# Entropy and Information

Entropy and Information in the Universe

When these probabilities are substituted into the above expression for the Gibbs entropy or equivalently k B times the Shannon entropy , Boltzmann's equation results.

In information theoretic terms, the information entropy of a system is the amount of "missing" information needed to determine a microstate, given the macrostate. In the view of Jaynes , thermodynamic entropy, as explained by statistical mechanics , should be seen as an application of Shannon's information theory: the thermodynamic entropy is interpreted as being proportional to the amount of further Shannon information needed to define the detailed microscopic state of the system, that remains uncommunicated by a description solely in terms of the macroscopic variables of classical thermodynamics, with the constant of proportionality being just the Boltzmann constant.

Adding heat to a system increases its thermodynamic entropy because it increases the number of possible microscopic states of the system that are consistent with the measurable values of its macroscopic variables, making any complete state description longer.

See article: maximum entropy thermodynamics. 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 and co-workers have shown, to function the demon himself must increase thermodynamic entropy in the process, by at least the amount of Shannon information he proposes to first acquire and store; and so the total thermodynamic entropy does not decrease which resolves the paradox.

Landauer's principle imposes a lower bound on the amount of heat a computer must generate to process a given amount of information, though modern computers are far less efficient. 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 B's has an entropy of 0, since the next character will always be a 'B'.

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 between 0. 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 see caveat below in italics. 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.

Entropy is one of several ways to measure diversity. Specifically, Shannon entropy is the logarithm of 1 D , the true diversity index with parameter equal to 1. Entropy effectively bounds the performance of the strongest lossless compression possible, which can be realized in theory by using the typical set or in practice using Huffman , Lempel—Ziv or arithmetic coding.

See also Kolmogorov complexity. In practice, compression algorithms deliberately include some judicious redundancy in the form of checksums to protect against errors.

A Short Introduction to Entropy, Cross-Entropy and KL-Divergence

A study in Science estimates the world's technological capacity to store and communicate optimally compressed information normalized on the most effective compression algorithms available in the year , therefore estimating the entropy of the technologically available sources. The authors estimate humankind technological capacity to store information fully entropically compressed in and again in They break the information into three categories—to store information on a medium, to receive information through a one-way broadcast networks, or to exchange information through two-way telecommunication networks.

There are a number of entropy-related concepts that mathematically quantify information content in some way:. 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. Often it is only clear from context which one is meant.

For example, when someone says that the "entropy" of the English language is about 1 bit per character, they are actually modeling the English language as a stochastic process and talking about its entropy rate. Shannon himself used the term in this way. If very large blocks were used, the estimate of per-character entropy rate may become artificially low.

If one considers the text of every book ever published as a sequence, with each symbol being the text of a complete book.

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Information entropy is the average rate at which information is produced by a stochastic source of data. The measure of information entropy associated with each. A layman's introduction to information theory. Shannon had a mathematical formula for the 'entropy' of a probability distribution, which outputs the minimum.

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.

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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.

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, In cryptanalysis , entropy is often roughly used as a measure of the unpredictability of a cryptographic key, though its real uncertainty is unmeasurable. An example would be a bit key which is uniformly and randomly generated has bits of entropy. Entropy fails to capture the number of guesses required if the possible keys are not chosen uniformly. Other problems may arise from non-uniform distributions used in cryptography. For example, a 1,,digit binary one-time pad using exclusive or. If the pad has 1,, bits of entropy, it is perfect.

If the pad has , bits of entropy, evenly distributed each individual bit of the pad having 0. But if the pad has , bits of entropy, where the first bit is fixed and the remaining , bits are perfectly random, the first bit of the ciphertext will not be encrypted at all. 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:. 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:.

Note: the b in " b -ary entropy" is the number of different symbols of the ideal alphabet used as a standard yardstick to measure source alphabets. In information theory, two symbols are necessary and sufficient for an alphabet to encode information. 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: "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. A source alphabet with non-uniform distribution will have less entropy than if those symbols had uniform distribution i.

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This deficiency in entropy can be expressed as a ratio called efficiency [ This quote needs a citation ] :. Efficiency has utility in quantifying the effective use of a communication channel. Furthermore, the efficiency is indifferent to choice of positive base b , as indicated by the insensitivity within the final logarithm above thereto. Shannon entropy is characterized by a small number of criteria, listed below. Any definition of entropy satisfying these assumptions has the form.

The measure should be continuous , so that changing the values of the probabilities by a very small amount should only change the entropy by a small amount. The measure should be unchanged if the outcomes x i are re-ordered. The measure should be maximal if all the outcomes are equally likely uncertainty is highest when all possible events are equiprobable. For continuous random variables, the multivariate Gaussian is the distribution with maximum differential entropy.

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 b 1 , 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. The Shannon entropy satisfies the following properties, for some of which it is useful to interpret entropy as the amount of information learned or uncertainty eliminated by revealing the value of a random variable X :. The Shannon entropy is restricted to random variables taking discrete values. This 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?

In order to obtain a generally finite measure as the bin size goes to zero. In the discrete case, the bin size is the implicit width of each of the n finite or infinite bins whose probabilities are denoted by p n. As the continuous domain is generalised, the width must be made explicit. By the mean-value theorem there exists a value x i in each bin such that. Rather, it differs from the limit of the Shannon entropy by an infinite offset see also the article on information dimension.

It turns out as a result that, unlike the Shannon entropy, the differential entropy is not in general a good measure of uncertainty or information. For example, the differential entropy can be negative; also it is not invariant under continuous co-ordinate transformations. This problem may be illustrated by a change of units when x is a dimensioned variable.

The argument of the logarithm must be dimensionless, otherwise it is improper, so that the differential entropy as given above will be improper. This is expected, continuous variables would typically have infinite 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.

Another useful measure of entropy that works equally well in the discrete and the continuous case is the relative entropy of a distribution. It is defined as the Kullback—Leibler divergence from the distribution to a reference measure m as follows.