Binary entropy function
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The logarithms in this formula are usually taken as shown in the graph to the base 2. This is the derivative binary entropy function of an unbiased coin flip. In terms of information derivative binary entropy function, entropy is considered to be a measure of the uncertainty in a message.
At this probability, the event is certain never to occur, and so there is no uncertainty at all, leading to an entropy of 0. In this case, the entropy is maximum at a value of 1 bit. The derivative of the binary entropy function may be expressed as the negative of the logit function:.
Bernoulli trial — It is named after Jacob Bernoulli, a 17th century Swiss mathematician. The mathematical formalisation of the Bernoulli trial is known as the Bernoulli process and this article offers an elementary introduction to the concept, whereas the article on the Bernoulli process offers a more advanced treatment. Since a Bernoulli trial has only two outcomes, it can be framed as some yes or no question. For example, Is the top card of a shuffled deck an ace, was the newborn child a girl.
Therefore, success and failure are merely labels for the two outcomes, and should not be construed literally, the term success in this sense consists in the result meeting specified conditions, not in any moral judgement. More generally, given any probability space, for any event, one can define a Bernoulli trial, examples of Bernoulli trials include, Flipping a coin.
In this context, obverse derivative binary entropy function denotes success and reverse denotes failure, a derivative binary entropy function coin has the probability of success 0. In this case there are exactly two outcomes, rolling a dice, where a six is success and everything else a failure.
In this case there are six outcomes, and the event is a six, in conducting a political opinion poll, choosing a voter at random to ascertain whether that voter will vote yes in an upcoming referendum.
Independent repeated trials of an experiment with two possible outcomes are called Bernoulli trials. Call one of the success and the other outcome failure. Let p be the probability of success in a Bernoulli trial, then the probability of success and the probability of failure sum to unity, since these are complementary events, success and failure are mutually exclusive and exhaustive. When multiple Bernoulli trials are performed, each with its probability of success, consider the simple experiment where a fair coin is tossed four times.
Find the probability that two of the tosses result in heads. Information theory — Information theory studies the quantification, storage, and communication of information. A key derivative binary entropy function in information theory is entropy, 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 coin flip provides less information than specifying the outcome from a roll of a die. Some other important derivative binary entropy function in information theory are mutual information, channel capacity, error exponents, applications derivative binary entropy function fundamental topics of information theory include lossless data compression, lossy data compression, and channel coding. The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, Information theory studies the transmission, processing, utilization, and extraction of information.
Abstractly, information can be thought of as the resolution of uncertainty, Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of coding theory.
These codes can be subdivided into data compression and error-correction techniques. In the latter case, it took years to find the methods Shannons work proved were possible.
A third class of information theory codes are cryptographic algorithms, concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis.
See the article ban for a historical application, Information theory is also used in information retrieval, intelligence gathering, gambling, statistics, and even in musical composition. Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, the unit of information was therefore the decimal digit, much later renamed the hartley in his honour as a unit or scale or measure of information.
Alan Turing in used similar ideas as part of the 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, Information theory is based on probability theory and statistics.
Information theory often concerns itself with measures of information of the associated with random variables. Derivative binary entropy function quantities of information are entropy, a measure of information in a random variable, and mutual information. The choice of 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 hartley, which is based on the common logarithm.
Entropy information theory — In information theory, systems are modeled by a transmitter, channel, and receiver. The transmitter produces messages that are sent through the channel, the channel modifies the message in some way.
The receiver attempts to infer which message was sent, in this context, entropy is the expected value of the derivative binary entropy function contained in each message. Messages can be modeled by any flow of information, in a more technical sense, there are reasons to define information as the negative of the logarithm of the probability distribution of possible events or messages.
The amount of information of every event forms a random variable whose expected value, units of entropy are the shannon, nat, or hartley, depending on the base of the logarithm used to define it, though the shannon is commonly referred to as a bit. The logarithm of the probability distribution is useful as a measure of entropy because it is additive for independent sources, for instance, the entropy of a coin toss is 1 shannon, whereas of m tosses it is m shannons.
Generally, you need log2 bits to represent a variable that can take one of n if n is a power derivative binary entropy function 2.
If these values are equally probable, the entropy is equal to the number of bits, equality between number of bits and shannons holds only while all outcomes are equally probable. If one of the events is more derivative binary entropy function than others, observation of event is less informative.
Conversely, rarer events provide more information when observed, since observation of less probable events occurs more rarely, the net effect is that the entropy received from non-uniformly distributed data is less than log2. Entropy is zero when one outcome is certain, Shannon derivative binary entropy function quantifies all these considerations exactly when a probability distribution of the source is known. The meaning of the events observed does not matter in the definition of entropy, generally, entropy refers to disorder or uncertainty.
Derivative binary entropy function entropy was introduced by Claude E. Shannon in his paper A Mathematical Theory of Communication, Shannon entropy provides an absolute limit on the best possible average length of lossless encoding or compression of an information source. Entropy is a measure of unpredictability of the state, or equivalently, to get an intuitive understanding of these terms, consider the example of a political poll. Usually, such polls happen because the outcome of the poll is not already known, now, consider the case that the same poll is performed a second time shortly derivative binary entropy function the first poll.
Now consider the example of a coin toss, assuming the probability of heads is the same as the probability of tails, then the entropy of the coin toss is derivative binary entropy function high as it could be.
Derivative binary entropy function a coin toss has one shannon of entropy since there are two possible outcomes that occur with probability, and learning the actual outcome contains one shannon of information. Contrarily, a toss with a coin that has two heads and no tails has zero entropy since the coin will always come up heads.
Probability — Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1, the higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair coin, since the coin derivative binary entropy function unbiased, the two outcomes are both equally probable, the probability of head equals the probability of tail.
Probability theory is used to describe the underlying mechanics and regularities of complex systems. For example, tossing a coin twice will yield head-head, head-tail, tail-head. A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, subjectivists assign numbers per subjective probability, i.
The degree of belief has been interpreted as, the price at which you would buy or sell a bet that pays 1 unit of utility derivative binary entropy function E,0 if not Derivative binary entropy function.
The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as data to produce probabilities. The expert knowledge is represented by some prior probability distribution and these data are incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a probability distribution that incorporates all the information known to date.
The scientific study of probability is a development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, there are reasons of course, for the slow development of the mathematics of probability. Whereas derivative binary entropy function of chance provided the impetus for the study of probability. According to Richard Jeffrey, Before the middle of the century, the term probable meant approvable.
A probable action or opinion was one such as people would undertake or hold. However, in legal contexts especially, probable could also apply to propositions for which there was good evidence, the sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes.
Random variable — In probability and statistics, a derivative binary entropy function variable, random quantity, aleatory variable, or stochastic variable is a variable quantity whose value depends on possible outcomes.
It is common that these outcomes depend on physical variables that are not well understood. For example, when you toss a coin, the outcome of heads or tails depends on the uncertain physics. Which outcome will be observed is not certain, of course the coin could get caught in a crack in the floor, but such a possibility is excluded from consideration.
The domain of a variable is the set of possible outcomes. In the case of the coin, there are derivative binary entropy function possible outcomes, namely heads or tails. Since one of these outcomes must occur, thus either the event that the coin lands heads or the event that the coin lands tails must have non-zero probability, a random variable is defined as a function that maps outcomes to numerical quantities, typically real numbers.
In this sense, it is a procedure for assigning a numerical quantity to each outcome, and, contrary to its name. What is random is the physics that describes how the coin lands. A random variables possible values might represent the possible outcomes of a yet-to-be-performed experiment and they may also conceptually represent either the results of an objectively random process or the subjective randomness that results from incomplete knowledge of a quantity. The mathematics works the same regardless of the interpretation in use.
A random variable has a probability distribution, which specifies the probability that its value falls in any given interval, two random variables with the same probability distribution can still differ in terms of their associations with, or independence from, other random variables. The realizations of a variable, that is, the results of randomly choosing values according to the variables probability distribution function, are called random variates. The formal mathematical treatment of random variables is a topic in probability theory, in that context, a random variable is understood as a function defined on a sample space whose outputs are numerical values.
Binary logarithm — In mathematics, the binary logarithm is the power to which the number 2 must be raised to obtain the value derivative binary entropy function. For example, the logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2. The binary logarithm is the logarithm to the base 2, the binary logarithm function is the inverse function of the power of two function. As well as log2, alternative notations for the binary logarithm include lg, ld, lb, and log.
Binary logarithms can be used to calculate the length of the representation of a number in the numeral system. In computer science, they count the number of steps needed for binary search, other areas in which the binary logarithm is frequently used include combinatorics, bioinformatics, the design of sports tournaments, and photography.
Binary logarithms are included in the standard C mathematical functions and other software packages. The integer part of a binary logarithm can be using the find first set operation on an integer value.
The fractional part of the logarithm can be calculated efficiently, the powers of two have been known since antiquity, for instance they appear in Euclids Elements, Props.
And the binary logarithm of a power of two is just its position in the sequence of powers of two. On this basis, Michael Stifel has been credited with publishing the first known table of binary logarithms in and his book Arthmetica Integra contains several tables that show the integers with their corresponding powers of two.
Reversing the rows of these allow them derivative binary entropy function be interpreted as tables of binary logarithms. Earlier than Stifel, the 8th century Jain mathematician Virasena is credited with a precursor to the binary logarithm, virasenas concept of ardhacheda has been defined as the number of times a given number can be divided evenly by two.