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Claude Shannon

Claude Shannon is properly described as "the father of information theory" although he described his work as "communication theory." While others had loosely connected the idea of information to its opposite, entropy, it was Shannon who put the communication of signals in the presence of noise on a sound mathematical basis.

In 1871, James Clerk Maxwell showed how an intelligent being could in principle sort out the disorder in a gas of randomly moving molecules, by gathering information about their speeds and sorting them into hot and cold gases, in apparent violation of the second law of thermodynamics. William Thomson (Lord Kelvin) called this being "Maxwell's intelligent demon."

As early as the 1890's, Ludwig Boltzmann, who established the statistical physics foundation of thermodynamics, had described entropy as "missing information." Boltzmann chose the logarithm of the number of equiprobable microstates as the measure for his entropy, because he wanted entropy to be an additive quantity. If one system can be in one thousand possible states and another system also in a thousand possible states, the combined system has a million possible states. In a base 10 system, log101000 = 3, and 3 + 3 is 6 = log101000000.

In 1929, Leo Szilard imagined a gas with but a single molecule in a container. He then devised a mechanism that could behave like Maxwell's demon. It would insert a partition into the middle of the container, then gather the information about which of the two sides of the partition the molecule was in. This was a binary decision and it allowed Szilard to develop the mathematical form for the amount of entropy S produced by a one-bit measurement, which Szilard identified as the acquisition of information and storage in the "memory" of a physical device or of a human observer.

S = k log 2

where k is Boltzmann's constant. The base-2 logarithm reflects the binary decision.

The amount of entropy generated by the measurement may, of course, always be greater than this fundamental amount, but not smaller, or the second law would be violated.

The earlier work of Maxwell, Boltzmann, and Szilard did not figure directly in Shannon's work. Shannon studied the design of early analog computers (specifically Vannevar Bush's differential analyzer at MIT, which was used by Coolidge and James to calculate the wave functions of the hydrogen molecule in 1936). Then, with John von Neumann and Alan Turing, he helped design the first digital computers, based on the Boolean logic of 1's and 0's and binary arithmetic.

Shannon analyzed telephone switching circuits that used electromagnetic relay switches, then realized that the switches could solve some problems in Boolean algebra.

During World War II, Shannon worked at Bell Labs on cryptography and sending control signals in the presence of noise. Alan Turing visited the labs for a couple of months and showed Shannon his 1936 ideas for a universal computer (the "Turing Machine").

Shannon's work on communications, control systems, and cryptography were initially classified, but they contained almost all of the mathematics that eventually appeared in his landmark 1948 article "A Mathematical Theory of Communication," that is the basis for modern information theory.

Norbert Wiener's work on probability theory in Cybernetics had an important influence on Shannon. There can be no new information in a world of certainty. Probability and statistics are at the heart of both information theory and quantum theory.

Shannon developed his expression for an information (Shannon) entropy, which he showed has the same mathematical form as thermodynamic (Boltzmann) entropy. He wrote:

Suppose we have a set of possible events whose probabilities of occurrence are p1, p2, • • • , pn. These probabilities are known but that is all we know concerning which event will occur. Can we find a measure of how much "choice" is involved in the selection of the event or of how uncertain we are of the outcome?

If there is such a measure, say H(p1, p2, • • • , pn), it is reasonable to require of it the following properties:

1. H should be continuous in the pn. 2. If all the pn are equal, pi = 1/n, then H should be a monotonic increasing function of n. With equally likely events there is more choice, or uncertainty, when there are more possible events.

3. If a choice be broken down into two successive choices, the original H should be the weighted sum of the individual values of H. The meaning of this is illustrated in Fig. 6.

Fig. 6.— Decomposition of a choice from three possibilities.

At the left we have three possibilities p1 = 1/2, p2 = 1/3, p3 = 1/6. On the right we first choose between two possibilities each with probability 1/2, and if the second occurs make another choice with probabilities 2/3, 1/3. The final results have the same probabilities as before. We require, in this special case, that

H(1/2, 1/3, 1/6) = H(1/2, 1/2) + 1/2 H(2/3, 1/3)

The coefficient 1/2 is the weighting factor introduced because this second choice only occurs half the time.

The only H satisfying the three above assumptions is of the form:

H = K Σ pi log pi

where K is a positive constant.

Quantities, of the form H = Σ pi log pi (the constant K merely amounts to a choice of a unit of measure) play a central role in information theory as measures of information, choice and uncertainty. The form of H will be recognized as that of entropy as defined in certain formulations of statistical mechanics where pi is the probability of a system being in cell i of its phase space.

H is then, for example, the H in Boltzmann's famous H theorem. We shall call H = pi log pi the entropy of the set of probabilities p1, p2, • • • , pn.

Shannon Entropy and Boltzmann Entropy

Shannon entropy is the average (expected) value of the information contained in a received message. If there are many possible messages, we get a lot more information than when there are only two possibilities (one bit of information). It is the base 2 logarithm of the number of possibilities. Entropy thus characterizes our uncertainty about the information in an incoming message, and increases for more possibilities with greater randomness. The less likely an event is, the more information it provides when it occurs. Shannon defined his entropy or information as the negative of the logarithm of the probability distribution. One bit of information is also known as one "shannon."

Boltzmann entropy is maximized when the particle distribution is maximally random among positions in phase space, when the number of microstates W corresponding to a given macrostate is as large as possible. An improbable macrostate might be when every particle is in the same microstate. Finding all the particles in a corner of the possible volume is information in the same sense as receiving one of the possible messages. An equilibrium macrostate is when particles are as randomly distributed as possible. Any information is gone.

Counterintuitively, maximum Boltzmann entropy (no information) is maximal uncertainty before a message is received and then maximal Shannon entropy (information), after a message is received, making the two entropies hard to compare.

Historical Background
Information in physical systems was connected to a measure of the structural order in a system as early as the nineteenth century by William Thomson (Lord Kelvin) and Ludwig Boltzmann, who described an increase in the thermodynamic entropy as “lost information.”

In 1877, Boltzmann proved his “H-Theorem” that the entropy or disorder in the universe always increases. He defined entropy S as the logarithm of the number W of possible states of a physical system, an equation now known as Boltzmann’s Principle,

S = k log W.

In 1929, Leo Szilard showed the mean value of the quantity of information produced by a 1-bit, two-possibility (“yes/no”) measurement as S = k log 2, where k is Boltzmann’s constant, connecting information directly to entropy.

Following Szilard, Ludwig von Bertalanffy, Erwin Schrödinger, Norbert Wiener, Claude Shannon, Warren Weaver, John von Neumann, and Leon Brillouin, all expressed similar views on the connection between physical entropy and abstract “bits” of information.

Schrödinger said the information in a living organism is the result of “feeding on negative entropy” from the sun. Wiener said “The quantity we define as amount of information is the negative of the quantity usually defined as entropy in similar situations.”

Brillouin created the term “negentropy” because he said, “One of the most interesting parts in Wiener’s Cybernetics is the discussion on “Time series, information, and communication,” in which he specifies that a certain “amount of information is the negative of the quantity usually defined as entropy in similar situations.”

Shannon, with a nudge from von Neumann, used the term entropy to describe his estimate of the amount of information that can be communicated over a channel, because his mathematical theory of the communication of information produced a mathematical formula identical to Boltzmann’s equation for entropy, except for a minus sign (the negative in negative entropy).

Shannon described a set of i messages, each with probability pi. He then defined a quantity H,

H = k Σ pi log pi,

where k is a positive constant. Since H looked like the H in Boltzmann’s H-Theorem, Shannon called it the entropy of the set of probabilities p1, p2, . . . , pn.

To see the connection between the two entropies, we can note that Boltzmann assumed that all his probabilities were equal. For n equal states, the probability of each state is p = 1/n.

The sum over n states, Σ pi log pi, is then n x 1/n x log (1/n) = log (1/n) = - log n.

If we set Shannon's number of possible messages n equal to Boltzmann's number of possible microstates W, we get Boltzmann’s entropy with a minus sign,

H = - k log W.

Shannon’s entropy H is the negative of Boltzmann’s S.

Shannon showed that a communication that is certain to tell you something you already know (one of the messages has probability unity) contains no new information.

For Teachers
For Scholars
The Mathematical Theory of Communication (excerpts)
Introduction
The recent development of various methods of modulation such as PCM and PPM which exchange bandwidth for signal-to-noise ratio has intensified the interest in a general theory of communication. A basis for such a theory is contained in the important papers of Nyquist1 and Hartley2 on this subject. In the present paper we will extend the theory to include a number of new factors, in particular the effect of noise in the channel, and the savings possible due to the statistical structure of the original message and due to the nature of the final destination of the information.
The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages. The system must be designed to operate for each possible selection, not just the one which will actually be chosen since this is unknown at the time of design.

If the number of messages in the set is finite then this number or any monotonic function of this number can be regarded as a measure of the information produced when one message is chosen from the set, all choices being equally likely. As was pointed out by Hartley [and Szilard and Boltzmann] the most natural choice is the logarithmic function. Although this definition must be generalized considerably when we consider the influence of the statistics of the message and when we have a continuous range of messages, we will in all cases use an essentially logarithmic measure.

The logarithmic measure is more convenient for various reasons:

1. It is practically more useful. Parameters of engineering importance such as time, bandwidth, number of relays, etc., tend to vary linearly with the logarithm of the number of possibilities. For example, adding one relay to a group doubles the number of possible states of the relays. It adds 1 to the base 2 logarithm of this number. Doubling the time roughly squares the number of possible messages, or doubles the logarithm, etc.

2. It is nearer to our intuitive feeling as to the proper measure. This is closely related to (1) since we intuitively measure entities by linear comparison with common standards. One feels, for example, that two punched cards should have twice the capacity of one for information storage, and two identical channels twice the capacity of one for transmitting information.

3. It is mathematically more suitable. Many of the limiting operations are simple in terms of the logarithm but would require clumsy restatement in terms of the number of possibilities.

The choice of a logarithmic base corresponds to the choice of a unit for measuring information. If the base 2 is used the resulting units may be called binary digits, or more briefly bits, a word suggested by J. W. Tukey. A device with two stable positions, such as a relay or a flip-flop circuit, can store one bit of information. N such devices can store N bits, since the total number of possible states is 2N and log2 2N = N. If the base 10 is used the units may be called decimal digits. Since

log2 M = log10M/log102
= 3.32 log10 M,

a decimal digit is about 3+1/2 bits. A digit wheel on a desk computing machine has ten stable positions and therefore has a storage capacity of one decimal digit. In analytical work where integration and differentiation are involved the base e is sometimes useful. The resulting units of information will be called natural units. Change from the base a to base b merely requires multiplication by logba.

By a communication system we will mean a system of the type indicated schematically in Fig. 1. It consists of essentially five parts:

Fig. 1 Schematic diagram of a general communication system.

1. An information source which produces a message or sequence of messages to be communicated to the receiving terminal. The message may be of various types: (a) A sequence of letters as in a telegraph or teletype system; (b) A single function of time f(t) as in radio or telephony; (c) A function of time and other variables as in black and white television — here the message may be thought of as a function f(x, y, t) of two space coordinates and time, the light intensity at point (x, y) and time t on a pickup tube plate; (d) Two or more functions of time, say f t), g(t), h(t) — this is the case in "three-dimensional" sound transmission or if the system is intended to service several individual channels in multiplex; (e) Several functions of several variables — in color television the message consists of three functions f(x, y, t), g(x, y, t), h(x, y, t) defined in a three-dimensional continuum -- we may also think of these three functions as components of a vector field defined in the region — similarly, several black and white television sources would produce "messages" consisting of a number of functions of three variables; (f) Various combinations also occur, for example in television with an associated audio channel.

2. A transmitter which operates on the message in some way to produce a signal suitable for transmission over the channel. In telephony this operation consists merely of changing sound pressure into a proportional electrical current. In telegraphy we have an encoding operation which produces a sequence of dots, dashes and spaces on the channel corresponding to the message. In a multiplex PCM system the different speech functions must be sampled, compressed, quantized and encoded, and finally interleaved properly to construct the signal. Vocoder systems, television and frequency modulation are other examples of complex operations applied to the message to obtain the signal. 3. The channel is merely the medium used to transmit the signal from transmitter to receiver. It may be a pair of wires, a coaxial cable, a band of radio frequencies, a beam of light, etc. During transmission, or at one of the terminals, the signal may be perturbed by noise. This is indicated schematically in Fig. 1 by the noise source acting on the transmitted signal to produce the received signal.

4. The receiver ordinarily performs the inverse operation of that done by the transmitter, reconstructing the message from the signal.

5. The destination is the person (or thing) for whom the message is intended.

We wish to consider certain general problems involving communication systems. To do this it is first necessary to represent the various elements involved as mathematical entities, suitably idealized from their physical counterparts. We may roughly classify communication systems into three main categories: discrete, continuous and mixed. By a discrete system we will mean one in which both the message and the signal are a sequence of discrete symbols. A typical case is telegraphy where the message is a sequence of letters and the signal a sequence of dots, dashes and spaces. A continuous system is one in which the message and signal are both treated as continuous functions, e.g., radio or television. A mixed system is one in which both discrete and continuous variables appear, e.g., PCM transmission of speech.

We first consider the discrete case. This case has applications not only in communication theory, but also in the theory of computing machines, the design of telephone exchanges and other fields. In addition the discrete case forms a foundation for the continuous and mixed cases which will be treated in the second half of the paper.

6. Choice, Uncertainty and Entropy
We have represented a discrete information source as a Markoff process. Can we define a quantity which will measure, in some sense, how much information is "produced" by such a process, or better, at what rate information is produced?

Suppose we have a set of possible events whose probabilities of occurrence are p1, p2, • • • , pn. These probabilities are known but that is all we know concerning which event will occur. Can we find a measure of how much "choice" is involved in the selection of the event or of how uncertain we are of the outcome?

If there is such a measure, say H(p1, p2, • • • , pn), it is reasonable to require of it the following properties:

1. H should be continuous in the pn. 2. If all the pn are equal, pi = 1/n, then H should be a monotonic increasing function of n. With equally likely events there is more choice, or uncertainty, when there are more possible events.

3. If a choice be broken down into two successive choices, the original H should be the weighted sum of the individual values of H. The meaning of this is illustrated in Fig. 6.

Fig. 6.— Decomposition of a choice from three possibilities.

At the left we have three possibilities p1 = 1/2, p2 = 1/3, p3 = 1/6. On the right we first choose between two possibilities each with probability 1/2, and if the second occurs make another choice with probabilities 2/3, 1/3. The final results have the same probabilities as before. We require, in this special case, that

H(1/2, 1/3, 1/6) = H(1/2, 1/2) + 1/2 H(2/3, 1/3)

The coefficient 1/2 is the weighting factor introduced because this second choice only occurs half the time.

In Appendix 2, the following result is established:

Theorem 2: The only H satisfying the three above assumptions is of the form:

H = K Σ pi log pi

where K is a positive constant. This theorem, and the assumptions required for its proof, are in no way necessary for the present theory. It is given chiefly to lend a certain plausibility to some of our later definitions. The real justification of these definitions, however, will reside in their implications.

Quantities, of the form H = Σ pi log pi (the constant K merely amounts to a choice of a unit of measure) play a central role in information theory as measures of information, choice and uncertainty. The form of H will be recognized as that of entropy as defined in certain formulations of statistical mechanics8 where pi is the probability of a system being in cell i of its phase space.

H is then, for example, the H in Boltzmann's famous H theorem. We shall call H = pi log pi the entropy of the set of probabilities p1, p2, • • • , pn. If x is a chance variable we will write H(x) for its entropy; thus x is not an argument of a function but a label for a number, to differentiate it from H(y) say, the entropy of the chance variable y.

The quantity H has a number of interesting properties which further substantiate it as a reasonable measure of choice or information.

1. H = 0 if and only if all the pi but one are zero, this one having the value unity. Thus only when we are certain of the outcome does H vanish. Otherwise H is positive.

2. For a given n, H is a maximum and equal to log n when all the pi are equal, i.e., 1/n. This is also intuitively the most uncertain situation.


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