Information theory

Information theory is the mathematical study of the quantification, storage, and communication of information. It was established by Claude Shannon in the 1940s, with early work by Harry Nyquist and Ralph Hartley. Though it began with telecommunication, it now connects mathematics, statistics, and computer science, helping many fields like electrical engineering and neurobiology.

One simple example is flipping a fair coin. Before looking, you don’t know if it’s heads or tails, so you lack some information. After it lands, you gain that information. For a fair coin, this amount of information is measured as 1 bit.

Information theory has shaped many modern technologies. It helps us compress data for ZIP files, correct errors in DSL connections, and even made the Voyager missions, compact discs, mobile phones, the Internet, and artificial intelligence possible. It also plays roles in areas like cryptography, bioinformatics, and even the study of black holes.

Overview

Information theory, created by Claude Shannon, looks at how we process and use information when there is uncertainty. Imagine trying to send a message through a noisy radio channel — Shannon showed we can send messages with very few mistakes if we know the channel's limits.

This field also helps us create special codes to make data smaller (like zipping a file) and to fix mistakes during sending. These ideas are also used in secret codes and solving puzzles to keep information safe.

Historical background

Main article: History of information theory

The field of information theory began with a major paper published by Claude Shannon in 1948, titled "A Mathematical Theory of Communication." This paper introduced new ways to understand how information can be sent, stored, and measured. Before Shannon's work, some ideas related to information had been explored by others, such as Harry Nyquist and Ralph Hartley, but Shannon brought everything together into a single, powerful theory.

Shannon's work showed how to measure information using a unit called the bit. He also explained how to make communication more efficient and how to overcome problems that occur when information is sent over noisy or imperfect channels. His ideas have had a huge impact on many areas, from telecommunications to computer science.

Quantities of information

Main article: Quantities of information

Information theory is based on probability and statistics, where measured information is usually described in terms of bits. It often focuses on measures of information related to random variables. One important measure is called entropy, which helps us understand how much information is in a single variable.

Another useful idea is mutual information, which looks at two variables to see how much they depend on each other. This helps us understand how well we can send information through a channel with noise. The way we calculate this depends on the type of numbers we use, like bits or other units.

Entropy can tell us how hard it is to predict an outcome. For example, if we know each bit being sent is either a 0 or 1 with equal chance, we can say how much information is being shared. Entropy is highest when all outcomes are equally likely, making them hardest to predict.

Coding theory

Main article: Coding theory

Coding theory is a key part of information theory. It helps us understand how to store and send information properly. Information theory tells us how many bits we need to describe data, which is called information entropy.

There are two main parts to coding theory:

• Data compression (source coding): This involves making data smaller. There are two types:
  • Lossless data compression: The data is reconstructed exactly as it was.
  • Lossy data compression: Some detail is lost, but the data is still useful. This is called rate–distortion theory.
• Error-correcting codes (channel coding): This adds extra information to data so that mistakes can be fixed when the data is sent over a noisy connection.

Coding theory splits into compression and transmission to make sure data can be sent clearly. But this only works well when one person is sending to one person. When there are more senders or receivers, things get more complicated.

Source theory

Any process that creates messages can be a source of information. A memoryless source is one where each message is random and not related to the others. These sources are studied in information theory.

Channel capacity

Main article: Channel capacity

When sending messages, we often deal with noise or problems that change the message. A channel can be thought of as a path for sending messages, which might sometimes make mistakes.

The goal is to send as much information as possible over this channel without too many errors. The maximum amount of information that can be sent safely is called the channel capacity. This depends on how noisy the channel is.

Different types of channels have different capacities. For example:

• A channel with Gaussian noise has a certain capacity, described by the Shannon–Hartley theorem.
• A binary symmetric channel (BSC) can flip bits with some probability.
• A binary erasure channel (BEC) can sometimes erase bits completely.

Fungible information

Fungible information is information where how it is encoded doesn’t matter. This is the kind of information most scientists study. It is sometimes called speakable information.

Applications to other fields

Information theory has been used in many different areas beyond its original use in communication. It helps us understand how living systems share and process information, such as how the brain and body work together. It also plays a role in keeping information safe, like in secret codes used during wars.

Other uses include creating better random numbers for computers, improving how we find oil underground, and even studying how we think and make decisions. Information theory also helps scientists search for signs of life beyond Earth and understand complex natural phenomena like black holes.