Firsthand Accounts of Events Can Be Viewed As______________ of Information.

Expected amount of information needed to specify the output of a stochastic data source

Ii $.25 of entropy: In the case of two fair money tosses, the information entropy in bits is the base of operations-2 logarithm of the number of possible outcomes; with two coins at that place are iv possible outcomes, and 2 bits of entropy. Generally, information entropy is the average corporeality of information conveyed by an issue, when considering all possible outcomes.

In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable'south possible outcomes. Given a discrete random variable $X$ , with possible outcomes $x_{one},...,x_{n}$ , which occur with probability $\mathrm {P} (x_{1}),...,\mathrm {P} (x_{north}),$ the entropy of $X$ is formally divers as:

$\mathrm {H} (X)=-\sum _{i=i}^{n}{\mathrm {P} (x_{i})\log \mathrm {P} (x_{i})}$

where $\Sigma$ denotes the sum over the variable's possible values. The choice of base for $\log$ , the logarithm, varies for different applications. Base of operations 2 gives the unit of bits (or "shannons"), while base e gives "natural units" nat, and base of operations 10 gives units of "dits", "bans", or "hartleys". An equivalent definition of entropy is the expected value of the self-information of a variable.^[ane]

The concept of data entropy was introduced past Claude Shannon in his 1948 paper "A Mathematical Theory of Communication",^[2] ^[iii] and is also referred to equally 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" – equally 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.^[2] ^[iii] Shannon considered various ways to encode, compress, and transmit letters from a data source, and proved in his famous source coding theorem that the entropy represents an absolute mathematical limit on how well data from the source tin be losslessly compressed onto a perfectly noiseless channel. Shannon strengthened this effect considerably for noisy channels in his noisy-channel coding theorem.

Entropy in information theory is direct coordinating to the entropy in statistical thermodynamics. 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 and machine learning. The definition can be derived from a set of axioms establishing that entropy should be a measure of how "surprising" the average outcome of a variable is. For a continuous random variable, differential entropy is coordinating to entropy.

Introduction [edit]

The core idea of information theory is that the "informational value" of a communicated bulletin depends on the caste to which the content of the message is surprising. If a highly likely issue occurs, the bulletin carries very petty information. On the other paw, if a highly unlikely effect occurs, the message is much more informative. For instance, the knowledge that some detail number will not be the winning number of a lottery provides very little data, because any detail chosen number will near certainly not win. However, noesis that a item number will win a lottery has high informational value because it communicates the outcome of a very low probability issue.

The data content, also called the surprisal or cocky-information, of an result $Due east$ is a function which increases as the probability $p(Due east)$ of an event decreases. When $P(Eastward)$ is close to i, the surprisal of the effect is low, but if $P(E)$ is close to 0, the surprisal of the effect is high. This human relationship is described by the function

$\log \left({\frac {1}{P(E)}}\right)$

where $\log$ is the logarithm, which gives 0 surprise when the probability of the event is 1.^[4] In fact, the $\log$ is the merely office that satisfies this specific gear up of characterization.

Hence, we can define the information, or surprisal, of an effect $E$ by

$I(E)=-\log _{two}(p(E))$ , or equivalently,

$I(E)=\log _{2}\left({\frac {1}{p(E)}}\correct)$ .

Entropy measures the expected (i.e., average) amount of information conveyed by identifying the outcome of a random trial.^[5] ^: 67 This implies that casting a die has higher entropy than tossing a money considering each outcome of a die toss has smaller probability (about $p=1/6$ ) than each event of a coin toss ( $p=ane/two$ ).

Consider a biased coin with probability $p$ of landing on heads and probability $1 - p$ of landing on tails. The maximum surprise is when $p = ane/2$ , for which one outcome is non expected over the other. In this case a coin flip has an entropy of ane bit. (Similarly, 1 trit with equiprobable values contains $\log _{2}3$ (about 1.58496) bits of information because it tin can have one of three values.) The minimum surprise is when $p = 0$ or $p = one$ , when the event outcome is known alee of time, and the entropy is zero bits. When the entropy is goose egg bits, this is sometimes referred to as unity, where there is no uncertainty at all - no freedom of choice - no information. Other values of p requite entropies between cipher and one $.25.

Data theory is useful to calculate the smallest amount of information required to convey a message, equally in data compression. For example, consider the manual of sequences comprising the iv characters 'A', 'B', 'C', and 'D' over a binary channel. If all 4 letters are every bit likely (25%), one can't practise better than using two bits to encode each letter. 'A' might code as '00', 'B' equally '01', 'C' every bit '10', and 'D' every bit 'xi'. Yet, if the probabilities of each letter are diff, say 'A' occurs with 70% probability, 'B' with 26%, and 'C' and 'D' with two% each, one could assign variable length codes. In this instance, 'A' would be coded as '0', 'B' as '10', 'C' as '110', and D every bit '111'. With this representation, lxx% of the time just i bit needs to be sent, 26% of the time ii bits, and only 4% of the time three bits. On average, fewer than 2 bits are required since the entropy is lower (owing to the loftier prevalence of 'A' followed by 'B' – together 96% of characters). The adding 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 sure that, for example, 'due east' will be far more than mutual than 'z', that the combination 'qu' will be much more than common than whatsoever other combination with a 'q' in information technology, and that the combination 'thursday' will be more common than 'z', 'q', or 'qu'. After the showtime few messages one can often guess the rest of the word. English language text has betwixt 0.half-dozen and i.3 $.25 of entropy per character of the message.^[6] ^: 234

Definition [edit]

Named after Boltzmann's Η-theorem, Shannon defined the entropy $Η$ (Greek upper-case letter eta) of a discrete random variable ${\textstyle X}$ with possible values ${\textstyle \left\{x_{i},\ldots ,x_{n}\right\}}$ and probability mass office ${\textstyle \mathrm {P} (X)}$ as:

\mathrm {H} (10)=\operatorname {E} [\operatorname {I} (X)]=\operatorname {E} [-\log(\mathrm {P} (X))].

Here $\operatorname {Eastward}$ is the expected value operator, and $I$ is the information content of $X$ .^[7] ^: xi ^[8] ^: xix–xx $\operatorname {I} (X)$ is itself a random variable.

The entropy can explicitly exist written equally:

$\mathrm {H} (Ten)=-\sum _{i=i}^{due north}{\mathrm {P} (x_{i})\log _{b}\mathrm {P} (x_{i})}$

where $b$ is the base of operations of the logarithm used. Mutual values of $b$ are 2, Euler's number $e$ , and 10, and the corresponding units of entropy are the bits for $b = two$ , nats for $b = due east$ , and bans for $b = ten$ .^[9]

In the instance of $P(x i) = 0$ for some $i$ , the value of the corresponding summand $0 log b (0)$ is taken to be $0$ , which is consistent with the limit:^[10] ^: thirteen

\lim _{p\to 0^{+}}p\log(p)=0.

1 may also define the conditional entropy of two variables $X$ and $Y$ taking values $x_{i}$ and $y_{j}$ respectively, equally:^[x] ^: 16

\mathrm {H} (X|Y)=-\sum _{i,j}p(x_{i},y_{j})\log {\frac {p(x_{i},y_{j})}{p(y_{j})}}

where $p(x_{i},y_{j})$ is the probability that $X=x_{i}$ and $Y=y_{j}$ . This quantity should be understood every bit the amount of randomness in the random variable $X$ given the random variable $Y$ .

Measure theory [edit]

Entropy can be formally defined in the language of mensurate theory as follows:^[11] Allow $(Ten,\Sigma ,\mu )$ be a probability infinite. Let $A\in \Sigma$ be an outcome. The surprisal of $A$ is

\sigma _{\mu }(A)=-\ln \mu (A)

The expected surprisal of $A$ is

h_{\mu }(A)=\mu (A)\sigma _{\mu }(A)

A $\mu$ -well-nigh partition is a fix family $P\subseteq {\mathcal {P}}(X)$ such that $\mu (\mathop {\cup } P)=1$ and $\mu (A\cap B)=0$ for all distinct $A,B\in P$ . (This is a relaxation of the usual conditions for a sectionalization.) The entropy of $P$ is

H_{\mu }(P)=\sum _{A\in P}h_{\mu }(A)

Let $Thousand$ exist a sigma-algebra on $X$ . The entropy of $M$ is

H_{\mu }(M)=\sup _{P\subseteq One thousand}H_{\mu }(P)

Finally, the entropy of the probability space is $H_{\mu }(\Sigma )$ , that is, the entropy with respect to $\mu$ of the sigma-algebra of all measurable subsets of $X$ .

Example [edit]

Entropy $Η(X)$ (i.e. the expected surprisal) of a coin flip, measured in bits, graphed versus the bias of the coin $Pr(X = one)$ , where $X = i$ represents a result of heads.^[10] ^: xiv–xv

Here, the entropy is at near 1 flake, and to communicate the effect of a coin flip (2 possible values) volition require an average of at virtually ane bit (exactly one flake for a fair coin). The result of a fair die (6 possible values) would take entropy log₂six bits.

Consider tossing a money with known, not necessarily fair, probabilities of coming up heads or tails; this can exist modelled as a Bernoulli process.

The entropy of the unknown result of the side by side toss of the coin is maximized if the coin is fair (that is, if heads and tails both have equal probability 1/ii). This is the state of affairs of maximum doubtfulness as information technology is near hard to predict the outcome of the next toss; the result of each toss of the coin delivers one full scrap of information. This is because

{\begin{aligned}\mathrm {H} (X)&=-\sum _{i=1}^{north}{\mathrm {P} (x_{i})\log _{b}\mathrm {P} (x_{i})}\\&=-\sum _{i=one}^{2}{{\frac {ane}{2}}\log _{two}{\frac {1}{2}}}\\&=-\sum _{i=1}^{2}{{\frac {1}{2}}\cdot (-1)}=ane\terminate{aligned}}

All the same, if we know the coin is not fair, but comes up heads or tails with probabilities $p$ and $q$ , where $p \neq q$ , then at that place is less uncertainty. Every fourth dimension information technology is tossed, one side is more than probable to come up than the other. The reduced dubiety is quantified in a lower entropy: on average each toss of the money delivers less than i full chip of data. For instance, if $p$ =0.vii, then

{\begin{aligned}\mathrm {H} (X)&=-p\log _{ii}(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 (-i.737)\\&=0.8816<one\end{aligned}}

Uniform probability yields maximum uncertainty and therefore maximum entropy. Entropy, and so, tin can only decrease from the value associated with uniform probability. The extreme example is that of a double-headed coin that never comes up tails, or a double-tailed money that never results in a head. Then there is no uncertainty. The entropy is cipher: each toss of the coin delivers no new information equally the upshot of each coin toss is always sure.^[ten] ^{: 14–fifteen}

Entropy can be normalized by dividing it by information length. This ratio is chosen metric entropy and is a measure of the randomness of the information.

Characterization [edit]

To understand the significant of $-Σ p i log(p i)$ , first define an data function $I$ in terms of an event $i$ with probability $p i$ . The amount of information acquired due to the observation of result $i$ follows from Shannon'south solution of the primal properties of data:^[12]

$I(p)$ is monotonically decreasing in $p$ : an increase in the probability of an event decreases the information from an observed event, and vice versa.
$I(p) \geq 0$ : information is a non-negative quantity.
$I(one) = 0$ : events that always occur do non communicate information.
$I(p 1 \cdot p 2) = I(p 1) + I(p 2)$ : the information learned from contained events is the sum of the information learned from each event.

Given 2 independent events, if the first result tin can yield one of $n$ equiprobable outcomes and another has one of $g$ equiprobable outcomes and so there are $mn$ equiprobable outcomes of the joint issue. This means that if $log 2 (n)$ bits are needed to encode the get-go value and $log two (yard)$ to encode the 2d, one needs $log 2 (mn) = log 2 (g) + log two (north)$ to encode both.

Shannon discovered that a suitable pick of $\operatorname {I}$ is given by:^[thirteen]

\operatorname {I} (p)=\log \left({\tfrac {ane}{p}}\correct)=-\log(p)

In fact, the just possible values of $\operatorname {I}$ are $\operatorname {I} (u)=1000\log u$ for $yard<0$ . Additionally, choosing a value for $one thousand$ is equivalent to choosing a value $x>one$ for $one thousand=-one/\log ten$ , and so that $x$ corresponds to the base for the logarithm. Thus, entropy is characterized by the above four backdrop.

Proof

Allow

{\textstyle \operatorname {I} }

exist the data function which one assumes to be twice continuously differentiable, 1 has:

{\begin{aligned}&\operatorname {I} (p_{1}p_{2})&=\ &\operatorname {I} (p_{one})+\operatorname {I} (p_{two})&&\quad {\text{Starting from holding 4}}\\&p_{two}\operatorname {I} '(p_{1}p_{2})&=\ &\operatorname {I} '(p_{1})&&\quad {\text{taking the derivative w.r.t}}\ p_{1}\\&\operatorname {I} '(p_{1}p_{two})+p_{1}p_{2}\operatorname {I} ''(p_{ane}p_{2})&=\ &0&&\quad {\text{taking the derivative w.r.t}}\ p_{ii}\\&\operatorname {I} '(u)+u\operatorname {I} ''(u)&=\ &0&&\quad {\text{introducing}}\,u=p_{ane}p_{2}\\&(u\mapsto u\operatorname {I} '(u))'&=\ &0\end{aligned}}

This differential equation leads to the solution $\operatorname {I} (u)=k\log u+c$ for some $thousand,c\in \mathbb {R}$ . Property three gives $c=0$ , and Property 2 leads to $k<0$ . Property 1 then holds too.

The unlike units of information ($.25 for the binary logarithm $log 2$ , nats for the natural logarithm $ln$ , bans for the decimal logarithm $log ten$ so on) are abiding multiples of each other. For instance, in instance of a fair coin toss, heads provides $log two (ii) = 1$ bit of data, which is approximately 0.693 nats or 0.301 decimal digits. Because of additivity, $due north$ tosses provide $n$ $.25 of information, which is approximately $0.693 northward$ nats or $0.301 n$ decimal digits.

The meaning of the events observed (the significant of messages) does not matter in the definition of entropy. Entropy only takes into account the probability of observing a specific consequence, so the information information technology encapsulates is information about the underlying probability distribution, not the meaning of the events themselves.

Alternate characterization [edit]

Another characterization of entropy uses the following properties. We denote $p i = Pr(10 = x i)$ and $Η n (p 1, ..., p n) = Η(X)$ .

Continuity: $H$ should be continuous, and then that irresolute the values of the probabilities by a very pocket-sized amount should but alter the entropy past a small amount.
Symmetry: $H$ should exist unchanged if the outcomes $x i$ are re-ordered. That is, $\mathrm {H} _{n}\left(p_{ane},p_{2},\ldots p_{north}\right)=\mathrm {H} _{due north}\left(p_{i_{i}},p_{i_{2}},\ldots ,p_{i_{n}}\right)$ for any permutation $\{i_{1},...,i_{north}\}$ of $\{1,...,n\}$ .
Maximum: $\mathrm {H} _{due north}$ should be maximal if all the outcomes are every bit likely i.e. $\mathrm {H} _{n}(p_{1},\ldots ,p_{n})\leq \mathrm {H} _{n}\left({\frac {1}{n}},\ldots ,{\frac {1}{n}}\right)$ .
Increasing number of outcomes: for equiprobable events, the entropy should increase with the number of outcomes i.due east. $\mathrm {H} _{n}{\bigg (}\underbrace {{\frac {i}{n}},\ldots ,{\frac {1}{n}}} _{due north}{\bigg )}<\mathrm {H} _{n+1}{\bigg (}\underbrace {{\frac {1}{n+1}},\ldots ,{\frac {one}{northward+i}}} _{n+1}{\bigg )}.$
Additivity: given an ensemble of $due north$ uniformly distributed elements that are divided into $g$ boxes (sub-systems) with $b one, ..., b grand$ elements each, the entropy of the whole ensemble should be equal to the sum of the entropy of the arrangement of boxes and the individual entropies of the boxes, each weighted with the probability of being in that detail box.

The rule of additivity has the following consequences: for positive integers $b i$ where $b one + ... + b yard = n$ ,

\mathrm {H} _{due north}\left({\frac {one}{north}},\ldots ,{\frac {ane}{n}}\right)=\mathrm {H} _{k}\left({\frac {b_{i}}{north}},\ldots ,{\frac {b_{thousand}}{north}}\right)+\sum _{i=1}^{1000}{\frac {b_{i}}{n}}\,\mathrm {H} _{b_{i}}\left({\frac {1}{b_{i}}},\ldots ,{\frac {1}{b_{i}}}\right).

Choosing $k = northward$ , $b 1 = ... = b n = 1$ this implies that the entropy of a sure outcome is zero: $Η 1 (one) = 0$ . This implies that the efficiency of a source alphabet with $north$ symbols can be defined just as beingness equal to its $n$ -ary entropy. See also Back-up (information theory).

Further backdrop [edit]

The Shannon entropy satisfies the following properties, for some of which it is useful to interpret entropy as the corporeality of information learned (or dubiousness eliminated) by revealing the value of a random variable $X$ :

Adding or removing an event with probability cypher does not contribute to the entropy:

\mathrm {H} _{north+ane}(p_{ane},\ldots ,p_{n},0)=\mathrm {H} _{north}(p_{1},\ldots ,p_{due north})

It can be confirmed using the Jensen inequality that

\mathrm {H} (X)=\operatorname {E} \left[\log _{b}\left({\frac {i}{p(Ten)}}\right)\correct]\leq \log _{b}\left(\operatorname {E} \left[{\frac {1}{p(Ten)}}\correct]\correct)=\log _{b}(n)

.^[10] ^: 29

This maximal entropy of

log b (north)

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 $(X, Y)$ (that is, evaluating $Ten$ and $Y$ simultaneously) is equal to the information revealed by conducting 2 consecutive experiments: offset evaluating the value of $Y$ , then revealing the value of $Ten$ given that you know the value of $Y$ . This may be written equally:^[10] ^: 16

\mathrm {H} (X,Y)=\mathrm {H} (Ten|Y)+\mathrm {H} (Y)=\mathrm {H} (Y|Ten)+\mathrm {H} (X).

\mathrm {H} (X)+\mathrm {H} (f(X)|X)=\mathrm {H} (f(Ten))+\mathrm {H} (Ten|f(Ten)),

H(f(X))\leq H(10)

, the entropy of a variable can only decrease when the latter is passed through a function.

If $X$ and $Y$ are two independent random variables, so knowing the value of $Y$ doesn't influence our knowledge of the value of $X$ (since the two don't influence each other past independence):

\mathrm {H} (X|Y)=\mathrm {H} (X).

More than generally, for whatever random variables $X$ and $Y$ , we have

\mathrm {H} (10|Y)\leq \mathrm {H} (X)

.^[ten] ^: 29

\mathrm {H} (\lambda p_{ane}+(1-\lambda )p_{2})\geq \lambda \mathrm {H} (p_{1})+(1-\lambda )\mathrm {H} (p_{ii})

for all probability mass functions

p_{1},p_{2}

and

0\leq \lambda \leq 1

.^[ten] ^: 32

Accordingly, the negative entropy (negentropy) function is convex, and its convex conjugate is LogSumExp.

Aspects [edit]

Relationship to thermodynamic entropy [edit]

The inspiration for adopting the give-and-take entropy in information theory came from the shut resemblance between Shannon's formula and very similar known formulae from statistical mechanics.

In statistical thermodynamics the most general formula for the thermodynamic entropy $S$ of a thermodynamic system is the Gibbs entropy,

S=-k_{\text{B}}\sum p_{i}\ln p_{i}\,

where $k B$ is the Boltzmann abiding, and $p i$ is the probability of a microstate. The Gibbs entropy was divers by J. Willard Gibbs in 1878 after earlier work past Boltzmann (1872).^[14]

The Gibbs entropy translates over almost unchanged into the world of quantum physics to give the von Neumann entropy, introduced by John von Neumann in 1927,

Southward=-k_{\text{B}}\,{\rm {Tr}}(\rho \ln \rho )\,

where ρ is the density matrix of the quantum mechanical organization and Tr is the trace.^[15]

At an everyday applied level, the links between data 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 2nd police force of thermodynamics, rather than an unchanging probability distribution. As the minuteness of Boltzmann's constant $1000 B$ indicates, the changes in $South / yard B$ for even tiny amounts of substances in chemical and physical processes correspond amounts of entropy that are extremely big compared to annihilation in data compression or betoken processing. In classical thermodynamics, entropy is divers in terms of macroscopic measurements and makes no reference to whatever probability distribution, which is central to the definition of data entropy.

The connectedness betwixt thermodynamics and what is now known as information theory was first made by Ludwig Boltzmann and expressed by his famous equation:

S=k_{\text{B}}\ln(W)

where $S$ is the thermodynamic entropy of a detail macrostate (defined by thermodynamic parameters such as temperature, volume, free energy, etc.), W is the number of microstates (various combinations of particles in various free energy states) that can yield the given macrostate, and grand_B is Boltzmann'south abiding.^[sixteen] Information technology is causeless that each microstate is every bit likely, then that the probability of a given microstate is p_i = one/W. When these probabilities are substituted into the above expression for the Gibbs entropy (or equivalently one thousand_B times the Shannon entropy), Boltzmann's equation results. In information theoretic terms, the information entropy of a system is the amount of "missing" data needed to determine a microstate, given the macrostate.

In the view of Jaynes (1957),^[17] thermodynamic entropy, as explained by statistical mechanics, should be seen as an awarding of Shannon'southward information theory: the thermodynamic entropy is interpreted as existence proportional to the amount of further Shannon information needed to define the detailed microscopic land of the system, that remains uncommunicated past a description solely in terms of the macroscopic variables of classical thermodynamics, with the abiding of proportionality being but the Boltzmann abiding. Calculation heat to a arrangement increases its thermodynamic entropy because it increases the number of possible microscopic states of the system that are consequent with the measurable values of its macroscopic variables, making any complete state clarification longer. (See article: maximum entropy thermodynamics). Maxwell's demon tin (hypothetically) reduce the thermodynamic entropy of a system by using information about u.s. of private molecules; but, every bit Landauer (from 1961) and co-workers^[18] have shown, to function the demon himself must increase thermodynamic entropy in the process, by at least the corporeality of Shannon information he proposes to first acquire and store; and so the total thermodynamic entropy does non decrease (which resolves the paradox). Landauer's principle imposes a lower jump on the amount of rut a computer must generate to process a given amount of information, though mod computers are far less efficient.

Data pinch [edit]

Shannon'due south definition of entropy, when applied to an data source, tin make up one's mind the minimum channel capacity required to reliably transmit the source as encoded binary digits. Shannon's entropy measures the data contained in a message equally opposed to the portion of the message that is adamant (or predictable). Examples of the latter include redundancy in language construction or statistical backdrop relating to the occurrence frequencies of letter or give-and-take pairs, triplets etc. The minimum aqueduct capacity can be realized in theory by using the typical set or in practice using Huffman, Lempel–Ziv or arithmetic coding. (See too Kolmogorov complexity.) In practise, compression algorithms deliberately include some judicious back-up in the form of checksums to protect against errors. The entropy charge per unit of a data source is the boilerplate number of bits per symbol needed to encode information technology. Shannon'southward experiments with human predictors show an information rate between 0.6 and ane.3 bits per character in English;^[19] the PPM compression algorithm can attain a compression ratio of 1.v bits per character in English text.

If a compression scheme is lossless – one in which you can e'er recover the entire original message by decompression – then a compressed bulletin has the same quantity of data every bit the original only communicated in fewer characters. It has more information (higher entropy) per grapheme. A compressed message has less redundancy. Shannon's source coding theorem states a lossless compression scheme cannot shrink messages, on average, to accept more than one flake of information per bit of message, but that any value less than one bit of information per bit of message tin 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 full data the message contains. Shannon's theorem as well implies that no lossless compression scheme can shorten all messages. If some letters come out shorter, at least one must come out longer due to the pigeonhole principle. In practical use, this is generally not a trouble, because i is usually only interested in compressing certain types of messages, such as a document in English language, 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 estimates the world'due south 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.^[20] ^{: sixty–65}

All figures in entropically compressed exabytes
Blazon of Information	1986	2007
Storage	2.6	295
Broadcast	432	1900
Telecommunications	0.281	65

The authors guess humankind technological chapters to shop information (fully entropically compressed) in 1986 and again in 2007. They break the information into three categories—to store data on a medium, to receive data through one-way broadcast networks, or to exchange information through two-way telecommunications networks.^[20]

Entropy as a mensurate of diversity [edit]

Entropy is one of several ways to measure out biodiversity, and is applied in the form of the Shannon alphabetize.^[21] A diversity index is a quantitative statistical measure of how many dissimilar types exist in a dataset, such as species in a community, accounting for ecological richness, evenness, and say-so. Specifically, Shannon entropy is the logarithm of $ane D$ , the truthful diversity index with parameter equal to 1. The Shannon alphabetize is related to the proportional abundances of types.

Limitations of entropy [edit]

There are a number of entropy-related concepts that mathematically quantify information content in some way:

the self-information of an individual message or symbol taken from a given probability distribution,
the entropy of a given probability distribution of letters or symbols, and
the entropy rate of a stochastic procedure.

(The "rate of self-information" can also be defined for a item sequence of messages or symbols generated by a given stochastic process: this will always be equal to the entropy charge per unit in the instance of a stationary process.) Other quantities of information are likewise used to compare or chronicle different sources of information.

It is of import not to confuse the higher up concepts. Often it is but clear from context which one is meant. For example, when someone says that the "entropy" of the English linguistic communication is about 1 flake per character, they are actually modeling the English linguistic communication every bit a stochastic procedure and talking about its entropy rate. Shannon himself used the term in this style.

If very large blocks are used, the estimate of per-grapheme entropy rate may become artificially low considering the probability distribution of the sequence is not known exactly; it is only an estimate. If one considers the text of every book e'er published as a sequence, with each symbol being the text of a consummate book, and if there are $N$ published books, and each volume is merely published one time, the estimate of the probability of each book is $1/ N$ , and the entropy (in $.25) is $-log 2 (1/ Due north) = log 2 (Due north)$ . As a practical lawmaking, this corresponds to assigning each volume a unique identifier 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 near books, but it is not so useful for characterizing the data content of an private book, or of language in general: it is non possible to reconstruct the book from its identifier without knowing the probability distribution, that is, the complete text of all the books. The fundamental thought is that the complexity of the probabilistic model must be considered. Kolmogorov complexity is a theoretical generalization of this idea that allows the consideration of the information content of a sequence independent of any detail probability model; information technology considers the shortest program for a universal computer 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 i such program, but it may not be the shortest.

The Fibonacci sequence is 1, 1, 2, three, v, 8, xiii, .... treating the sequence equally a bulletin and each number as a symbol, there are nearly equally many symbols as there are characters in the message, giving an entropy of approximately $log 2 (n)$ . The offset 128 symbols of the Fibonacci sequence has an entropy of approximately vii $.25/symbol, merely the sequence can exist expressed using a formula [ $F(n) = F(northward -1) + F(n -two)$ for $n = iii, iv, 5, ...$ , $F(ane) =ane$ , $F(ii) = ane$ ] and this formula has a much lower entropy and applies to any length of the Fibonacci sequence.

Limitations of entropy in cryptography [edit]

In cryptanalysis, entropy is frequently roughly used as a measure of the unpredictability of a cryptographic key, though its existent uncertainty is unmeasurable. For case, a 128-bit cardinal that is uniformly and randomly generated has 128 bits of entropy. It also takes (on boilerplate) $2^{127}$ guesses to break by beast force. Entropy fails to capture the number of guesses required if the possible keys are not chosen uniformly.^[22] ^[23] Instead, a measure called guesswork can be used to mensurate the try required for a animate being forcefulness attack.^[24]

Other problems may arise from not-compatible distributions used in cryptography. For example, a 1,000,000-digit binary one-fourth dimension pad using sectional or. If the pad has one,000,000 bits of entropy, information technology is perfect. If the pad has 999,999 bits of entropy, evenly distributed (each private flake of the pad having 0.999999 $.25 of entropy) information technology may provide adept security. But if the pad has 999,999 $.25 of entropy, where the starting time chip is fixed and the remaining 999,999 $.25 are perfectly random, the outset bit of the ciphertext will not be encrypted at all.

Information every bit a Markov process [edit]

A mutual way to ascertain entropy for text is based on the Markov model of text. For an order-0 source (each graphic symbol is selected independent of the concluding characters), the binary entropy is:

\mathrm {H} ({\mathcal {S}})=-\sum p_{i}\log p_{i},

where $p i$ is the probability of $i$ . For a outset-order Markov source (one in which the probability of selecting a grapheme is dependent merely on the immediately preceding graphic symbol), the entropy rate is:

\mathrm {H} ({\mathcal {S}})=-\sum _{i}p_{i}\sum _{j}\ p_{i}(j)\log p_{i}(j),

^{[ citation needed ]}

where $i$ is a land (sure preceding characters) and $p_{i}(j)$ is the probability of $j$ given $i$ as the previous character.

For a second order Markov source, the entropy rate is

\mathrm {H} ({\mathcal {S}})=-\sum _{i}p_{i}\sum _{j}p_{i}(j)\sum _{k}p_{i,j}(thousand)\ \log \ p_{i,j}(thou).

Efficiency (normalized entropy) [edit]

A source alphabet with non-uniform distribution will have less entropy than if those symbols had uniform distribution (i.east. the "optimized alphabet"). This deficiency in entropy can be expressed equally a ratio called efficiency^{[ This quote needs a citation ]}:

\eta (Ten)={\frac {H}{H_{max}}}=-\sum _{i=1}^{north}{\frac {p(x_{i})\log _{b}(p(x_{i}))}{\log _{b}(n)}}

Applying the basic backdrop of the logarithm, this quantity tin as well exist expressed every bit:

\eta (X)=-\sum _{i=1}^{due north}{\frac {p(x_{i})\log _{b}(p(x_{i}))}{\log _{b}(n)}}=\sum _{i=1}^{n}{\frac {\log _{b}(p(x_{i})^{-p(x_{i})})}{\log _{b}(north)}}=\sum _{i=i}^{n}\log _{due north}(p(x_{i})^{-p(x_{i})})=\log _{north}(\prod _{i=one}^{northward}p(x_{i})^{-p(x_{i})})

Efficiency has utility in quantifying the effective use of a communication channel. This formulation is also referred to as the normalized entropy, as the entropy is divided by the maximum entropy ${\log _{b}(n)}$ . Furthermore, the efficiency is indifferent to choice of (positive) base of operations $b$ , as indicated by the insensitivity inside the terminal logarithm in a higher place thereto.

Entropy for continuous random variables [edit]

Differential entropy [edit]

The Shannon entropy is restricted to random variables taking detached values. The corresponding formula for a continuous random variable with probability density function $f (x)$ with finite or infinite support $\mathbb {Ten}$ on the real line is defined by analogy, using the higher up grade of the entropy equally an expectation:^[10] ^: 224

h[f]=\operatorname {Eastward} [-\ln(f(x))]=-\int _{\mathbb {10} }f(ten)\ln(f(x))\,dx.

This is the differential entropy (or continuous entropy). A precursor of the continuous entropy $h [f]$ is the expression for the functional $Η$ in the H-theorem of Boltzmann.

Although the analogy between both functions is suggestive, the following question must exist 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 tin even be negative – and corrections accept been suggested, notably limiting density of discrete points.

To answer this question, a connection must be established between the two functions:

In club to obtain a generally finite measure as the bin size goes to naught. In the discrete case, the bin size is the (implicit) width of each of the $north$ (finite or space) bins whose probabilities are denoted by $p n$ . Every bit the continuous domain is generalized, the width must be made explicit.

To exercise this, start with a continuous function $f$ discretized into bins of size $\Delta$ . Past the mean-value theorem there exists a value $10 i$ in each bin such that

f(x_{i})\Delta =\int _{i\Delta }^{(i+i)\Delta }f(ten)\,dx

the integral of the role $f$ tin be approximated (in the Riemannian sense) by

\int _{-\infty }^{\infty }f(10)\,dx=\lim _{\Delta \to 0}\sum _{i=-\infty }^{\infty }f(x_{i})\Delta

where this limit and "bin size goes to zero" are equivalent.

Nosotros will denote

\mathrm {H} ^{\Delta }:=-\sum _{i=-\infty }^{\infty }f(x_{i})\Delta \log \left(f(x_{i})\Delta \correct)

and expanding the logarithm, we have

\mathrm {H} ^{\Delta }=-\sum _{i=-\infty }^{\infty }f(x_{i})\Delta \log(f(x_{i}))-\sum _{i=-\infty }^{\infty }f(x_{i})\Delta \log(\Delta ).

As Δ → 0, we accept

{\begin{aligned}\sum _{i=-\infty }^{\infty }f(x_{i})\Delta &\to \int _{-\infty }^{\infty }f(10)\,dx=1\\\sum _{i=-\infty }^{\infty }f(x_{i})\Delta \log(f(x_{i}))&\to \int _{-\infty }^{\infty }f(x)\log f(ten)\,dx.\end{aligned}}

Note; $log(Δ) \to -\infty$ as $Δ \to 0$ , requires a special definition of the differential or continuous entropy:

h[f]=\lim _{\Delta \to 0}\left(\mathrm {H} ^{\Delta }+\log \Delta \right)=-\int _{-\infty }^{\infty }f(x)\log f(x)\,dx,

which is, every bit said before, referred to as the differential entropy. This means that the differential entropy is not a limit of the Shannon entropy for $n \to \infty$ . Rather, it differs from the limit of the Shannon entropy past an space offset (run across also the article on information dimension).

Limiting density of discrete points [edit]

It turns out as a upshot that, unlike the Shannon entropy, the differential entropy is not in general a good measure out of incertitude or information. For example, the differential entropy can be negative; also it is not invariant under continuous according transformations. This problem may be illustrated by a change of units when ten is a dimensioned variable. f(x) will then accept the units of 1/x. The argument of the logarithm must be dimensionless, otherwise it is improper, and so that the differential entropy as given in a higher place will be improper. If Δ is some "standard" value of x (i.e. "bin size") and therefore has the aforementioned units, then a modified differential entropy may exist written in proper form every bit:

H=\int _{-\infty }^{\infty }f(10)\log(f(x)\,\Delta )\,dx

and the result will be the same for whatever option of units for 10. In fact, the limit of detached entropy every bit $N\rightarrow \infty$ would also include a term of $\log(North)$ , which would in general be infinite. This is expected: continuous variables would typically accept space entropy when discretized. The limiting density of discrete points 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 [edit]

Another useful mensurate of entropy that works equally well in the discrete and the continuous case is the relative entropy of a distribution. Information technology is defined as the Kullback–Leibler difference from the distribution to a reference measure $g$ equally follows. Presume that a probability distribution $p$ is absolutely continuous with respect to a measure $m$ , i.e. is of the form $p (dx) = f (x) 1000 (dx)$ for some non-negative $m$ -integrable role $f$ with $grand$ -integral one, then the relative entropy tin exist defined every bit

D_{\mathrm {KL} }(p\|g)=\int \log(f(ten))p(dx)=\int f(x)\log(f(x))m(dx).

In this course the relative entropy generalizes (upward to modify in sign) both the discrete entropy, where the measure $thou$ is the counting measure, and the differential entropy, where the measure $g$ is the Lebesgue mensurate. If the measure $m$ is itself a probability distribution, the relative entropy is non-negative, and zero if $p = thousand$ equally measures. It is defined for whatsoever mensurate infinite, hence coordinate independent and invariant nether co-ordinate reparameterizations if one properly takes into business relationship the transformation of the measure $m$ . The relative entropy, and (implicitly) entropy and differential entropy, exercise depend on the "reference" measure $one thousand$ .

Use in combinatorics [edit]

Entropy has get a useful quantity in combinatorics.

Loomis–Whitney inequality [edit]

A simple case of this is an alternate proof of the Loomis–Whitney inequality: for every subset $A \subseteq Z d$ , nosotros have

|A|^{d-1}\leq \prod _{i=i}^{d}|P_{i}(A)|

where $P i$ is the orthogonal projection in the $i$ th coordinate:

P_{i}(A)=\{(x_{1},\ldots ,x_{i-1},x_{i+1},\ldots ,x_{d}):(x_{1},\ldots ,x_{d})\in A\}.

The proof follows as a simple corollary of Shearer's inequality: if $10 ane, ..., X d$ are random variables and $Due south 1, ..., S northward$ are subsets of ${1, ..., d$ } such that every integer between 1 and $d$ lies in exactly $r$ of these subsets, then

\mathrm {H} [(X_{1},\ldots ,X_{d})]\leq {\frac {1}{r}}\sum _{i=ane}^{due north}\mathrm {H} [(X_{j})_{j\in S_{i}}]

where $(X_{j})_{j\in S_{i}}$ is the Cartesian product of random variables $X j$ with indexes $j$ in $S i$ (so the dimension of this vector is equal to the size of $S i$ ).

We sketch how Loomis–Whitney follows from this: Indeed, allow $X$ be a uniformly distributed random variable with values in $A$ and so that each signal in $A$ occurs with equal probability. And then (by the further properties of entropy mentioned above) $Η(10) = log| A |$ , where $| A |$ denotes the cardinality of $A$ . Let $S i = {1, 2, ..., i -1, i +i, ..., d$ }. The range of $(X_{j})_{j\in S_{i}}$ is contained in $P i (A)$ and hence $\mathrm {H} [(X_{j})_{j\in S_{i}}]\leq \log |P_{i}(A)|$ . At present employ this to jump the right side of Shearer'south inequality and exponentiate the opposite sides of the resulting inequality you obtain.

Approximation to binomial coefficient [edit]

For integers $0 < k < n$ let $q = g / north$ . Then

{\frac {two^{n\mathrm {H} (q)}}{n+1}}\leq {\tbinom {northward}{thou}}\leq two^{northward\mathrm {H} (q)},

where

\mathrm {H} (q)=-q\log _{2}(q)-(i-q)\log _{2}(1-q).

^[25] ^: 43

Proof (sketch)

Note that

{\tbinom {due north}{k}}q^{qn}(1-q)^{northward-nq}

is 1 term of the expression

\sum _{i=0}^{n}{\tbinom {north}{i}}q^{i}(i-q)^{n-i}=(q+(ane-q))^{n}=ane.

Rearranging gives the upper bound. For the lower bound 1 first shows, using some algebra, that it is the largest term in the summation. Simply and then,

{\binom {n}{g}}q^{qn}(1-q)^{n-nq}\geq {\frac {1}{n+1}}

since there are $n + ane$ terms in the summation. Rearranging gives the lower bound.

A prissy estimation of this is that the number of binary strings of length $n$ with exactly $m$ many 1's is approximately $2^{n\mathrm {H} (one thousand/n)}$ .^[26]

Use in machine learning [edit]

Car 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 dubiousness.

Conclusion tree learning algorithms utilize relative entropy to determine the decision rules that govern the data at each node.^[27] The Data proceeds in determination copse $IG(Y,X)$ , which is equal to the difference between the entropy of $Y$ and the conditional entropy of $Y$ given $Ten$ , quantifies the expected information, or the reduction in entropy, from additionally knowing the value of an attribute $X$ . The data gain is used to place which attributes of the dataset provide the near information and should be used to dissever the nodes of the tree optimally.

Bayesian inference models often apply the Principle of maximum entropy to obtain Prior probability distributions.^[28] The idea is that the distribution that best represents the current state of knowledge of a organisation is the one with the largest entropy, and is therefore suitable to be the prior.

Nomenclature in machine learning performed by Logistic regression or Artificial neural networks frequently employs a standard loss function, called Cross entropy loss, that minimizes the average cantankerous entropy between footing truth and predicted distributions.^[29] In general, cross entropy is a measure of the differences between two datasets similar to the KL departure (or relative entropy).

References [edit]

^ Pathria, R. K.; Beale, Paul (2011). Statistical Mechanics (Third ed.). Academic Printing. p. 51. ISBN978-0123821881.
^ ^a ^b Shannon, Claude E. (July 1948). "A Mathematical Theory of Communication". Bell System Technical Journal. 27 (3): 379–423. doi:10.1002/j.1538-7305.1948.tb01338.10. hdl:10338.dmlcz/101429. (PDF, archived from here)
^ ^a ^b Shannon, Claude E. (Oct 1948). "A Mathematical Theory of Communication". Bong Organization Technical Journal. 27 (4): 623–656. doi:10.1002/j.1538-7305.1948.tb00917.10. hdl:11858/00-001M-0000-002C-4317-B. (PDF, archived from hither)
^ "Entropy (for information scientific discipline) Clearly Explained!!!". YouTube.
^ MacKay, David J.C. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge Academy Press. ISBN0-521-64298-one.
^ Schneier, B: Applied Cryptography, Second edition, John Wiley and Sons.
^ Borda, Monica (2011). Fundamentals in Information Theory and Coding. Springer. ISBN978-3-642-20346-6.
^ Han, Te Sun & Kobayashi, Kingo (2002). Mathematics of Information and Coding. American Mathematical Order. ISBN978-0-8218-4256-0. {{cite book}}: CS1 maint: uses authors parameter (link)
^ Schneider, T.D, Information theory primer with an appendix on logarithms, National Cancer Establish, 14 April 2007.
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^m Thomas M. Cover; Joy A. Thomas (1991). Elements of Information Theory. Hoboken, New Jersey: Wiley. ISBN978-0-471-24195-9.
^ Entropy in nLab
^ Carter, Tom (March 2014). An introduction to information theory and entropy (PDF). Santa Fe. Retrieved four August 2017.
^ Chakrabarti, C. Thou., and Indranil Chakrabarty. "Shannon entropy: evident label and application." International Journal of Mathematics and Mathematical Sciences 2005.17 (2005): 2847-2854 url
^ Compare: Boltzmann, Ludwig (1896, 1898). Vorlesungen über Gastheorie : 2 Volumes – Leipzig 1895/98 UB: O 5262-6. English version: Lectures on gas theory. Translated past Stephen K. Brush (1964) Berkeley: University of California Press; (1995) New York: Dover ISBN 0-486-68455-5
^ Życzkowski, Karol (2006). Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge Academy Press. p. 301.
^ Sharp, Kim; Matschinsky, Franz (2015). "Translation of Ludwig Boltzmann'southward Paper "On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium"". Entropy. 17: 1971–2009. doi:10.3390/e17041971.
^ Jaynes, E. T. (15 May 1957). "Data Theory and Statistical Mechanics". Physical Review. 106 (iv): 620–630. Bibcode:1957PhRv..106..620J. doi:x.1103/PhysRev.106.620.
^ Landauer, R. (July 1961). "Irreversibility and Oestrus Generation in the Computing Process". IBM Journal of Inquiry and Evolution. 5 (3): 183–191. doi:ten.1147/rd.53.0183. ISSN 0018-8646.
^ Marking Nelson (24 Baronial 2006). "The Hutter Prize". Retrieved 27 Nov 2008.
^ ^a ^b "The Earth'south Technological Capacity to Store, Communicate, and Compute Data", Martin Hilbert and Priscila López (2011), Scientific discipline, 332(6025); free access to the article through here: martinhilbert.net/WorldInfoCapacity.html
^ Spellerberg, Ian F.; Fedor, Peter J. (2003). "A tribute to Claude Shannon (1916–2001) and a plea for more rigorous apply of species richness, species diversity and the 'Shannon–Wiener' Index". Global Ecology and Biogeography. 12 (3): 177–179. doi:10.1046/j.1466-822X.2003.00015.x. ISSN 1466-8238.
^ Massey, James (1994). "Guessing and Entropy" (PDF). Proc. IEEE International Symposium on Information Theory . Retrieved 31 December 2013.
^ Malone, David; Sullivan, Wayne (2005). "Guesswork is not a Substitute for Entropy" (PDF). Proceedings of the Information technology & Telecommunications Conference . Retrieved 31 December 2013.
^ Pliam, John (1999). "Guesswork and variation distance as measures of cipher security". International Workshop on Selected Areas in Cryptography. doi:10.1007/three-540-46513-8_5.
^ Aoki, New Approaches to Macroeconomic Modeling.
^ Probability and Computing, M. Mitzenmacher and E. Upfal, Cambridge University Press
^ Batra, Mridula; Agrawal, Rashmi (2018). Panigrahi, Bijaya Ketan; Hoda, Chiliad. N.; Sharma, Vinod; Goel, Shivendra (eds.). "Comparative Assay of Decision Tree Algorithms". Nature Inspired Computing. Advances in Intelligent Systems and Calculating. Singapore: Springer. 652: 31–36. doi:10.1007/978-981-10-6747-1_4. ISBN978-981-10-6747-i.
^ Jaynes, Edwin T. (September 1968). "Prior Probabilities". IEEE Transactions on Systems Science and Cybernetics. 4 (3): 227–241. doi:ten.1109/TSSC.1968.300117. ISSN 2168-2887.
^ Rubinstein, Reuven Y.; Kroese, Dirk P. (nine March 2013). The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning. Springer Scientific discipline & Business organisation Media. ISBN978-1-4757-4321-0.

This article incorporates material from Shannon's entropy on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

External links [edit]

"Entropy", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
"Entropy" at Rosetta Lawmaking—repository of implementations of Shannon entropy in different programming languages.
Entropy an interdisciplinary journal on all aspects of the entropy concept. Open access.

northcutttheavalogy.blogspot.com

Source: https://en.wikipedia.org/wiki/Entropy_(information_theory)