, multiplied by their probabilities, represents the average uncertainty of the information source.
In discussions about information entropy, we often encounter two terms: “information entropy” and “Shannon entropy.” In reality, they refer to the same mathematical concept.
Binary memoryless
A Binary Memoryless Source (BMS) is a fundamental model in Information Theory used to describe sources that generate independent binary symbols (0s and 1s) without memory.
The characteristics of a BMS:
It produces only two possible symbols: 0 and 1.
The probability of each symbol occurring is fixed and independent of previous symbols.
No memory means that previous outputs do not influence the current symbol generation.
If p is the probability of generating a 1, and (1 - p) is the probability of generating a 0, then the entropy of a BMS is given by:
H(X) = plog2p - (1-p)log2(1-p)
How BMS plays a significant role:
It helps model digital communication channels.
Forms the basis for error detection and correction techniques.
Essential in data compression and coding theory.
Information theory has a lot of applications across multiple fields:
Machine Learning & AI
Concepts like cross-entropy loss are used in training machine learning models.
It helps optimize information flow in deep learning networks.
Error Detection & Correction
Helps design error-correcting codes like Hamming Codes, Reed-Solomon Codes, and Turbo Codes.
Ensures reliable data transmission over noisy communication channels.
Data Compression
Used in Huffman Coding, Arithmetic Coding, and Lempel-Ziv (LZ) Compression.
Enables efficient storage and transmission of large datasets (e.g., ZIP files, MP3, JPEG).
Conclusion
The concept of entropy quantifies uncertainty, while Binary Memoryless Sources (BMS) serve as a foundational model for digital information systems.
From internet communication to artificial intelligence, Shannon’s work continues to shape the way we handle information in the digital age.