Information content

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In information theory, information content, self-information, or surprisal of a random variable or signal is the amount of information gained when it is sampled. Formally, information content is a random variable defined for any event in probability theory regardless of whether a random variable is being measured or not.

Information content is expressed in a unit of information, as explained below. The expected value of self-information is information theoretic entropy, the average amount of information an observer would expect to gain about a system when sampling the random variable.[1]

Definition[edit]

Given a random variable with probability mass function , the self-information of measuring as outcome is defined as [2]

Broadly given an event with probability , information content is defined analogously:

In general, the base of the logarithmic chosen does not matter for most information-theoretic properties; however, different units of information are assigned based on popular choices of base.

If the logarithmic base is 2, the unit is named the Shannon but "bit" is also used. If the base of the logarithm is the natural logarithm (logarithm to base Euler's number e ≈ 2.7182818284), the unit is called the nat, short for "natural". If the logarithm is to base 10, the units are called hartleys or decimal digits.

The Shannon entropy of the random variable above is defined as

by definition equal to the expected information content of measurement of .[3]:11[4]:19-20

Properties[edit]

Antitonicity for probability[edit]

For a given probability space, measurement of rarer events will yield more information content than more common values. Thus, self-information is antitonic in probability for events under observation.

  • 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.

Additivity of independent events[edit]

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 with probability mass functions and respectively. The joint probability mass function is

because and are independent. The information content of the outcome is

See § Two independent, identically distributed dice below for an example.

Notes[edit]

This measure has also been called surprisal, as it represents the "surprise" of seeing the outcome (a highly improbable outcome is very surprising). This term (as a log-probability measure) was coined by Myron Tribus in his 1961 book Thermostatics and Thermodynamics.[5][6]

When the event is a random realization (of a variable) the self-information of the variable is defined as the expected value of the self-information of the realization.

Self-information is an example of a proper scoring rule.[clarification needed]

Examples[edit]

Fair coin toss[edit]

Consider the Bernoulli trial of tossing a fair coin . The probabilities of the events of the coin landing as heads and tails (see fair coin and obverse and reverse) are one half each, . Upon measuring the variable as heads, the associated information gain is

so the information gain of a fair coin landing as heads is 1 shannon.[2] Likewise, the information gain of measuring tails is

Fair dice roll[edit]

Suppose we have a fair six-sided dice. The value of a dice roll is a discrete uniform random variable with probability mass function

The probability of rolling a 4 is , as for any other valid roll. The information content of rolling a 4 is thus
of information.

Two independent, identically distributed dice[edit]

Suppose we have two independent, identically distributed random variables each corresponding to an independent fair 6-sided dice roll. The joint distribution of and is

The information content of the random variate is

just as

as explained in § Additivity of independent events.

Information from frequency of rolls[edit]

If we receive information about the value of the dice without knowledge of which die had which value, we can formalize the approach with so-called counting variables

for , then and the counts have the multinomial distribution

To verify this, the 6 outcomes correspond to the event and a total probability of 1/6. These are the only events that are faithfully preserved with identity of which dice rolled which outcome because the outcomes are the same. Without knowledge to distinguish the dice rolling the other numbers, the other combinations correspond to one die rolling one number and the other die rolling a different number, each having probability 1/18. Indeed, , as required.

Unsurprisingly, the information content of learning that both dice were rolled as the same particular number is more than the information content of learning that one dice was one number and the other was a different number. Take for examples the events and for . For example, and .

The information contents are

Let be the event that both dice rolled the same value and be the event that the dice differed. Then and . The information contents of the events are

Information from sum of die[edit]

The probability mass or density function (collectively probability measure) of the sum of two independent random variables is the convolution of each probability measure. In the case of independent fair 6-sided dice rolls, the random variable has probability mass function , where represents the discrete convolution. The outcome has probability . Therefore, the information asserted is

General discrete uniform distribution[edit]

Generalizing the § Fair dice roll example above, consider a general discrete uniform random variable (DURV) For convenience, define . The p.m.f. is

In general, the values of the DURV need not be integers, or for the purposes of information theory even uniformly spaced; they need only be equiprobable.[2] The information gain of any observation is

Special case: constant random variable[edit]

If above, degenerates to a constant random variable with probability distribution deterministically given by and probability measure the Dirac measure . The only value can take is deterministically , so the information content of any measurement of is

In general, there is no information gained from measuring a known value.[2]

Categorical distribution[edit]

Generalizing all of the above cases, consider a categorical discrete random variable with support and p.m.f. given by

For the purposes of information theory, the values do not even have to be numbers at all; they can just be mutually exclusive events on a measure space of finite measure that has been normalized to a probability measure . Without loss of generality, we can assume the categorical distribution is supported on the set ; the mathematical structure is isomorphic in terms of probability theory and therefore information theory as well.

The information of the outcome is given

From these examples, it is possible to calculate the information of any set of independent DRVs with known distributions by additivity.

Relationship to entropy[edit]

The entropy is the expected value of the information content of the discrete random variable, with expectation taken over the discrete values it takes. Sometimes, the entropy itself is called the "self-information" of the random variable, possibly because the entropy satisfies , where is the mutual information of with itself.[7]

Derivation[edit]

By definition, information is transferred from an originating entity possessing the information to a receiving entity only when the receiver had not known the information a priori. If the receiving entity had previously known the content of a message with certainty before receiving the message, the amount of information of the message received is zero.

For example, quoting a character (the Hippy Dippy Weatherman) of comedian George Carlin, “Weather forecast for tonight: dark. Continued dark overnight, with widely scattered light by morning.” Assuming one does not reside near the Earth's poles or polar circles, the amount of information conveyed in that forecast is zero because it is known, in advance of receiving the forecast, that darkness always comes with the night.

When the content of a message is known a priori with certainty, with probability of 1, there is no actual information conveyed in the message. Only when the advance knowledge of the content of the message by the receiver is less than 100% certain does the message actually convey information.

Accordingly, the amount of self-information contained in a message conveying content informing an occurrence of event, , depends only on the probability of that event.

for some function to be determined below. If , then . If , then .

Further, by definition, the measure of self-information is nonnegative and additive. If a message informing of event is the intersection of two independent events and , then the information of event occurring is that of the compound message of both independent events and occurring. The quantity of information of compound message would be expected to equal the sum of the amounts of information of the individual component messages and respectively:

.

Because of the independence of events and , the probability of event is

.

However, applying function results in

The class of function having the property such that

is the logarithm function of any base. The only operational difference between logarithms of different bases is that of different scaling constants.

Since the probabilities of events are always between 0 and 1 and the information associated with these events must be nonnegative, that requires that .

Taking into account these properties, the self-information associated with outcome with probability is defined as:

The smaller the probability of event , the larger the quantity of self-information associated with the message that the event indeed occurred. If the above logarithm is base 2, the unit of is bits. This is the most common practice. When using the natural logarithm of base , the unit will be the nat. For the base 10 logarithm, the unit of information is the hartley.

As a quick illustration, the information content associated with an outcome of 4 heads (or any specific outcome) in 4 consecutive tosses of a coin would be 4 bits (probability 1/16), and the information content associated with getting a result other than the one specified would be ~0.09 bits (probability 15/16). See below for detailed examples.

See also[edit]

References[edit]

  1. ^ Jones, D.S., Elementary Information Theory, Vol., Clarendon Press, Oxford pp 11-15 1979
  2. ^ a b c d McMahon, David M. (2008). Quantum Computing Explained. Hoboken, NJ: Wiley-Interscience. ISBN 9780470181386. OCLC 608622533.
  3. ^ Borda, Monica (2011). Fundamentals in Information Theory and Coding. Springer. ISBN 978-3-642-20346-6.
  4. ^ Han, Te Sun & Kobayashi, Kingo (2002). Mathematics of Information and Coding. American Mathematical Society. ISBN 978-0-8218-4256-0.
  5. ^ R. B. Bernstein and R. D. Levine (1972) "Entropy and Chemical Change. I. Characterization of Product (and Reactant) Energy Distributions in Reactive Molecular Collisions: Information and Entropy Deficiency", The Journal of Chemical Physics 57, 434-449 link.
  6. ^ Myron Tribus (1961) Thermodynamics and Thermostatics: An Introduction to Energy, Information and States of Matter, with Engineering Applications (D. Van Nostrand, 24 West 40 Street, New York 18, New York, U.S.A) Tribus, Myron (1961), pp. 64-66 borrow.
  7. ^ Thomas M. Cover, Joy A. Thomas; Elements of Information Theory; p. 20; 1991.

Further reading[edit]

  • C.E. Shannon, A Mathematical Theory of Communication, Bell Systems Technical Journal, Vol. 27, pp 379–423, (Part I), 1948.

External links[edit]