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Conditional Information
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable Y given that the value of another random variable X is known. Here, information is measured in shannons, nats, or hartleys. The ''entropy of Y conditioned on X'' is written as \Eta(Y, X). Definition The conditional entropy of Y given X is defined as :\Eta(Y, X)\ = -\sum_p(x,y)\log \frac where \mathcal X and \mathcal Y denote the support sets of X and Y. ''Note:'' Here, the convention is that the expression 0 \log 0 should be treated as being equal to zero. This is because \lim_ \theta\, \log \theta = 0. Intuitively, notice that by definition of expected value and of conditional probability, \displaystyle H(Y, X) can be written as H(Y, X) = \mathbb (X,Y)/math>, where f is defined as \displaystyle f(x,y) := -\log\left(\frac\right) = -\log(p(y, x)). One can think of \displaystyle f as associating each pair \displaystyle (x, y) with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayes' Rule
Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting conditional probabilities, allowing one to find the probability of a cause given its effect. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to someone of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the person is typical of the population as a whole. Based on Bayes' law, both the prevalence of a disease in a given population and the error rate of an infectious disease test must be taken into account to evaluate the meaning of a positive test result and avoid the '' base-rate fallacy''. One of Bayes' theorem's many applications is Bayesian inference, an approach to statistical inference, where it is used to invert the probability of observations given a model configuration (i.e., the likelihood function) to obtain the probability o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variation Of Information
In probability theory and information theory, the variation of information or shared information distance is a measure of the distance between two clusterings ( partitions of elements). It is closely related to mutual information; indeed, it is a simple linear expression involving the mutual information. Unlike the mutual information, however, the variation of information is a true metric, in that it obeys the triangle inequality. Definition Suppose we have two partitions X and Y of a set A, namely X = \ and Y = \. Let: :n = \sum_ , X_, = \sum_ , Y_, =, A, :p_ = , X_, / n and q_ = , Y_, / n :r_ = , X_i\cap Y_, / n Then the variation of information between the two partitions is: :\mathrm(X; Y ) = - \sum_ r_ \left log(r_/p_i)+\log(r_/q_j) \right/math>. This is equivalent to the shared information distance between the random variables ''i'' and ''j'' with respect to the uniform probability measure on A defined by \mu(B):=, B, /n for B\subseteq A. Explicit information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Quantum Entropy
The conditional quantum entropy is an entropy measure used in quantum information theory. It is a generalization of the conditional entropy of classical information theory. For a bipartite state \rho^, the conditional entropy is written S(A, B)_\rho, or H(A, B)_\rho, depending on the notation being used for the von Neumann entropy. The quantum conditional entropy was defined in terms of a conditional density operator \rho_ by Nicolas Cerf and Chris Adami, who showed that quantum conditional entropies can be negative, something that is forbidden in classical physics. The negativity of quantum conditional entropy is a sufficient criterion for quantum non-separability. In what follows, we use the notation S(\cdot) for the von Neumann entropy, which will simply be called "entropy". Definition Given a bipartite quantum state \rho^, the entropy of the joint system AB is S(AB)_\rho \ \stackrel\ S(\rho^), and the entropies of the subsystems are S(A)_\rho \ \stackrel\ S(\rho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutual Information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual Statistical dependence, dependence between the two variables. More specifically, it quantifies the "Information content, amount of information" (in Units of information, units such as shannon (unit), shannons (bits), Nat (unit), nats or Hartley (unit), hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of Entropy (information theory), entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable. Not limited to real-valued random variables and linear dependence like the Pearson correlation coefficient, correlation coefficient, MI is more general and determines how different the joint distribution of the pair (X,Y) is from the product of the marginal distributions of X and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropy (information Theory)
In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across all potential states. Given a discrete random variable X, which may be any member x within the set \mathcal and is distributed according to p\colon \mathcal\to[0, 1], the entropy is \Eta(X) := -\sum_ p(x) \log p(x), where \Sigma denotes the sum over the variable's possible values. The choice of base for \log, the logarithm, varies for different applications. Base 2 gives the unit of bits (or "shannon (unit), shannons"), while base Euler's number, ''e'' gives "natural units" nat (unit), nat, and base 10 gives units of "dits", "bans", or "Hartley (unit), hartleys". An equivalent definition of entropy is the expected value of the self-information of a v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Quantum Entropy
The conditional quantum entropy is an entropy measure used in quantum information theory. It is a generalization of the conditional entropy of classical information theory. For a bipartite state \rho^, the conditional entropy is written S(A, B)_\rho, or H(A, B)_\rho, depending on the notation being used for the von Neumann entropy. The quantum conditional entropy was defined in terms of a conditional density operator \rho_ by Nicolas Cerf and Chris Adami, who showed that quantum conditional entropies can be negative, something that is forbidden in classical physics. The negativity of quantum conditional entropy is a sufficient criterion for quantum non-separability. In what follows, we use the notation S(\cdot) for the von Neumann entropy, which will simply be called "entropy". Definition Given a bipartite quantum state \rho^, the entropy of the joint system AB is S(AB)_\rho \ \stackrel\ S(\rho^), and the entropies of the subsystems are S(A)_\rho \ \stackrel\ S(\rho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Information Theory
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term. It is an interdisciplinary field that involves quantum mechanics, computer science, information theory, philosophy and cryptography among other fields. Its study is also relevant to disciplines such as cognitive science, psychology and neuroscience. Its main focus is in extracting information from matter at the microscopic scale. Observation in science is one of the most important ways of acquiring information and measurement is required in order to quantify the observation, making this crucial to the scientific method. In quantum mechanics, due to the uncertainty principle, non-commuting observables cannot be precisely measured simultaneously, as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncertainty Principle
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known. More formally, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the product of the accuracy of certain related pairs of measurements on a quantum system, such as position, ''x'', and momentum, ''p''. Such paired-variables are known as complementary variables or canonically conjugate variables. First introduced in 1927 by German physicist Werner Heisenberg, the formal inequality relating the standard deviation of position ''σx'' and the standard deviation of momentum ''σp'' was derived by Earle Hesse Kennard later that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the sample mean is a commonly used estimator of the population mean. There are point estimator, point and interval estimators. The point estimators yield single-valued results. This is in contrast to an interval estimator, where the result would be a range of plausible values. "Single value" does not necessarily mean "single number", but includes vector valued or function valued estimators. ''Estimation theory'' is concerned with the properties of estimators; that is, with defining properties that can be used to compare different estimators (different rules for creating estimates) for the same quantity, based on the same data. Such properties can be used to determine the best rules to use under given circumstances. Howeve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joint Probability Density Function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a '' relative likelihood'' that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. More precisely, the PDF is used to specify the probability of the random variable falling ''within a particular range o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
