Entropy (information theory)

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This article is about the Shannon entropy in information theory. For other uses, see Entropy (disambiguation).

In information theory, entropy (sometimes known as self-information) is a measure of the uncertainty associated with a random variable. The term by itself in this context usually refers to the Shannon entropy, which quantifies, in the sense of an expected value, the information contained in a message, usually in units such as bits. Equivalently, the Shannon entropy is a measure of the average information content one is missing when one does 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".

Shannon's entropy represents an absolute limit on the best possible lossless compression of any communication: treating messages to be encoded as a sequence of independent and identically-distributed random variables, Shannon's source coding theorem shows that, in the limit, the average length of the shortest possible representation to encode the messages in a given alphabet is their entropy divided by the logarithm of the number of symbols in the target alphabet.

A fair coin has an entropy of one bit. However, if the coin is not fair, then the uncertainty is lower (if asked to bet on the next outcome, we would bet preferentially on the most frequent result), and thus the Shannon entropy is lower. Mathematically, a coin flip is an example of a Bernoulli trial, and its entropy is given by the binary entropy function. A long string of repeating characters has an entropy rate of 0, since every character is predictable. The entropy rate of English text is between 1.0 and 1.5 bits per letter,^[1] or as low as 0.6 to 1.3 bits per letter, according to estimates by Shannon based on human experiments.^[2]

[edit] Definition

The information entropy of a discrete random variable X with possible values {x₁, …, x_n} is

$H(X) = \operatorname{E}(I(X)).$

Here E is the expected value function, and I(X) is the information content or self-information of X.

I(X) is itself a random variable. If p denotes the probability mass function of X then the information entropy can explicitly be written as

$H(X) = \sum_{i=1}^n {p(x_i)\,I(x_i)} = -\sum_{i=1}^n {p(x_i) \log_b p(x_i)},$

where b is the base of the logarithm used. Common values of b are 2, Euler's number e, and 10, and the unit of the information entropy is bit for b = 2, nat for b = e, and dit (or digit) for b = 10.^[3]

In the case of p_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

$\lim_{p\to0}p\log p = 0.$

[edit] Characterization

Information entropy is characterized by a small number of criteria, listed below. It can be shown that any definition of entropy satisfying these assumptions has the form

$-K\sum_{i=1}^np_i\log p_i\,\!$

where K is a constant corresponding to a choice of measurement units.

In the following, $p_i=\Pr(X=x_i)$ and $H_n(p_1,\ldots,p_n)=H(X)$ .

[edit] Continuity

The measure should be continuous, so that changing the value of one of the probabilities by a very small amount should only change the entropy by a small amount.

[edit] Symmetry

The measure should be unchanged if the outcomes x_i are re-ordered.

$H_n\left(p_1, p_2, \ldots \right) = H_n\left(p_2, p_1, \ldots \right)$ etc.

[edit] Maximum

The measure should be maximal if all the outcomes are equally likely (uncertainty is highest when all possible events are equiprobable).

$H_n(p_1,\ldots,p_n) \le H_n\left(\frac{1}{n}, \ldots, \frac{1}{n}\right).$

For equiprobable events the entropy should increase with the number of outcomes.

$H_n\bigg(\underbrace{\frac{1}{n}, \ldots, \frac{1}{n}}_{n}\bigg) < H_{n+1}\bigg(\underbrace{\frac{1}{n+1}, \ldots, \frac{1}{n+1}}_{n+1}\bigg).$

[edit] Additivity

The amount of entropy should be independent 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 entropies of its sub-systems if the interactions between the sub-systems are known.

Given an ensemble of n uniformly distributed elements that are divided into k boxes (sub-systems) with b₁, b₂, … , b_k 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.

For positive integers b_i where b₁ + … + b_k = n,

$H_n\left(\frac{1}{n}, \ldots, \frac{1}{n}\right) = H_k\left(\frac{b_1}{n}, \ldots, \frac{b_k}{n}\right) + \sum_{i=1}^k \frac{b_i}{n} \, H_{b_i}\left(\frac{1}{b_i}, \ldots, \frac{1}{b_i}\right).$

Choosing k = n, b₁ = … = b_n = 1 this implies that the entropy of a certain outcome is zero:

$H_1\left(1\right) = 0\,$

[edit] Information entropy explained

For a random variable $X$ with $n$ outcomes $\{ x_i : i = 1, \cdots , n\}$ , the Shannon information entropy, a measure of uncertainty (see further below) and denoted by $H (X)$ , is defined as

$\displaystyle H(X) \equiv - \sum_{i=1}^np(x_i)\log_b p(x_i)$

(1)

where $p (x i)$ is the probability mass function of outcome $x i$ .

To understand the meaning of Eq. (1), let's first consider a set of $n$ possible outcomes (events) $\left\{ x_i : i = 1 , \cdots , n \right\}$ , with equal probability $p (x i) = 1 / n$ . An example would be a fair die with $n$ values, from $1$ to $n$ . The uncertainty for such set of $n$ outcomes is defined by

$\displaystyle u = \log_b (n).$

(2)

The logarithm is used so to provide the additivity characteristic for independent uncertainty. For example, consider appending to each value of the first die the value of a second die, which has $m$ possible outcomes $\left\{ y_j : j = 1 , \cdots , m \right\}$ . There are thus $m n$ possible outcomes $\left\{ x_i y_j : i = 1 , \cdots , n , j = 1 , \cdots , m \right\}$ . The uncertainty for such set of $m n$ outcomes is then

$\displaystyle u = \log_b (nm) = \log_b (n) + \log_b (m).$

(3)

Thus the uncertainty of playing with two dice is obtained by adding the uncertainty of the second die $log b (m)$ to the uncertainty of the first die $log b (n)$ .

Now return to the case of playing with one die only (the first one). Since the probability of each event is $1 / n$ , we can write

$\displaystyle u = \log_b \left(\frac{1}{p(x_i)}\right) = - \log_b (p(x_i)) \ \forall i = 1, \cdots , n.$

In the case of a non-uniform probability mass function (or density in the case of continuous random variables), we let

$\displaystyle u_i := - \log_b (p(x_i))$

(4)

which is also called a surprisal; the lower the probability $p (x i)$ , i.e. $p(x_i) \rightarrow 0$ , the higher the uncertainty or the surprise, i.e. $u_i \rightarrow \infty$ , for the outcome $x i$ .

The average uncertainty $\langle u \rangle$ , with $\langle \cdot \rangle$ being the average operator, is obtained by

$\displaystyle \langle u \rangle = \sum_{i=1}^{n} p(x_i) u_i = - \sum_{i=1}^{n} p(x_i) \log_b (p(x_i))$

(5)

and is used as the definition of the information entropy $H (X)$ in Eq. (1). The above also explained why information entropy and information uncertainty can be used interchangeably.^[4]

[edit] Example

Entropy of a Coin toss as a function of the probability of it coming up heads

Main article: Binary entropy function

Consider tossing a coin with known, not necessarily fair, probabilities of coming up heads or tails.

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, but comes up heads or tails with probabilities p and q, then 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.

[edit] Further properties

The Shannon entropy satisfies the following properties:

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

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

It can be confirmed using the Jensen inequality that

$H(X) = \operatorname{E}\left[\log_2 \left( \frac{1}{p(X)}\right) \right] \leq \log_2 \left[ \operatorname{E}\left( \frac{1}{p(X)} \right) \right] = \log_2(n)$ .

This maximal entropy of $log 2 (n)$ is effectively attained by a source alphabet having a uniform probability distribution: uncertainty is maximal when all possible events are equiprobable.

[edit] Aspects

[edit] Relationship to thermodynamic entropy

Main article: 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,

$S = - k_B \sum p_i \ln p_i \,$

where k_B is the Boltzmann constant, defined by J. Willard Gibbs in 1878 after earlier work by Boltzmann (1872).^[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,

$S = - k_B \,\,{\rm Tr}(\rho \ln \rho) \,$

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 so large as to be 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 Jaynes (1957), thermodynamics should be seen as an application of Shannon's information theory: the thermodynamic entropy is interpreted as being an estimate of 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. For example, adding heat to a system increases its thermodynamic entropy because it increases the number of possible microscopic states that it could be in, thus making any complete state description longer. (See article: maximum entropy thermodynamics). Maxwell's demon can (hypothetically) reduce the thermodynamic entropy of a system by using information about the states of individual molecules; but, as Landauer (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 entropy does not decrease (which resolves the paradox).

[edit] Entropy as information content

Main article: 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 bits per symbol needed to encode it. Shannon's experiments with human predictors show an information rate of between 0.6 and 1.3 bits per character,^[6] depending on the experimental setup; the PPM compression algorithm 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.

[edit] 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.^[7]^[8]

[edit] Limitations of entropy as information content

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 information entropy of a given probability distribution of messages or symbols, and
the entropy rate of a stochastic process.

(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.) Other quantities of information are also used to compare or relate different sources of information.

It is important not to confuse the above concepts. Oftentimes it is only clear from context which one is meant. For example, when someone says that the "entropy" of the English language is about 1.5 bits per character, he/she is actually modelling the English language as a stochastic process and talking about its entropy rate.

It is also important to note that, because all these concepts are defined in terms of the mathematics of probability distributions, none of them can be considered a measure of the philosophical significance, importance to society, or emotional impact of information.

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 rate 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 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 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 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.

For example, 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 log₂(n). So the first 128 symbols of the Fibonacci sequence has an entropy of approximately 7 bits/symbol. However, the sequence can be expressed using a formula [F(n) = F(n-1) + F(n-2) for n={3,4,5,...}, F(1)=1, F(2)=1] and this formula has a much lower entropy and applies to any length of the Fibonacci sequence.

[edit] 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.

For a second order Markov source, the entropy rate is

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

[edit] b-ary entropy

In general the b-ary entropy of a source $\mathcal{S}$ = (S,P) with source alphabet S = {a₁, …, a_n} and discrete probability distribution P = {p₁, …, p_n} where p_i is the probability of a_i (say p_i = p(a_i)) 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 $\log_b \, n$ .

[edit] 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).

[edit] Extending discrete entropy to the continuous case: differential entropy

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

$h[f] = -\int\limits_{-\infty}^{\infty} f(x) \log f(x)\, dx,\quad (*)$

where f denotes a probability density function on the real line, is analogous to the Shannon entropy and could thus be viewed as an extension of the Shannon entropy to the domain of real numbers.

A precursor of the continuous information entropy $h [f]$ given in (*) is the expression for the functional $H$ in the H-theorem of Boltzmann.

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 differential entropy a valid extension of the Shannon discrete 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 p_n. 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 x_i 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

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

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

$\begin{align} H^{\Delta} &= - \sum_{i=-\infty}^{\infty} \Delta f(x_i) \log \Delta f(x_i) \\ &= - \sum_{i=-\infty}^{\infty} \Delta f(x_i) \log f(x_i) -\sum_{i=-\infty}^{\infty} f(x_i) \Delta \log \Delta. \end{align}$

As $\Delta \to 0$ , we have

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

and so

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

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

$h[f] = \lim_{\Delta \to 0} \left[H^{\Delta} + \log \Delta\right] = -\int_{-\infty}^{\infty} f(x) \log f(x)\,dx,$

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 $n \to \infty$ .

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.

Another useful measure of entropy 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),

$D_{\mathrm{KL}}(f(x)\|m(x)) = \int f(x)\log\frac{f(x)}{m(x)}\,dx.$

The relative entropy carries over directly from discrete to continuous distributions, and is invariant under co-ordinate reparametrisations.

[edit] References

^ Schneier, B: Applied Cryptography, Second edition, page 234. John Wiley and Sons.
^ Shannon, Claude E.: Prediction and entropy of printed English, The Bell System Technical Journal, 30:50-64, 1950.
^ Schneider, T.D, Information theory primer with an appendix on logarithms, National Cancer Institute, 14 Apr 2007.
^ Jaynes, E.T. (May 1957). "Information Theory and Statistical Mechanics". Physical Review Vol. 106 (No. 4): pp. 620–630. http://prola.aps.org/pdf/PR/v106/i4/p620_1.
^ 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
^ Mark Nelson (2006-08-24). "The Hutter Prize". Retrieved on 2008-11-27.
^ T. Schürmann and P. Grassberger, Entropy Estimation of Symbol Sequences, CHAOS,Vol. 6, No. 3 (1996) 414-427
^ T. Schürmann, Bias Analysis in Entropy Estimation J. Phys. A: Math. Gen. 37 (2004) L295-L301.

This article incorporates material from Shannon's entropy on PlanetMath, which is licensed under the GFDL.

[edit] See also

Binary entropy function – the entropy of a Bernoulli trial with probability of success p
Conditional entropy
Differential entropy
Entropy Estimation
Entropy rate
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 encoding - a coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols.
Hamming distance
Joint entropy – is the measure how much entropy is contained in a joint system of two random variables.
Kolmogorov-Sinai entropy in dynamical systems
Levenshtein distance
Limiting density of discrete points - encompassing discrete distibutions
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 generalisation of information entropy; it is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system.
Theil index

[edit] External links

Information is not entropy, information is not uncertainty ! - a discussion of the use of the terms "information" and "entropy".
I'm Confused: How Could Information Equal Entropy? - a similar discussion on the bionet.info-theory FAQ.
Description of information entropy from "Tools for Thought" by Howard Rheingold
A java applet representing Shannon's Experiment to Calculate the Entropy of English
Image Segmentation using 2D Multistage Entropy
Slides on information gain and entropy
CertainKey's password strength meter, which includes information entropy (in bits) as a mathematical criterion of sufficient randomness

[0] Schneier, B: Applied Cryptography, Second edition, page 234. John Wiley and Sons.

[1] Shannon, Claude E.: Prediction and entropy of printed English, The Bell System Technical Journal, 30:50-64, 1950.

[2] Schneider, T.D, Information theory primer with an appendix on logarithms, National Cancer Institute, 14 Apr 2007.

[3] Jaynes, E.T. (May 1957). "Information Theory and Statistical Mechanics". Physical Review Vol. 106 (No. 4): pp. 620–630. http://prola.aps.org/pdf/PR/v106/i4/p620_1.

[4] 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

[5] Mark Nelson (2006-08-24). "The Hutter Prize". Retrieved on 2008-11-27.

[6] T. Schürmann and P. Grassberger, Entropy Estimation of Symbol Sequences, CHAOS,Vol. 6, No. 3 (1996) 414-427

[7] T. Schürmann, Bias Analysis in Entropy Estimation J. Phys. A: Math. Gen. 37 (2004) L295-L301.

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