Shannon (unit) in the context of "Nat (unit)"

In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across all potential states. Given a discrete random variable ${\displaystyle X}$, which may be any member ${\displaystyle x}$ within the set ${\displaystyle {\mathcal {X}}}$ and is distributed according to ${\displaystyle p\colon {\mathcal {X}}\to [0,1]}$, the entropy is$${\displaystyle \mathrm {H} (X):=-\sum _{x\in {\mathcal {X}}}p(x)\log p(x),}$$where ${\displaystyle \Sigma }$ denotes the sum over the variable's possible values. The choice of base for ${\displaystyle \log }$, the logarithm, varies for different applications. Base 2 gives the unit of bits (or "shannons"), while base e gives "natural units" nat, and base 10 gives units of "dits", "bans", or "hartleys". An equivalent definition of entropy is the expected value of the self-information of a variable.

The concept of information entropy was introduced by Claude Shannon in his 1948 paper "A Mathematical Theory of Communication", and is also referred to as Shannon entropy. Shannon's theory defines a data communication system composed of three elements: a source of data, a communication channel, and a receiver. The "fundamental problem of communication" – as expressed by Shannon – is for the receiver to be able to identify what data was generated by the source, based on the signal it receives through the channel. Shannon considered various ways to encode, compress, and transmit messages from a data source, and proved in his source coding theorem that the entropy represents an absolute mathematical limit on how well data from the source can be losslessly compressed onto a perfectly noiseless channel. Shannon strengthened this result considerably for noisy channels in his noisy-channel coding theorem.

The hartley (symbol Hart), also called a ban, or a dit (short for "decimal digit"), is a logarithmic unit that measures information or entropy, based on base 10 logarithms and powers of 10. One hartley is the information content of an event if the probability of that event occurring is 1⁄10. It is therefore equal to the information contained in one decimal digit (or dit), assuming a priori equiprobability of each possible value. It is named after Ralph Hartley.

If base 2 logarithms and powers of 2 are used instead, then the unit of information is the shannon or bit, which is the information content of an event if the probability of that event occurring is 1⁄2. Natural logarithms and powers of e define the nat.

In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as shannons (bits), nats or hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable.

Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair ${\displaystyle (X,Y)}$ is from the product of the marginal distributions of ${\displaystyle X}$ and ${\displaystyle Y}$. MI is the expected value of the pointwise mutual information (PMI).