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Log Sum Inequality
The log sum inequality is used for proving theorems in information theory. Statement Let a_1,\ldots,a_n and b_1,\ldots,b_n be nonnegative numbers. Denote the sum of all a_is by a and the sum of all b_is by b. The log sum inequality states that :\sum_^n a_i\log\frac\geq a\log\frac, with equality if and only if \frac are equal for all i, in other words a_i =c b_i for all i. (Take a_i\log \frac to be 0 if a_i=0 and \infty if a_i>0, b_i=0. These are the limiting values obtained as the relevant number tends to 0.) Proof Notice that after setting f(x)=x\log x we have : \begin \sum_^n a_i\log\frac & = \sum_^n b_i f\left(\frac\right) = b\sum_^n \frac f\left(\frac\right) \\ & \geq b f\left(\sum_^n \frac\frac\right) = b f\left(\frac\sum_^n a_i\right) = b f\left(\frac\right) \\ & = a\log\frac, \end where the inequality follows from Jensen's inequality In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure 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 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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jensen's Inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations. Jensen's inequality generalizes the statement that the secant line of a convex function lies ''above'' the graph of the function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function (for ''t'' ∈ ,1, :t f(x_1) + (1-t) f(x_2), whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gibbs' Inequality
200px, Josiah Willard Gibbs In information theory, Gibbs' inequality is a statement about the information entropy of a discrete probability distribution. Several other bounds on the entropy of probability distributions are derived from Gibbs' inequality, including Fano's inequality. It was first presented by J. Willard Gibbs in the 19th century. Gibbs' inequality Suppose that : P = \ is a discrete probability distribution. Then for any other probability distribution : Q = \ the following inequality between positive quantities (since pi and qi are between zero and one) holds: : - \sum_^n p_i \log p_i \leq - \sum_^n p_i \log q_i with equality if and only if : p_i = q_i for all ''i''. Put in words, the information entropy of a distribution P is less than or equal to its cross entropy with any other distribution Q. The difference between the two quantities is the Kullback–Leibler divergence or relative entropy, so the inequality can also be written: : D_(P\, Q) \equiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequalities
Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * Spatial inequality, the unequal distribution of income and resources across geographical regions * Income inequality metrics, used to measure income and economic inequality among participants in a particular economy * International inequality, economic differences between countries Healthcare * Health equity, the study of differences in the quality of health and healthcare across different populations Mathematics * Inequality (mathematics), a relation between two values when they are different Social sciences * Educational inequality, the unequal distribution of academic resources to socially excluded communities * Gender inequality, unequal treatment or perceptions of individuals due to their gender * Participation inequality, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure 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 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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
