Information content

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In information theory, the information content, self-information, surprisal, or Shannon information is a basic quantity derived from the probability of a particular event occurring from a random variable. It can be thought of as an alternative way of expressing probability, much like odds or log-odds, but which has particular mathematical advantages in the setting of information theory.

The Shannon information can be interpreted as quantifying the level of "surprise" of a particular outcome. As it is such a basic quantity, it also appears in several other settings, such as the length of a message needed to transmit the event given an optimal source coding of the random variable.

The Shannon information is closely related to information theoretic entropy, which is the expected value of the self-information of a random variable, quantifying how surprising the random variable is "on average." This is the average amount of self-information an observer would expect to gain about a random variable when measuring it.

The information content can be expressed in various units of information, of which the most common is the "bit" (sometimes also called the "shannon"), as explained below.

Definition

Claude Shannon's definition of self-information was chosen to meet several axioms:

1. An event with probability 100% is perfectly unsurprising and yields no information.
2. The less probable an event is, the more surprising it is and the more information it yields.
3. If two independent events are measured separately, the total amount of information is the sum of the self-informations of the individual events.

The detailed derivation is below, but it can be shown that there is a unique function of probability that meets these three axioms, up to a multiplicative scaling factor. Broadly given an event $x$ with probability $P$ , the information content is defined as follows:

$\operatorname {I} (x):=-\log _{b}{\left[\Pr {\left(x\right)}\right]}=-\log _{b}{\left(P\right)}.$ The base of the log is left unspecified, which corresponds to the scaling factor above. Different choices of base correspond to different units of information: if the logarithmic base is 2, the unit is a bit or shannon; if the logarithm is the natural logarithm (base e), the unit is the nat, short for "natural"; and if the base is 10, the units are hartleys, decimal digits, or occasionally dits.

Formally, given a random variable $X$ with probability mass function $p_{X}{\left(x\right)}$ , the self-information of measuring $X$ as outcome $x$ is defined as

$\operatorname {I} _{X}(x):=-\log {\left[p_{X}{\left(x\right)}\right]}=\log {\left({\frac {1}{p_{X}{\left(x\right)}}}\right)}.$ The Shannon entropy of the random variable $X$ above is defined as

{\begin{alignedat}{2}\mathrm {H} (X)&=\sum _{x}{-p_{X}{\left(x\right)}\log {p_{X}{\left(x\right)}}}\\&=\sum _{x}{p_{X}{\left(x\right)}\operatorname {I} _{X}(x)}\\&{\overset {\underset {\mathrm {def} }{}}{=}}\ \operatorname {E} {\left[\operatorname {I} _{X}(X)\right]},\end{alignedat}} by definition equal to the expected information content of measurement of $X$ .: 11 : 19–20

The use of the notation $I_{X}(x)$ for self-information above is not universal. Since the notation $I(X;Y)$ is also often used for the related quantity of mutual information, many authors use a lowercase $h_{X}(x)$ for self-entropy instead, mirroring the use of the capital $H(X)$ for the entropy.

Properties

Monotonically decreasing function of probability

For a given probability space, the measurement of rarer events are intuitively more "surprising," and yield more information content, than more common values. Thus, self-information is a strictly decreasing monotonic function of the probability, or sometimes called an "antitonic" function.

While standard probabilities are represented by real numbers in the interval $[0,1]$ , self-informations are represented by extended real numbers in the interval $[0,\infty ]$ . In particular, we have the following, for any choice of logarithmic base:

• If a particular event has a 100% probability of occurring, then its self-information is $-\log(1)=0$ : its occurrence is "perfectly non-surprising" and yields no information.
• If a particular event has a 0% probability of occurring, then its self-information is $-\log(0)=\infty$ : its occurrence is "infinitely surprising."

From this, we can get a few general properties:

• Intuitively, more information is gained from observing an unexpected event—it is "surprising".
• For example, if there is a one-in-a-million chance of Alice winning the lottery, her friend Bob will gain significantly more information from learning that she won than that she lost on a given day. (See also: Lottery mathematics.)
• This establishes an implicit relationship between the self-information of a random variable and its variance.

Relationship to log-odds

The Shannon information is closely related to the log-odds. In particular, given some event $x$ , suppose that $p(x)$ is the probability of $x$ occurring, and that $p(\lnot x)=1-p(x)$ is the probability of $x$ not occurring. Then we have the following definition of the log-odds:

${\text{log-odds}}(x)=\log \left({\frac {p(x)}{p(\lnot x)}}\right)$ This can be expressed as a difference of two Shannon informations:

${\text{log-odds}}(x)=I(\lnot x)-I(x)$ In other words, the log-odds can be interpreted as the level of surprise if the event 'doesn't' happen, minus the level of surprise if the event 'does' happen.

The information content of two independent events is the sum of each event's information content. This property is known as additivity in mathematics, and sigma additivity in particular in measure and probability theory. Consider two independent random variables ${\textstyle X,\,Y}$ with probability mass functions $p_{X}(x)$ and $p_{Y}(y)$ respectively. The joint probability mass function is
$p_{X,Y}\!\left(x,y\right)=\Pr(X=x,\,Y=y)=p_{X}\!(x)\,p_{Y}\!(y)$ because ${\textstyle X}$ and ${\textstyle Y}$ are independent. The information content of the outcome $(X,Y)=(x,y)$ is
{\begin{aligned}\operatorname {I} _{X,Y}(x,y)&=-\log _{2}\left[p_{X,Y}(x,y)\right]=-\log _{2}\left[p_{X}\!(x)p_{Y}\!(y)\right]\\[5pt]&=-\log _{2}\left[p_{X}{(x)}\right]-\log _{2}\left[p_{Y}{(y)}\right]\\[5pt]&=\operatorname {I} _{X}(x)+\operatorname {I} _{Y}(y)\end{aligned}} 