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In Information Theory , the Shannon entropy or '''information entropy''' is a measure of the uncertainty associated with a Random Variable .

Entropy quantifies information in a piece of data. of any communication: the shortest average number of bits that can be sent to communicate one message out of all the possibilities is the Shannon entropy.

Equivalently, the Shannon entropy is a measure of the average Information Content the recipient is ''missing'' when they do ''not'' know the value of the random variable.

The concept was introduced by Claude E. Shannon in his 1948 paper " A Mathematical Theory Of Communication ".

DEFINITION


The information entropy of a Discrete Random Variable ''X'', that can take on possible values {''x''1...''x''n} is

:
\begin{matrix}
H(X) = \operatorname{E}( I(X) ) & = & \displaystyle{\sum_{i=1}^np(x_i)\log_2 \left(1/p(x_i) ight)} \
& = & - \displaystyle{\sum_{i=1}^np(x_i)\log_2 p(x_i)} \qquad
\end{matrix}


where

I

p



Characterization


Information entropy is Characterised by these desiderata:

(Define p_i=\Pr(X=x_i) and H_n(p_1,\ldots,p_n)=H(X))

;Continuity
:The measure should be Continuous — i.e., changing the value of one of the probabilities by a very small amount should only change the entropy by a small amount.

;Symmetry
:The measure should be unchanged if the outcomes ''x''''i'' are re-ordered.
:
H_n\left(p_1, p_2, \ldots ight) = H_n\left(p_2, p_1, \ldots ight)
etc.

;Maximum
:If all the outcomes are equally likely, then entropy should be maximal. In this case, the entropy increases with the number of outcomes.
:
H_n(p_1,\ldots,p_n) \le H_n\left( rac{1}{n}, \ldots, rac{1}{n} ight) < H_{n+1}\left( rac{1}{n+1}, \ldots, rac{1}{n+1} ight).


;Additivity
:The amount of entropy should be the same independently of how the process is regarded as being divided into parts.
:This last functional relationship characterizes the entropy of a system with sub-systems. It demands that the entropy of a system can be calculated from the entropy of its sub-systems if we know how the sub-systems interact with each other.

:Assume that we have an ensemble of ''n'' elements with a uniform distribution on them. If we mentally divide this ensemble into ''k'' boxes (sub-systems) with ''bi'' elements in each, the entropy can be calculated as a sum of individual entropies of the boxes weighed by the probability of finding oneself in that particular box PLUS the entropy of the system of boxes.

:For Positive Integers ''bi'' where ''b''1 + … + ''bk'' = ''n'',
:
H_n\left( rac{1}{n}, \ldots, rac{1}{n} ight) = H_k\left( rac{b_1}{n}, \ldots, rac{b_k}{n} ight) + \sum_{i=1}^k rac{b_i}{n} H_{b_i}\left( rac{1}{b_i}, \ldots, rac{1}{b_i} ight).


:This implies that the entropy of a certain outcome is zero:
:
H_1(1.0) = 0\,


Any definition of entropy satisfying these assumptions has the form:
::-K\sum_{i=1}^np(x_i)\log p(x_i)\,\!
where ''K'' is a constant corresponding to a choice of measurement units.


EXAMPLE


See Also: binary entropy function


Consider tossing a coin which may or may not be fair.

The entropy of the unknown result of the next toss of the coin is maximised 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 a full 1 Bit of information.

However, if we know the coin is not fair, there is less uncertainty. Every time, 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 a full 1 bit of information.

The extreme case is that of a double-headed coin which never comes up tails. Then there is no uncertainty. The entropy is zero: each toss of the coin delivers no information.


FURTHER PROPERTIES

The Shannon entropy satisfies the following properties:

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

    • :H_{n+1}(p_1,\ldots,p_n,0) = H_n(p_1,\ldots,p_n).




ASPECTS



Relationship to thermodynamic entropy

See Also: Entropy in thermodynamics and information theory


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 Thermodynamics .

In Statistical Thermodynamics the most general formula for the Thermodynamic Entropy ''S'' of a Thermodynamic System is the Gibbs Entropy ,
:S = - k_B \sum p_i \ln p_i \,
defined by J. Willard Gibbs in 1878 after earlier work by Boltzmann ( 1872 ).Compare: Boltzmann, Ludwig (1896, 1898). Vorlesungen über Gastheorie : 2 Volumes - Leipzig 1895/98 UB: O 5262-6. English version: Lectures on gas theory. Translated by Stephen G. Brush (1964) Berkeley: University of California Press; (1995) New York: Dover ISBN 0-486-68455-5

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 ,
:S = - k_B \,\,{ m Tr}( ho \ln ho) \,
where ρ is the Density Matrix of the quantum mechanical system.

At an everyday practical level the links between information entropy and thermodynamic entropy are not close. 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 , rather than an unchanging probability distribution. And, as the numerical smallness of Boltzmann's Constant ''k''''B'' indicates, the changes in ''S''/''k''B for even minute amounts of substances in chemical and physical processes represent amounts of entropy which are large right off the scale compared to anything seen in data compression or signal processing.

But, at a more philosophical level, connections ''can'' be made between thermodynamic and informational entropy, although it took many years in the development of the theories of statistical mechanics and information theory to make the relationship fully apparent. In fact, in the view of ''). Maxwell's Demon (hypothetically) reduces the thermodynamic entropy of a system using information about the states of individual molecules; but, as Landauer (from 1961 ) and co-workers have shown, the demon himself must increase his own thermodynamic entropy in the process, by at least the amount of Shannon information he proposes to first acquire and store; and so the total entropy does not decrease (which resolves the paradox).


Entropy as information content


See Also: Shannon's source coding theorem


Entropy is defined in the context of a probabilistic model. Independent fair coin flips have an entropy of 1 bit per flip. A source that always generates a long string of A's has an entropy of 0, since the next character will always be an 'A'.

The entropy rate of a data source means the average number of can achieve a compression ratio of 1.5 bits per character.

From the preceding example, note the following points:

# The amount of entropy is not always an integer number of bits.
# Many data bits may not convey information. For example, data structures often store information redundantly, or have identical sections regardless of the information in the data structure.

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. The formula can be derived by calculating the mathematical expectation of the ''amount of information'' contained in a digit from the information source. ''See also'' Shannon-Hartley Theorem .

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. See Markov Chain .


Data compression

Entropy effectively bounds the performance of the strongest lossless (or nearly lossless) compression possible, which can be realized in theory by using the Typical Set or in practice using Huffman , Lempel-Ziv or Arithmetic Coding . The performance of existing data compression algorithms is often used as a rough estimate of the entropy of a block of data.


Limitations of entropy as information content

Although entropy is often used as a characterization of the information content of a data source, this information content is not absolute: it depends crucially on the probabilistic model. A source that always generates the same symbol has an entropy of 0, but the definition of what a symbol is depends on the alphabet. Consider a source that produces the string ABABABABAB... in which A is always followed by B and vice versa. If the probabilistic model considers individual letters as Independent , the entropy rate of the sequence is 1 bit per character. But if the sequence is considered as "AB AB AB AB AB..." with symbols as two-character blocks, then the entropy rate is 0 bits per character.

However, if we use very large blocks, then the estimate of per-character entropy rate may become artificially low. This is because in reality, the probability distribution of the sequence is not knowable exactly; it is only an estimate. For example, suppose one considers the text of every book ever published as a sequence, with each symbol being the text of a complete book. If there are ''N'' published books, and each book is only published once, the estimate of the probability of each book is 1/''N'', and the entropy (in bits) is -log2 1/''N''. As a practical code, this corresponds to assigning each book a 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 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 one such program, but it may not be the shortest.


Data as a Markov process

A common way to define entropy for text is based on the Markov Model of text. For an order-0 source (each character is selected independent of the last characters), the binary entropy is:

:H(\mathcal{S}) = - \sum p_i \log_2 p_i, \,\!

where ''p''''i'' is the probability of ''i''. 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 is:

:H(\mathcal{S}) = - \sum_i p_i \sum_j \ p_i (j) \log_2 p_i (j), \,\!

where ''i'' is a state (certain preceding characters) and p_i(j) is the probability of j given i as the previous character (s).

For a second order Markov source, the entropy rate is

:H(\mathcal{S}) = -\sum_i p_i \sum_j p_i(j) \sum_k p_{i,j}(k)\ \log_2 \ p_{i,j}(k). \,\!


''b''-ary entropy

In general the ''b''-ary entropy of a source \mathcal{S} = (''S'',''P'') with Source Alphabet ''S'' = {''a''1, …, ''an''} and Discrete Probability Distribution ''P'' = {''p''1, …, ''pn''} where ''pi'' is the probability of ''ai'' (say ''pi'' = ''p''(''ai'')) is defined by:

: H_b(\mathcal{S}) = - \sum_{i=1}^n p_i \log_b p_i, \,\!

Note: the ''b'' in "''b''-ary entropy" is the number of different symbols of the "ideal alphabet" which is being used as the standard yardstick to measure source alphabets. In information theory, two symbols are Necessary And Sufficient for an alphabet to be able to encode information, therefore the default is to let ''b'' = 2 ("binary entropy"). Thus, the entropy of the source alphabet, with its given empiric probability distribution, is a number equal to the number (possibly fractional) of symbols of the "ideal alphabet", with an optimal probability distribution, necessary to encode for each symbol of the source alphabet. Also note that "optimal probability distribution" here means a Uniform Distribution : a source alphabet with ''n'' symbols has the highest possible entropy (for an alphabet with ''n'' symbols) when the probability distribution of the alphabet is uniform. This optimal entropy turns out to be \log_b \, n .


EFFICIENCY

A source alphabet encountered in practice should be found to have a probability distribution which is less than optimal. If the source alphabet has ''n'' symbols, then it can be compared to an "optimized alphabet" with ''n'' symbols, whose probability distribution is uniform. The ratio of the entropy of the source alphabet with the entropy of its optimized version is the efficiency of the source alphabet, which can be expressed as a Percentage .

This implies that the efficiency of a source alphabet with ''n'' symbols can be defined simply as being equal to its ''n''-ary entropy. See also Redundancy (information Theory) .


EXTENDING DISCRETE ENTROPY TO THE CONTINUOUS CASE: DIFFERENTIAL ENTROPY


The Shannon entropy is restricted random variables taking discrete values. The formula

  • )


  • ) is usually referred to as the continuous entropy, or Differential Entropy . Although the analogy between both functions is suggestive, the following question must be set: is the Boltzmann entropy a valid extension of the Shannon entropy? To answer this question, we must establish a connection between the two functions:


We wish 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 ''n'' (finite or infinite) bins whose probabilities are denoted by ''pn''. As we generalize to the continuous domain, we must make this width explicit.

To do this, start with a continuous function ''f'' discretized as shown in the figure.

As the figure indicates, by the mean-value theorem there exists a value ''xi'' in each bin such that

:f(x_i) \Delta = \int_{i\Delta}^{(i+1)\Delta} f(x)\, dx

and thus the integral of the function ''f'' can be approximated (in the Riemannian sense) by

:\int_{-\infty}^{\infty} f(x)\, dx = \lim_{\Delta o 0} \sum_{i = -\infty}^{\infty} f(x_i) \Delta

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

We will denote

:H^{\Delta} :=- \sum_{i=-\infty}^{\infty} \Delta f(x_i) \log \Delta f(x_i)

and expanding the logarithm, we have

:H^{\Delta} = - \sum_{i=-\infty}^{\infty} \Delta f(x_i) \log \Delta f(x_i)

: = - \sum_{i=-\infty}^{\infty} \Delta f(x_i) \log f(x_i) -\sum_{i=-\infty}^{\infty} f(x_i) \Delta \log \Delta.

As \Delta o 0, we have

:\sum_{i=-\infty}^{\infty} f(x_i) \Delta o \int f(x)\, dx = 1

and so

:\sum_{i=-\infty}^{\infty} \Delta f(x_i) \log f(x_i) o \int f(x) \log f(x)\, dx.

But note that \log \Delta o -\infty as \Delta o 0, therefore we need a special definition of the differential or continuous entropy:

:h = \lim_{\Delta o 0} \left[H^{\Delta} + \log \Delta ight = -\int_{-\infty}^{\infty} f(x) \log f(x)\,dx,

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 ''n'' → ∞

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.

More useful for the continuous case is the relative entropy of a distribution, defined as the Kullback-Leibler Divergence from the distribution to a reference measure ''m''(''x''),