A First Course in Information Theory is an up-to-date introduction to information Shannon's information measures refer to entropy, conditional entropy, mutual. This book presents a succinct and mathematically rigorous treatment of the main pillars of Shannon's information theory, discussing the fundamental. provides the first comprehensive treatment of the theory of I-Measure, network coding theory, Shannon and non-Shannon type information inequalities, and a.
Entropie (Informationstheorie)Originally developed by Claude Shannon in the s, information theory laid the foundations for the digital revolution, and is now an essential tool in. A First Course in Information Theory is an up-to-date introduction to information Shannon's information measures refer to entropy, conditional entropy, mutual. In diesem Jahr veröffentlichte Shannon seine fundamentale Arbeit A Mathematical Theory of Communication und prägte damit die moderne Informationstheorie.
Shannon Information Theory Post navigation VideoInformation entropy - Journey into information theory - Computer Science - Khan Academy
A simple text is more like a quick statement, question, or request. These differences in communication style is what has made communication better through digital coding.
Instead of trying to figure out all of the variables in a communication effort like Morse Code, the 0s and 1s of digital coding allow for long strings of digits to be sent without the same levels of informational entropy.
A 0, for example, can be represented by a specific low-voltage signal. A 1 could then be represented by a high voltage signal. Because there are just two digits and each has a very specific state that can be recognized, even after the signal has experienced extensive entropy, it becomes possible to reconstruct the information with greater accuracy.
Using the information theory, a base 2 is used for the mathematical logarithms so that we can obtain total informational content.
In the instance of a coin flip, the value received is one bit. The same would be true when dice are rolled. This is another bit of information.
You could expand this to a twenty-sided die as well. This principle can then be used to communicate letters, numbers, and other informational concepts that we recognize.
Take the alphabet, for example. In reducing the uncertainty of the equation, multiple bits of information are generated.
This is because each character being transmitted either is or is not a specific letter of that alphabet. Indeed, a communication device has to be able to work with any information of the context.
This has led Shannon to re -define the fundamental concept of entropy , which talks about information of a context. You should call it entropy, for two reasons.
In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name.
In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage.
In , Ludwig Boltzmann shook the world of physics by defining the entropy of gases, which greatly confirmed the atomic theory.
He defined the entropy more or less as the logarithm of the number of microstates which correspond to a macrostate.
For instance, a macrostate would say that a set of particles has a certain volume, pressure, mass and temperature.
Meanwhile, a microstate defines the position and velocity of every particle. This is explained in the following figure, where each color stands for a possible message of the context:.
The average amount of information is therefore the logarithm of the number of microstates. This is another important interpretation of entropy.
For the average information to be high, the context must allow for a large number of unlikely events. Another way of phrasing this is to say that there is a lot of uncertainties in the context.
In other words, entropy is a measure of the spreading of a probability. In some sense, the second law of thermodynamics which states that entropy cannot decrease can be reinterpreted as the increasing impossibility of defining precise contexts on a macroscopic level.
It is essential! The most important application probably regards data compression. Indeed, the entropy provides the theoretical limit to the average number of bits to code a message of a context.
It also gives an insight into how to do so. Data compression has been applied to image, audio or file compressing, and is now essential on the Web.
Youtube videos can now be compressed enough to surf all over the Internet! For any given introduction, the message can be described with a conditional probability.
This defines a entropy conditional to the given introduction. Now, the conditional entropy is the average of this entropy conditional to the given introduction, when this given introduction follows the probabilistic distribution of introductions.
Roughly said, the conditional entropy is the average added information of the message given its introduction. I know! Common sense says that the added information of a message to its introduction should not be larger than the information of the message.
This translates into saying that the conditional entropy should be lower than the non-conditional entropy. This is a theorem proven by Shannon!
In fact, he went further and quantified this sentence: The entropy of a message is the sum of the entropy of its introduction and the entropy of the message conditional to its introduction!
Fortunately, everything can be more easily understood on a figure. The amount of information of the introduction and the message can be drawn as circles.
Because they are not independent, they have some mutual information, which is the intersection of the circles. On the left of the following figure is the entropies of two coins thrown independently.
On the right is the case where only one coin is thrown, and where the blue corresponds to a sensor which says which face the coin fell on.
The sensor has two positions heads or tails , but, now, all the information is mutual:. As you can see, in the second case, conditional entropies are nil.
Indeed, once we know the result of the sensor, then the coin no longer provides any information. Thus, in average, the conditional information of the coin is zero.
In other words, the conditional entropy is nil. It surely is! Indeed, if you try to encode a message by encoding each character individually, you will be consuming space to repeat mutual information.
In fact, as Shannon studied the English language, he noticed that the conditional entropy of a letter knowing the previous one is greatly decreased from its non-conditional entropy.
The structure of information also lies in the concatenation into longer texts. In fact, Shannon defined the entropy of each character as the limit of the entropy of messages of great size divided by the size.
Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip with two equally likely outcomes provides less information lower entropy than specifying the outcome from a roll of a die with six equally likely outcomes.
Some other important measures in information theory are mutual information , channel capacity, error exponents , and relative entropy.
Important sub-fields of information theory include source coding , algorithmic complexity theory , algorithmic information theory , and information-theoretic security.
Applications of fundamental topics of information theory include lossless data compression e. ZIP files , lossy data compression e. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc , the feasibility of mobile phones and the development of the Internet.
The theory has also found applications in other areas, including statistical inference ,  cryptography , neurobiology ,  perception ,  linguistics, the evolution  and function  of molecular codes bioinformatics , thermal physics ,  quantum computing , black holes, information retrieval , intelligence gathering , plagiarism detection ,  pattern recognition , anomaly detection  and even art creation.
Information theory studies the transmission, processing, extraction, and utilization of information. Abstractly, information can be thought of as the resolution of uncertainty.
In the case of communication of information over a noisy channel, this abstract concept was made concrete in by Claude Shannon in his paper "A Mathematical Theory of Communication", in which "information" is thought of as a set of possible messages, where the goal is to send these messages over a noisy channel, and then to have the receiver reconstruct the message with low probability of error, in spite of the channel noise.
Shannon's main result, the noisy-channel coding theorem showed that, in the limit of many channel uses, the rate of information that is asymptotically achievable is equal to the channel capacity, a quantity dependent merely on the statistics of the channel over which the messages are sent.
Information theory is closely associated with a collection of pure and applied disciplines that have been investigated and reduced to engineering practice under a variety of rubrics throughout the world over the past half-century or more: adaptive systems , anticipatory systems , artificial intelligence , complex systems , complexity science , cybernetics , informatics , machine learning , along with systems sciences of many descriptions.
Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of coding theory.
Coding theory is concerned with finding explicit methods, called codes , for increasing the efficiency and reducing the error rate of data communication over noisy channels to near the channel capacity.
These codes can be roughly subdivided into data compression source coding and error-correction channel coding techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible.
A third class of information theory codes are cryptographic algorithms both codes and ciphers. Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis.
See the article ban unit for a historical application. The landmark event that established the discipline of information theory and brought it to immediate worldwide attention was the publication of Claude E.
Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs , all implicitly assuming events of equal probability.
The unit of information was therefore the decimal digit , which has since sometimes been called the hartley in his honor as a unit or scale or measure of information.
Alan Turing in used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers.
Much of the mathematics behind information theory with events of different probabilities were developed for the field of thermodynamics by Ludwig Boltzmann and J.
Willard Gibbs. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the s, are explored in Entropy in thermodynamics and information theory.
In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of , Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion that.
Information theory is based on probability theory and statistics. Information theory often concerns itself with measures of information of the distributions associated with random variables.
Important quantities of information are entropy, a measure of information in a single random variable, and mutual information, a measure of information in common between two random variables.
The former quantity is a property of the probability distribution of a random variable and gives a limit on the rate at which data generated by independent samples with the given distribution can be reliably compressed.
The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy channel in the limit of long block lengths, when the channel statistics are determined by the joint distribution.
The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. A common unit of information is the bit, based on the binary logarithm.
Other units include the nat , which is based on the natural logarithm , and the decimal digit , which is based on the common logarithm.
Based on the probability mass function of each source symbol to be communicated, the Shannon entropy H , in units of bits per symbol , is given by.
This equation gives the entropy in the units of "bits" per symbol because it uses a logarithm of base 2, and this base-2 measure of entropy has sometimes been called the shannon in his honor.
Entropy is also commonly computed using the natural logarithm base e , where e is Euler's number , which produces a measurement of entropy in nats per symbol and sometimes simplifies the analysis by avoiding the need to include extra constants in the formulas.
Other bases are also possible, but less commonly used. Intuitively, the entropy H X of a discrete random variable X is a measure of the amount of uncertainty associated with the value of X when only its distribution is known.
If one transmits bits 0s and 1s , and the value of each of these bits is known to the receiver has a specific value with certainty ahead of transmission, it is clear that no information is transmitted.
If, however, each bit is independently equally likely to be 0 or 1, shannons of information more often called bits have been transmitted.
Between these two extremes, information can be quantified as follows. The special case of information entropy for a random variable with two outcomes is the binary entropy function, usually taken to the logarithmic base 2, thus having the shannon Sh as unit:.
The joint entropy of two discrete random variables X and Y is merely the entropy of their pairing: X , Y. This implies that if X and Y are independent , then their joint entropy is the sum of their individual entropies.
For example, if X , Y represents the position of a chess piece— X the row and Y the column, then the joint entropy of the row of the piece and the column of the piece will be the entropy of the position of the piece.
Despite similar notation, joint entropy should not be confused with cross entropy. The conditional entropy or conditional uncertainty of X given random variable Y also called the equivocation of X about Y is the average conditional entropy over Y : .
Because entropy can be conditioned on a random variable or on that random variable being a certain value, care should be taken not to confuse these two definitions of conditional entropy, the former of which is in more common use.
A basic property of this form of conditional entropy is that:. Mutual information measures the amount of information that can be obtained about one random variable by observing another.
It is important in communication where it can be used to maximize the amount of information shared between sent and received signals.
The mutual information of X relative to Y is given by:. Mutual information is symmetric :. Mutual information can be expressed as the average Kullback—Leibler divergence information gain between the posterior probability distribution of X given the value of Y and the prior distribution on X :.
In other words, this is a measure of how much, on the average, the probability distribution on X will change if we are given the value of Y.
This is often recalculated as the divergence from the product of the marginal distributions to the actual joint distribution:. Morse built a telegraph line between Washington, D.
Morse encountered many electrical problems when he sent signals through buried transmission lines, but inexplicably he encountered fewer problems when the lines were suspended on poles.
This attracted the attention of many distinguished physicists, most notably the Scotsman William Thomson Baron Kelvin.
Much of their work was done using Fourier analysis , a technique described later in this article, but in all of these cases the analysis was dedicated to solving the practical engineering problems of communication systems.
This view is in sharp contrast with the common conception of information, in which meaning has an essential role. Shannon also realized that the amount of knowledge conveyed by a signal is not directly related to the size of the message.
Similarly, a long, complete message in perfect French would convey little useful knowledge to someone who could understand only English.
Shannon thus wisely realized that a useful theory of information would first have to concentrate on the problems associated with sending and receiving messages, and it would have to leave questions involving any intrinsic meaning of a message—known as the semantic problem—for later investigators.
Clearly, if the technical problem could not be solved—that is, if a message could not be transmitted correctly—then the semantic problem was not likely ever to be solved satisfactorily.
Solving the technical problem was therefore the first step in developing a reliable communication system. It is no accident that Shannon worked for Bell Laboratories.Claude Shannon first proposed the information theory in The goal was to find the fundamental limits of communication operations and signal processing through an operation like data compression. It is a theory that has been extrapolated into thermal physics, quantum computing, linguistics, and even plagiarism detection. Information Theory was not just a product of the work of Claude Shannon. It was the result of crucial contributions made by many distinct individuals, from a variety of backgrounds, who took his ideas and expanded upon them. Indeed the diversity and directions of their perspectives and interests shaped the direction of Information flyknitsoldes.com Size: KB. 10/14/ · A year after he founded and launched information theory, Shannon published a paper that proved that unbreakable cryptography was possible. (He did this work in , but at that time it was. Concepts, methods Schießen Spiele Kostenlos results from coding theory and information theory are widely used in cryptography and cryptanalysis. Kakeguri, even though the noise is small, as you amplify the message over and over, the noise eventually gets bigger than the message. Sign Up. Natural or biological.