TheInfoList

In
information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...
, the entropy of a
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable $X$, with possible outcomes $x_1, ..., x_n$, which occur with probability $\mathrm\left(x_1\right), ..., \mathrm\left(x_n\right),$ the entropy of $X$ is formally defined as: where $\Sigma$ denotes the sum over the variable's possible values. The choice of base for $\log$, the
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, varies for different applications. Base 2 gives the unit of
bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algo ...
s (or " shannons"), while base ''e'' gives "natural units"
nat Nat or NAT may refer to: Computing * Network address translation (NAT), in computer networking Organizations * National Actors Theatre, New York City, U.S. * National AIDS trust, a British charity * National Archives of Thailand * National Asse ...
, and base 10 gives units of "dits", "bans", or " hartleys". An equivalent definition of entropy is the
expected value In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...
of the
self-information In information theory Information theory is the scientific study of the quantification, storage, and communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an apparent answer to the p ...
of a variable. The concept of information entropy was introduced by
Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbe ...
in his 1948 paper "
A Mathematical Theory of Communication "A Mathematical Theory of Communication" is an article by mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and nu ...
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and is also referred to as Shannon entropy. Shannon's theory defines a
data communication Data transmission and data reception or, more broadly, data communication or digital communications is the transfer and reception of data in the form of a Digital data, digital bitstream or a digitized analog signal over a Point-to-point (tele ...
system composed of three elements: a source of data, a
communication channel A communication channel refers either to a physical transmission medium A transmission medium is a system or substance that can mediate the propagation of signal In signal processing Signal processing is an electrical engineering ...
, 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 famous
source coding theorem In information theory, Shannon's source coding theorem (or noiseless coding theorem) establishes the limits to possible data compression, and the operational meaning of the Shannon entropy. Named after Claude Shannon, the source coding theorem ...
that the entropy represents an absolute mathematical limit on how well data from the source can be
lossless Lossless compression is a class of data compression In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image ...
ly compressed onto a perfectly noiseless channel. Shannon strengthened this result considerably for noisy channels in his
noisy-channel coding theorem In information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fund ...
. Entropy in information theory is directly analogous to the
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...
in
statistical thermodynamics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and forc ...
. The analogy results when the values of the random variable designate energies of microstates, so Gibbs formula for the entropy is formally identical to Shannon's formula. Entropy has relevance to other areas of mathematics such as
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
and
machine learning Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data ...

. The definition can be derived from a set of
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

s establishing that entropy should be a measure of how "surprising" the average outcome of a variable is. For a continuous random variable,
differential entropy Differential entropy (also referred to as continuous entropy) is a concept in information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication ...
is analogous to entropy.

# Introduction

The core idea of information theory is that the "informational value" of a communicated message depends on the degree to which the content of the message is surprising. If a highly likely event occurs, the message carries very little new information. On the other hand, if a highly unlikely event occurs, the message is much more informative. For instance, the knowledge that some particular number ''will not'' be the winning number of a lottery provides very little information, because any particular chosen number will almost certainly not win. However, knowledge that a particular number ''will'' win a lottery has high informational value because it communicates the outcome of a very low probability event. The ''
information content In information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamen ...
,'' also called the ''surprisal'' or ''self-information,'' of an event $E$ is a function which increases as the probability $p\left(E\right)$ of an event decreases. When $P\left(E\right)$ is close to 1, the surprisal of the event is low, but if $P\left(E\right)$ is close to 0, the surprisal of the event is high. This relationship is described by the function $\log\left\left(\frac\right\right)$ where $\log$ is the
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, which gives 0 surprise when the probability of the event is 1. In fact, the $\log$ is the only function that satisfies this specific set of
characterization Characterization or characterisation is the representation of persons (or other beings or creatures) in narrative A narrative, story or tale is any account of a series of related events or experiences, whether nonfiction Nonfiction (also spelle ...
. Hence, we can define the information, or surprisal, of an event $E$ by $I\left(E\right) = -\log_2\left(p\left(E\right)\right)$ , or equivalently, $I\left(E\right) = \log_2\left\left(\frac\right\right)$. Entropy measures the expected (i.e., average) amount of information conveyed by identifying the outcome of a random trial. This implies that casting a die has higher entropy than tossing a coin because each outcome of a die toss has smaller probability (about $p=1/6$) than each outcome of a coin toss ($p=1/2$). Consider a
biased coin In probability theory and statistics, a sequence of Independence (probability theory), independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is cal ...
with probability of landing on heads and probability of landing on tails. The maximum surprise is when , for which one outcome is not expected over the other. In this case a coin flip has an entropy of one
bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algo ...
. (Similarly, one with equiprobable values contains $\log_2 3$ (about 1.58496) bits of information because it can have one of three values.) The minimum surprise is when or , when the event outcome is known and the entropy is zero bits. When the entropy is zero bits, this is sometimes referred to as unity, where there is no uncertainty at all - no freedom of choice - no
information Information is processed, organised and structured data Data (; ) are individual facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to c ...
. Other values of ''p'' give entropies between zero and one bits. Information theory is useful to calculate the smallest amount of information required to convey a message, as in
data compression In signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electro ...
. For example, consider the transmission of sequences comprising the 4 characters 'A', 'B', 'C', and 'D' over a binary channel. If all 4 letters are equally likely (25%), one can't do better than using two bits to encode each letter. 'A' might code as '00', 'B' as '01', 'C' as '10', and 'D' as '11'. However, if the probabilities of each letter are unequal, say 'A' occurs with 70% probability, 'B' with 26%, and 'C' and 'D' with 2% each, one could assign variable length codes. In this case, 'A' would be coded as '0', 'B' as '10', 'C' as '110', and D as '111'. With this representation, 70% of the time only one bit needs to be sent, 26% of the time two bits, and only 4% of the time 3 bits. On average, fewer than 2 bits are required since the entropy is lower (owing to the high prevalence of 'A' followed by 'B' – together 96% of characters). The calculation of the sum of probability-weighted log probabilities measures and captures this effect. English text, treated as a string of characters, has fairly low entropy, i.e., is fairly predictable. We can be fairly certain that, for example, 'e' will be far more common than 'z', that the combination 'qu' will be much more common than any other combination with a 'q' in it, and that the combination 'th' will be more common than 'z', 'q', or 'qu'. After the first few letters one can often guess the rest of the word. English text has between 0.6 and 1.3 bits of entropy per character of the message.Schneier, B: ''Applied Cryptography'', Second edition, John Wiley and Sons.

# Definition

Named after Boltzmann's Η-theorem, Shannon defined the entropy (Greek capital letter
eta Eta (uppercase , lowercase ; grc, ἦτα ''ē̂ta'' or ell, ήτα ''ita'' ) is the seventh letter of the Greek alphabet The Greek alphabet has been used to write the Greek language since the late ninth or early eighth century BC. It is ...

) of a
discrete random variable A random variable is a Dependent and independent variables, variable whose values depend on Outcome (probability), outcomes of a Randomness, random event. Also called random quantity, aleatory variable, or stochastic variable. It is formally defi ...
$X$ with possible values $\left\$ and
probability mass function In probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which th ...
$\mathrm(X)$ as: : Here $\operatorname$ is the expected value operator, and is the
information content In information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamen ...
of . $\operatorname\left(X\right)$ is itself a random variable. The entropy can explicitly be written as: where is the
base Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider) Base (stylized as BASE) is the third largest of Belgium Belgium ( nl, België ; french: Belgique ; german: Belgien ), officially the Kingdom of Belgium, ...
of the
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

used. Common values of are 2, Euler's number , and 10, and the corresponding units of entropy are the
bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algo ...
s for , nats for , and bans for . In the case of for some , the value of the corresponding summand is taken to be , which is consistent with the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

: :$\lim_p\log \left(p\right) = 0.$ One may also define the
conditional entropy Image:Entropy-mutual-information-relative-entropy-relation-diagram.svg, 256px, Venn diagram showing additive and subtractive relationships various Quantities of information, information measures associated with correlated variables X and Y. The ar ...

of two variables $X$ and $Y$ taking values $x_i$ and $y_j$ respectively, as: :$\Eta\left(X, Y\right)=-\sum_p\left(x_,y_\right)\log\frac$ where $p\left(x_i,y_j\right)$ is the probability that $X=x_i$ and $Y=y_j$. This quantity should be understood as the amount of randomness in the random variable $X$ given the random variable $Y$.

# Example

Consider tossing a coin with known, not necessarily fair, probabilities of coming up heads or tails; this can be modelled as a
Bernoulli process In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The ...
. The entropy of the unknown result of the next toss of the coin is maximized if the coin is fair (that is, if heads and tails both have equal probability 1/2). This is the situation of maximum uncertainty as it is most difficult to predict the outcome of the next toss; the result of each toss of the coin delivers one full bit of information. This is because :$\begin \Eta\left(X\right) &= -\sum_^n \\ &= -\sum_^2 \\ &= -\sum_^2 = 1 \end$ However, if we know the coin is not fair, but comes up heads or tails with probabilities and , where , then there is less uncertainty. Every time it is tossed, one side is more likely to come up than the other. The reduced uncertainty is quantified in a lower entropy: on average each toss of the coin delivers less than one full bit of information. For example, if =0.7, then :$\begin \Eta(X) &= - p \log_2 (p) - q \log_2 (q) \\ &= - 0.7 \log_2 (0.7) - 0.3 \log_2 (0.3) \\ &\approx - 0.7 \cdot (-0.515) - 0.3 \cdot (-1.737) \\ &= 0.8816 < 1 \end$ Uniform probability yields maximum uncertainty and therefore maximum entropy. Entropy, then, can only decrease from the value associated with uniform probability. The extreme case is that of a double-headed coin that never comes up tails, or a double-tailed coin that never results in a head. Then there is no uncertainty. The entropy is zero: each toss of the coin delivers no new information as the outcome of each coin toss is always certain. Entropy can be normalized by dividing it by information length. This ratio is called metric entropy and is a measure of the randomness of the information.

# Characterization

To understand the meaning of , first define an information function in terms of an event with probability . The amount of information acquired due to the observation of event follows from Shannon's solution of the fundamental
properties Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to Consumable, consume, alter ...
of
information Information is processed, organised and structured data Data (; ) are individual facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to c ...
: # is monotonically decreasing in : an increase in the probability of an event decreases the information from an observed event, and vice versa. # : information is a
non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third ...
quantity. # : events that always occur do not communicate information. # : the information learned from
independent events Independence is a fundamental notion in probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related stru ...
is the sum of the information learned from each event. Given two independent events, if the first event can yield one of
equiprobable Equiprobability is a property for a collection of events that each have the same probability Probability is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number the ...
outcomes and another has one of equiprobable outcomes then there are equiprobable outcomes of the joint event. This means that if bits are needed to encode the first value and to encode the second, one needs to encode both. Shannon discovered that a suitable choice of $\operatorname$ is given by: :$\operatorname\left(p\right) = \log\left\left(\tfrac\right\right) = -\log\left(p\right)$ In fact, the only possible values of $\operatorname$ are $\operatorname\left(u\right) = k \log u$ for $k<0$. Additionally, choosing a value for is equivalent to choosing a value $x>1$ for $k = - 1/\log x$, so that corresponds to the base for the logarithm. Thus, entropy is characterized by the above four properties. : The different
units of information In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algorithm) for calculating the greatest commo ...

(
bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algo ...
s for the
binary logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, nats for the
natural logarithm The natural logarithm of a number is its logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
, bans for the decimal logarithm and so on) are constant multiples of each other. For instance, in case of a fair coin toss, heads provides bit of information, which is approximately 0.693 nats or 0.301 decimal digits. Because of additivity, tosses provide bits of information, which is approximately nats or decimal digits. The ''meaning'' of the events observed (the meaning of ''messages'') does not matter in the definition of entropy. Entropy only takes into account the probability of observing a specific event, so the information it encapsulates is information about the underlying probability distribution, not the meaning of the events themselves.

## Alternate characterization

Another characterization of entropy uses the following properties. We denote and . # Continuity: should be
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
, so that changing the values of the probabilities by a very small amount should only change the entropy by a small amount. # Symmetry: should be unchanged if the outcomes are re-ordered. That is, $\Eta_n\left\left(p_1, p_2, \ldots p_n \right\right) = \Eta_n\left\left(p_, p_, \ldots, p_ \right\right)$ for any
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

$\$ of $\$. # Maximum: $\Eta_n$ should be maximal if all the outcomes are equally likely i.e. $\Eta_n\left(p_1,\ldots,p_n\right) \le \Eta_n\left\left(\frac, \ldots, \frac\right\right)$. # Increasing number of outcomes: for equiprobable events, the entropy should increase with the number of outcomes i.e. $\Eta_n\bigg\left(\underbrace_\bigg\right) < \Eta_\bigg\left(\underbrace_\bigg\right).$ # Additivity: given an ensemble of uniformly distributed elements that are divided into boxes (sub-systems) with elements each, the entropy of the whole ensemble should be equal to the sum of the entropy of the system of boxes and the individual entropies of the boxes, each weighted with the probability of being in that particular box. The rule of additivity has the following consequences: for
positive integers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
where , :$\Eta_n\left\left(\frac, \ldots, \frac\right\right) = \Eta_k\left\left(\frac, \ldots, \frac\right\right) + \sum_^k \frac \, \Eta_\left\left(\frac, \ldots, \frac\right\right).$ Choosing , this implies that the entropy of a certain outcome is zero: . This implies that the efficiency of a source alphabet with symbols can be defined simply as being equal to its -ary entropy. See also
Redundancy (information theory) In Information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamen ...
.

# Further properties

The Shannon entropy satisfies the following properties, for some of which it is useful to interpret entropy as the amount of information learned (or uncertainty eliminated) by revealing the value of a random variable : * Adding or removing an event with probability zero does not contribute to the entropy: ::$\Eta_\left(p_1,\ldots,p_n,0\right) = \Eta_n\left(p_1,\ldots,p_n\right)$. * It can be confirmed using the
Jensen inequality In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen (mathematician), Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906 ...
that ::. :This maximal entropy of is effectively attained by a source alphabet having a uniform probability distribution: uncertainty is maximal when all possible events are equiprobable. * The entropy or the amount of information revealed by evaluating (that is, evaluating and simultaneously) is equal to the information revealed by conducting two consecutive experiments: first evaluating the value of , then revealing the value of given that you know the value of . This may be written as: ::$\Eta\left(X,Y\right)=\Eta\left(X, Y\right)+\Eta\left(Y\right)=\Eta\left(Y, X\right)+\Eta\left(X\right).$ * If $Y=f\left(X\right)$ where $f$ is a function, then $H\left(f\left(X\right), X\right) = 0$. Applying the previous formula to $H\left(X,f\left(X\right)\right)$ yields ::$\Eta\left(X\right)+\Eta\left(f\left(X\right), X\right)=\Eta\left(f\left(X\right)\right)+\Eta\left(X, f\left(X\right)\right),$ :so $H\left(f\left(X\right)\right) \le H\left(X\right)$, the entropy of a variable can only decrease when the latter is passed through a function. * If and are two independent random variables, then knowing the value of doesn't influence our knowledge of the value of (since the two don't influence each other by independence): ::$\Eta\left(X, Y\right)=\Eta\left(X\right).$ * More generally, for any random variables and , we have ::$\Eta\left(X, Y\right)\leq \Eta\left(X\right)$. * The entropy of two simultaneous events is no more than the sum of the entropies of each individual event i.e., $\Eta\left(X,Y\right)\leq \Eta\left(X\right)+\Eta\left(Y\right)$, with equality if and only if the two events are independent. * The entropy $\Eta\left(p\right)$ is
concave Concave means curving in or hollowed inward, as opposed to convex. Concave may refer to: * Concave function In , a concave function is the of a . A concave function is also ously called concave downwards, concave down, convex upwards, convex c ...

in the probability mass function $p$, i.e. ::$\Eta\left(\lambda p_1 + \left(1-\lambda\right) p_2\right) \ge \lambda \Eta\left(p_1\right) + \left(1-\lambda\right) \Eta\left(p_2\right)$ :for all probability mass functions $p_1,p_2$ and $0 \le \lambda \le 1$. :* Accordingly, the
negative entropy In information theory and statistics, negentropy is used as a measure of distance to normality. The concept and phrase "negative entropy" was introduced by Erwin Schrödinger in his 1944 popular-science book ''What is Life? (Schrödinger), What is ...
(negentropy) function is convex, and its
convex conjugate In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
is
LogSumExp The LogSumExp (LSE) (also called RealSoftMax or multivariable softplus) function is a smooth maximum – a smooth approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and ...
.

# Aspects

## Relationship to thermodynamic entropy

The inspiration for adopting the word ''entropy'' in information theory came from the close resemblance between Shannon's formula and very similar known formulae from
statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and forc ...
. In
statistical thermodynamics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and forc ...
the most general formula for the thermodynamic
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...

of a
thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
is the
Gibbs entropy The concept entropy was first developed by German physicist Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist A physicist is a scientist A scientist is a person who conducts Scie ...
, :$S = - k_\text \sum p_i \ln p_i \,$ where is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality, Multiplication, multiplicatively connected to a Constant (mathematics), c ...
, and is the probability of a
microstate A microstate or ministate is a sovereign state A sovereign state is a political entity A polity is an identifiable political entity—any group of people who have a collective identity, who are organized by some form of Institutionalisati ...
. The
Gibbs entropy The concept entropy was first developed by German physicist Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist A physicist is a scientist A scientist is a person who conducts Scie ...
was defined by J. Willard Gibbs in 1878 after earlier work by
Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...
(1872). The Gibbs entropy translates over almost unchanged into the world of
quantum physics Quantum mechanics is a fundamental theory A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, ...
to give the
von Neumann entropy In quantum statistical mechanics, the von Neumann entropy, named after John von Neumann, is the extension of classical Gibbs entropy concepts to the field of quantum mechanics. For a quantum-mechanical system described by a density matrix , the vo ...
, introduced by
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American Hungarian Americans (Hungarian language, Hungarian: ''amerikai magyarok'') are United States, Americans of Hungarian p ...

in 1927, :$S = - k_\text \,\left(\rho \ln \rho\right) \,$ where ρ is the
density matrix In quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking i ...
of the quantum mechanical system and Tr is the
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * The Trace (album), ''The Trace'' (album) Other ...
. At an everyday practical level, the links between information entropy and thermodynamic entropy are not evident. Physicists and chemists are apt to be more interested in ''changes'' in entropy as a system spontaneously evolves away from its initial conditions, in accordance with the
second law of thermodynamics The second law of thermodynamics establishes the concept of entropy Entropy is a scientific concept, as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term an ...
, rather than an unchanging probability distribution. As the minuteness of
Boltzmann's constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
indicates, the changes in for even tiny amounts of substances in chemical and physical processes represent amounts of entropy that are extremely large compared to anything in
data compression In signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electro ...
or
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

. In classical thermodynamics, entropy is defined in terms of macroscopic measurements and makes no reference to any probability distribution, which is central to the definition of information entropy. The connection between thermodynamics and what is now known as information theory was first made by
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austria Austria (, ; german: Österreich ), officially the Republic of Austria (german: Republik Österreich, links=no, ), is a landlocked Eastern Alps, East ...
and expressed by his famous equation: :$S=k_\text \ln\left(W\right)$ where $S$ is the thermodynamic entropy of a particular macrostate (defined by thermodynamic parameters such as temperature, volume, energy, etc.), ''W'' is the number of microstates (various combinations of particles in various energy states) that can yield the given macrostate, and ''kB'' is
Boltzmann's constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
. It is assumed that each microstate is equally likely, so that the probability of a given microstate is ''pi = 1/W''. When these probabilities are substituted into the above expression for the Gibbs entropy (or equivalently ''kB'' times the Shannon entropy), Boltzmann's equation results. In information theoretic terms, the information entropy of a system is the amount of "missing" information needed to determine a microstate, given the macrostate. In the view of
Jaynes Jaynes is a surname In some cultures, a surname, family name, or last name is the portion of one's personal name A personal name, or full name, in onomastic Onomastics or onomatology is the study of the etymology, history, and use of pro ...
(1957), thermodynamic entropy, as explained by
statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and forc ...
, should be seen as an ''application'' of Shannon's information theory: the thermodynamic entropy is interpreted as being proportional to the amount of further Shannon information needed to define the detailed microscopic state of the system, that remains uncommunicated by a description solely in terms of the macroscopic variables of classical thermodynamics, with the constant of proportionality being just the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor In mathematics, two varying quantities are said to be in a Binary relation, relation of proportionality, Multiplication, multiplicatively connected to a Constant (mathematics), c ...
. Adding heat to a system increases its thermodynamic entropy because it increases the number of possible microscopic states of the system that are consistent with the measurable values of its macroscopic variables, making any complete state description longer. (See article: ''
maximum entropy thermodynamics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
'').
Maxwell's demon Maxwell's demon is a thought experiment that would hypothetically violate the second law of thermodynamics. It was proposed by the physicist James Clerk Maxwell in 1867. In his first letter Maxwell called the Demon (thought experiment), demon a ...

can (hypothetically) reduce the thermodynamic entropy of a system by using information about the states of individual molecules; but, as
LandauerLandauer is a surname, originally referring to somebody from Landau. It may refer to: * Carl Landauer (1891–1983), German economist * Gustav Landauer (1870–1919), German anarchist, writer, and critic * Kurt Landauer (1884–1961), German footbal ...
(from 1961) and co-workers have shown, to function the demon himself must increase thermodynamic entropy in the process, by at least the amount of Shannon information he proposes to first acquire and store; and so the total thermodynamic entropy does not decrease (which resolves the paradox).
Landauer's principle Landauer's principle is a physical principle pertaining to the lower theoretical A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of c ...
imposes a lower bound on the amount of heat a computer must generate to process a given amount of information, though modern computers are far less efficient.

## Data compression

Shannon's definition of entropy, when applied to an information source, can determine the minimum channel capacity required to reliably transmit the source as encoded binary digits. Shannon's entropy measures the information contained in a message as opposed to the portion of the message that is determined (or predictable). Examples of the latter include redundancy in language structure or statistical properties relating to the occurrence frequencies of letter or word pairs, triplets etc. The minimum channel capacity can be realized in theory by using the
typical set In information theory, the typical set is a set of sequences whose probability is close to two raised to the negative power of the Information entropy, entropy of their source distribution. That this set has total probability close to one is a conse ...
or in practice using Huffman,
Lempel–Ziv LZ77 and LZ78 are the two lossless data compression Lossless compression is a class of data compression algorithms that allows the original data to be perfectly reconstructed from the compressed data. By contrast, lossy compression permits rec ...
or
arithmetic coding Arithmetic coding is a form of entropy encoding In information theory, an entropy coding (or entropy encoding) is a lossless compression , lossless data compression scheme that is independent of the specific characteristics of the medium. One of ...
Kolmogorov complexity In algorithmic information theory Algorithmic information theory (AIT) is a branch of theoretical computer science Theoretical computer science (TCS) is a subset of general computer science Computer science deals with the theoretical ...
.) In practice, compression algorithms deliberately include some judicious redundancy in the form of
checksum A checksum is a small-sized Block (data storage), block of data derived from another block of digital data for the purpose of error detection, detecting errors that may have been introduced during its telecommunication, transmission or computer ...

s to protect against errors. The
entropy rate In the mathematical theory of probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), sha ...
of a data source is the average number of bits per symbol needed to encode it. Shannon's experiments with human predictors show an information rate between 0.6 and 1.3 bits per character in English; the
PPM compression algorithm Prediction by partial matching (PPM) is an adaptive statistical Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
can achieve a compression ratio of 1.5 bits per character in English text. If a
compression Compression may refer to: Physical science *Compression (physics), size reduction due to forces *Compression member, a structural element such as a column *Compressibility, susceptibility to compression *Gas compression *Compression ratio, of a c ...
scheme is lossless – one in which you can always recover the entire original message by decompression – then a compressed message has the same quantity of information as the original but communicated in fewer characters. It has more information (higher entropy) per character. A compressed message has less redundancy.
Shannon's source coding theorem In information theory, Shannon's source coding theorem (or noiseless coding theorem) establishes the limits to possible data compression, and the operational meaning of the Shannon entropy. Named after Claude Shannon, the source coding theorem ...
states a lossless compression scheme cannot compress messages, on average, to have ''more'' than one bit of information per bit of message, but that any value ''less'' than one bit of information per bit of message can be attained by employing a suitable coding scheme. The entropy of a message per bit multiplied by the length of that message is a measure of how much total information the message contains. Shannon's theorem also implies that no lossless compression scheme can shorten ''all'' messages. If some messages come out shorter, at least one must come out longer due to the
pigeonhole principle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. In practical use, this is generally not a problem, because one is usually only interested in compressing certain types of messages, such as a document in English, as opposed to gibberish text, or digital photographs rather than noise, and it is unimportant if a compression algorithm makes some unlikely or uninteresting sequences larger. A 2011 study in ''
Science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ...
'' estimates the world's technological capacity to store and communicate optimally compressed information normalized on the most effective compression algorithms available in the year 2007, therefore estimating the entropy of the technologically available sources."The World's Technological Capacity to Store, Communicate, and Compute Information"
Martin Hilbert and Priscila López (2011),
Science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ...
The authors estimate humankind technological capacity to store information (fully entropically compressed) in 1986 and again in 2007. They break the information into three categories—to store information on a medium, to receive information through one-way
broadcast Broadcasting is the distributionDistribution may refer to: Mathematics *Distribution (mathematics) Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functi ...

networks, or to exchange information through two-way
telecommunication Telecommunication is the transmission of information by various types of technologies over wire A wire is a single usually cylindrical A cylinder (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Gr ...
networks.

## Entropy as a measure of diversity

Entropy is one of several ways to measure biodiversity, and is applied in the form of the
Shannon index A diversity index is a quantitative measure that reflects how many different types (such as species In biology, a species is the basic unit of biological classification, classification and a taxonomic rank of an organism, as well as a unit o ...
. A diversity index is a quantitative statistical measure of how many different types exist in a dataset, such as species in a community, accounting for ecological richness, evenness, and dominance. Specifically, Shannon entropy is the logarithm of , the true diversity index with parameter equal to 1. The Shannon index is related to the proportional abundances of types.

## Limitations of entropy

There are a number of entropy-related concepts that mathematically quantify information content in some way: * the
self-information In information theory Information theory is the scientific study of the quantification, storage, and communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an apparent answer to the p ...
of an individual message or symbol taken from a given probability distribution, * the entropy of a given probability distribution of messages or symbols, and * the
entropy rate In the mathematical theory of probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), sha ...
of a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and phen ...

. (The "rate of self-information" can also be defined for a particular sequence of messages or symbols generated by a given stochastic process: this will always be equal to the entropy rate in the case of a
stationary process In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
.) Other
quantities of information The Information theory, mathematical theory of information is based on probability theory and statistics, and measures information with several quantities of information. The choice of logarithmic base in the following formulae determines the unit ...
are also used to compare or relate different sources of information. It is important not to confuse the above concepts. Often it is only clear from context which one is meant. For example, when someone says that the "entropy" of the English language is about 1 bit per character, they are actually modeling the English language as a stochastic process and talking about its entropy ''rate''. Shannon himself used the term in this way. If very large blocks are used, the estimate of per-character entropy rate may become artificially low because the probability distribution of the sequence is not known exactly; it is only an estimate. If one considers the text of every book ever published as a sequence, with each symbol being the text of a complete book, and if there are published books, and each book is only published once, the estimate of the probability of each book is , and the entropy (in bits) is . As a practical code, this corresponds to assigning each book a
unique identifier A unique identifier (UID) is an identifier An identifier is a name that identifies (that is, labels the identity of) either a unique object or a unique ''class'' of objects, where the "object" or class may be an idea, physical countable In m ...

and using it in place of the text of the book whenever one wants to refer to the book. This is enormously useful for talking about books, but it is not so useful for characterizing the information content of an individual book, or of language in general: it is not possible to reconstruct the book from its identifier without knowing the probability distribution, that is, the complete text of all the books. The key idea is that the complexity of the probabilistic model must be considered.
Kolmogorov complexity In algorithmic information theory Algorithmic information theory (AIT) is a branch of theoretical computer science Theoretical computer science (TCS) is a subset of general computer science Computer science deals with the theoretical ...
is a theoretical generalization of this idea that allows the consideration of the information content of a sequence independent of any particular probability model; it considers the shortest
program Program, programme, programmer, or programming may refer to: Business and management * Program management, the process of managing several related projects * Time management * Program, a part of planning Arts and entertainment Audio * Program ...
for a
universal computer A Turing machine is a mathematical model of computation Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are ...
that outputs the sequence. A code that achieves the entropy rate of a sequence for a given model, plus the codebook (i.e. the probabilistic model), is one such program, but it may not be the shortest. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, .... treating the sequence as a message and each number as a symbol, there are almost as many symbols as there are characters in the message, giving an entropy of approximately . The first 128 symbols of the Fibonacci sequence has an entropy of approximately 7 bits/symbol, but the sequence can be expressed using a formula for , , and this formula has a much lower entropy and applies to any length of the Fibonacci sequence.

## Limitations of entropy in cryptography

In
cryptanalysis Cryptanalysis (from the Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...
, entropy is often roughly used as a measure of the unpredictability of a cryptographic key, though its real
uncertainty Uncertainty refers to epistemic Epistemology (; ) is the branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, ...

is unmeasurable. For example, a 128-bit key that is uniformly and randomly generated has 128 bits of entropy. It also takes (on average) $2^$ guesses to break by brute force. Entropy fails to capture the number of guesses required if the possible keys are not chosen uniformly. Instead, a measure called ''guesswork'' can be used to measure the effort required for a brute force attack. Other problems may arise from non-uniform distributions used in cryptography. For example, a 1,000,000-digit binary
one-time pad In cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia ''-logy'' is a suffix in the English language, used with words originally adapted from Ancient G ...

using exclusive or. If the pad has 1,000,000 bits of entropy, it is perfect. If the pad has 999,999 bits of entropy, evenly distributed (each individual bit of the pad having 0.999999 bits of entropy) it may provide good security. But if the pad has 999,999 bits of entropy, where the first bit is fixed and the remaining 999,999 bits are perfectly random, the first bit of the ciphertext will not be encrypted at all.

## Data as a Markov process

A common way to define entropy for text is based on the
Markov model In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in ...
of text. For an order-0 source (each character is selected independent of the last characters), the binary entropy is: :$\Eta\left(\mathcal\right) = - \sum p_i \log p_i ,$ where is the probability of . For a first-order Markov source (one in which the probability of selecting a character is dependent only on the immediately preceding character), the
entropy rate In the mathematical theory of probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), sha ...
is: :$\Eta\left(\mathcal\right) = - \sum_i p_i \sum_j \ p_i \left(j\right) \log p_i \left(j\right) ,$ where is a state (certain preceding characters) and $p_i\left(j\right)$ is the probability of given as the previous character. For a second order Markov source, the entropy rate is :$\Eta\left(\mathcal\right) = -\sum_i p_i \sum_j p_i\left(j\right) \sum_k p_\left(k\right)\ \log \ p_\left(k\right) .$

# Efficiency (normalized entropy)

A source alphabet with non-uniform distribution will have less entropy than if those symbols had uniform distribution (i.e. the "optimized alphabet"). This deficiency in entropy can be expressed as a ratio called efficiency: :$\eta\left(X\right) = \frac = -\sum_^n \frac$ Applying the basic properties of the logarithm, this quantity can also be expressed as: :$\eta\left(X\right) = -\sum_^n \frac = \sum_^n \frac = \sum_^n \log_n\left(p\left(x_i\right)^\right) = \log_n \left(\prod_^n p\left(x_i\right)^\right)$ Efficiency has utility in quantifying the effective use of a
communication channel A communication channel refers either to a physical transmission medium A transmission medium is a system or substance that can mediate the propagation of signal In signal processing Signal processing is an electrical engineering ...
. This formulation is also referred to as the normalized entropy, as the entropy is divided by the maximum entropy . Furthermore, the efficiency is indifferent to choice of (positive) base , as indicated by the insensitivity within the final logarithm above thereto.

# Entropy for continuous random variables

## Differential entropy

The Shannon entropy is restricted to random variables taking discrete values. The corresponding formula for a continuous random variable with
probability density function In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
with finite or infinite support $\mathbb X$ on the real line is defined by analogy, using the above form of the entropy as an expectation: : This is the differential entropy (or continuous entropy). A precursor of the continuous entropy is the expression for the functional in the
H-theorem In classical statistical mechanics In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature ...
of
Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...

. Although the analogy between both functions is suggestive, the following question must be set: is the differential entropy a valid extension of the Shannon discrete entropy? Differential entropy lacks a number of properties that the Shannon discrete entropy has – it can even be negative – and corrections have been suggested, notably
limiting density of discrete points In information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamen ...
. To answer this question, a connection must be established between the two functions: In order to obtain a generally finite measure as the bin size goes to zero. In the discrete case, the bin size is the (implicit) width of each of the (finite or infinite) bins whose probabilities are denoted by . As the continuous domain is generalized, the width must be made explicit. To do this, start with a continuous function discretized into bins of size $\Delta$. By the mean-value theorem there exists a value in each bin such that :$f\left(x_i\right) \Delta = \int_^ f\left(x\right)\, dx$ the integral of the function can be approximated (in the Riemannian sense) by :$\int_^ f\left(x\right)\, dx = \lim_ \sum_^ f\left(x_i\right) \Delta$ where this limit and "bin size goes to zero" are equivalent. We will denote :$\Eta^ := - \sum_^ f\left(x_i\right) \Delta \log \left\left( f\left(x_i\right) \Delta \right\right)$ and expanding the logarithm, we have :$\Eta^ = - \sum_^ f\left(x_i\right) \Delta \log \left(f\left(x_i\right)\right) -\sum_^ f\left(x_i\right) \Delta \log \left(\Delta\right).$ As Δ → 0, we have :$\begin \sum_^ f\left(x_i\right) \Delta &\to \int_^ f\left(x\right)\, dx = 1 \\ \sum_^ f\left(x_i\right) \Delta \log \left(f\left(x_i\right)\right) &\to \int_^ f\left(x\right) \log f\left(x\right)\, dx. \end$ Note; as , requires a special definition of the differential or continuous entropy: : which is, as said before, referred to as the differential entropy. This means that the differential entropy ''is not'' a limit of the Shannon entropy for . Rather, it differs from the limit of the Shannon entropy by an infinite offset (see also the article on information dimension).

## Limiting density of discrete points

It turns out as a result that, unlike the Shannon entropy, the differential entropy is ''not'' in general a good measure of uncertainty or information. For example, the differential entropy can be negative; also it is not invariant under continuous co-ordinate transformations. This problem may be illustrated by a change of units when ''x'' is a dimensioned variable. ''f(x)'' will then have the units of ''1/x''. The argument of the logarithm must be dimensionless, otherwise it is improper, so that the differential entropy as given above will be improper. If ''Δ'' is some "standard" value of ''x'' (i.e. "bin size") and therefore has the same units, then a modified differential entropy may be written in proper form as: :$H=\int_^\infty f\left(x\right) \log\left(f\left(x\right)\,\Delta\right)\,dx$ and the result will be the same for any choice of units for ''x''. In fact, the limit of discrete entropy as $N \rightarrow \infty$ would also include a term of $\log\left(N\right)$, which would in general be infinite. This is expected: continuous variables would typically have infinite entropy when discretized. The
limiting density of discrete points In information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamen ...
is really a measure of how much easier a distribution is to describe than a distribution that is uniform over its quantization scheme.

## Relative entropy

Another useful measure of entropy that works equally well in the discrete and the continuous case is the relative entropy of a distribution. It is defined as the Kullback–Leibler divergence from the distribution to a reference measure as follows. Assume that a probability distribution is absolutely continuous with respect to a measure , i.e. is of the form for some non-negative -integrable function with -integral 1, then the relative entropy can be defined as :$D_\left(p \, m \right) = \int \log \left(f\left(x\right)\right) p\left(dx\right) = \int f\left(x\right)\log \left(f\left(x\right)\right) m\left(dx\right) .$ In this form the relative entropy generalizes (up to change in sign) both the discrete entropy, where the measure is the counting measure, and the differential entropy, where the measure is the Lebesgue measure. If the measure is itself a probability distribution, the relative entropy is non-negative, and zero if as measures. It is defined for any measure space, hence coordinate independent and invariant under co-ordinate reparameterizations if one properly takes into account the transformation of the measure . The relative entropy, and (implicitly) entropy and differential entropy, do depend on the "reference" measure .

# Use in combinatorics

Entropy has become a useful quantity in
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
.

## Loomis–Whitney inequality

A simple example of this is an alternate proof of the Loomis–Whitney inequality: for every subset , we have :$, A, ^\leq \prod_^ , P_\left(A\right),$ where is the orthogonal projection in the th coordinate: :$P_\left(A\right)=\.$ The proof follows as a simple corollary of Shearer's inequality: if are random variables and are subsets of such that every integer between 1 and lies in exactly of these subsets, then :$\Eta\left[\left(X_, \ldots ,X_\right)\right]\leq \frac\sum_^\Eta\left[\left(X_\right)_\right]$ where $\left(X_\right)_$ is the Cartesian product of random variables with indexes in (so the dimension of this vector is equal to the size of ). We sketch how Loomis–Whitney follows from this: Indeed, let be a uniformly distributed random variable with values in and so that each point in occurs with equal probability. Then (by the further properties of entropy mentioned above) , where denotes the cardinality of . Let . The range of $\left(X_\right)_$ is contained in and hence $\Eta\left[\left(X_\right)_\right]\leq \log , P_\left(A\right),$. Now use this to bound the right side of Shearer's inequality and exponentiate the opposite sides of the resulting inequality you obtain.

## Approximation to binomial coefficient

For integers let . Then :$\frac \leq \tbinom nk \leq 2^,$ where :$\Eta\left(q\right) = -q \log_2\left(q\right) - \left(1-q\right) \log_2\left(1-q\right).$ : A nice interpretation of this is that the number of binary strings of length with exactly many 1's is approximately $2^$.

# Use in machine learning

Machine learning techniques arise largely from statistics and also information theory. In general, entropy is a measure of uncertainty and the objective of machine learning is to minimize uncertainty. Decision tree learning algorithms use relative entropy to determine the decision rules that govern the data at each node. The Information gain in decision trees $IG\left(Y,X\right)$, which is equal to the difference between the entropy of $Y$ and the conditional entropy of $Y$ given $X$, quantifies the expected information, or the reduction in entropy, from additionally knowing the value of an attribute $X$. The information gain is used to identify which attributes of the dataset provide the most information and should be used to split the nodes of the tree optimally. Bayesian inference models often apply the Principle of maximum entropy to obtain Prior probability distributions. The idea is that the distribution that best represents the current state of knowledge of a system is the one with the largest entropy, and is therefore suitable to be the prior. Classification in machine learning performed by Logistic regression or Artificial neural networks often employs a standard loss function, called Cross entropy loss, that minimizes the average cross entropy between ground truth and predicted distributions. In general, cross entropy is a measure of the differences between two datasets similar to the KL divergence (or relative entropy).

*Cross entropy – is a measure of the average number of bits needed to identify an event from a set of possibilities between two probability distributions *Entropy (arrow of time) *Entropy encoding – a coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols. *Entropy estimation *Entropy power inequality *Fisher information *Graph entropy *Hamming distance *History of entropy *History of information theory *Information fluctuation complexity *Information geometry *Kolmogorov–Sinai entropy in dynamical systems *Levenshtein distance *Mutual information *Perplexity *Qualitative variation – other measures of statistical dispersion for nominal distributions *Quantum relative entropy – a measure of distinguishability between two quantum states. *Rényi entropy – a generalization of Shannon entropy; it is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system. *Randomness *Shannon index *Theil index *Typoglycemia

# References

## Textbooks on information theory

* Thomas M. Cover, Cover, T.M., Joy A. Thomas, Thomas, J.A. (2006), ''Elements of Information Theory - 2nd Ed.'', Wiley-Interscience, * David J.C. MacKay, MacKay, D.J.C. (2003), ''Information Theory, Inference and Learning Algorithms'' , Cambridge University Press, * Arndt, C. (2004), ''Information Measures: Information and its Description in Science and Engineering'', Springer, * Gray, R. M. (2011), ''Entropy and Information Theory'', Springer. * * Claude Shannon, Shannon, C.E., Warren Weaver, Weaver, W. (1949) ''The Mathematical Theory of Communication'', Univ of Illinois Press. * Stone, J. V. (2014), Chapter 1 o
''Information Theory: A Tutorial Introduction''
University of Sheffield, England. .