An important theorem from information theory says that the mutual informa. Yao xie, ece587, information theory, duke university. The concept of information entropy was created by mathematician claude shannon. Compsci 650 applied information theory jan 21, 2016 instructor. Entropy and mutual information entropy mutual information dr. More generally, this can be used to quantify the information in an event and a random variable, called entropy, and is calculated using probability.
Why entropy is a fundamental measure of information content. To do so, the transmitter sends a series possibly just one partial messages that give clues towards the original message. If your image is an 8bit image, i would not recommend you do that due to loss of bin count. Nov 4, 2014 iftach haitner tau application of information theory, lecture 2 nov 4, 2014 1 26. Application of information theory, lecture 2 joint. When this is the case, the units of entropy are bits. Mutual informationbased registration of digitally reconstructed radiographs and electronic portal images by katherine anne bachman master of basic science, mathematics, university of colorado at denver, 2002 bachelor of science, chemistry, university of colorado at denver, 2000 a thesis submitted to the university of colorado at denver. The defining expression for entropy in the theory of information established by claude e. Shannons work form the underlying theme for the present course. Examples are entropy, mutual information, conditional entropy, conditional information, and. Consider you are designing a system to transmit information as efficiently as. In information theory, joint entropy is a measure of the uncertainty associated with a set of variables.
Relationship between entropy and mutual information. Information gain, mutual information and related measures. Information theory georgia institute of technology. Information theory is a subfield of mathematics concerned with transmitting data across a noisy channel. E logpx 1 the entropy measures the expected uncertainty in x. In our case we will be interested in natural language messages, but information theory applies to any form of messages.
We also say that hx is approximately equal to how much information we learn on average from one instance of the random variable x. Similarly, apart from kolmogorov complexity itself, the algorithmic mutual information is one of the most important concepts in kolmogorovs theory. It does not depend on the actual values taken by the r. In information theory, the major goal is for one person a transmitter to convey some message over a channel to another person the receiver. Image similarity based on joint entropy joint histogram image similarity is used in image registration, content based image retrieval, medical application etc. Casino i youre at a casino i you can bet on coins, dice, or roulette i coins 2 possible outcomes. A foundation of information theory information theory can be viewed as a way to measure and reason about the complexity of messages. Information entropy is a concept from information theory. Entropy and information theory stanford ee stanford university. Cross validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. In general, the more certain or deterministic the event is, the less information it will contain.
Entropy and mutual information in information theory 3 notice that hx is the expected value of the selfinformation. Differential entropy elements of information theory. Information entropy simple english wikipedia, the free. A cornerstone of information theory is the idea of quantifying how much information there is in a message. Marginal entropy, joint entropy, conditional entropy, and the chain rule for entropy. The entropy of a discrete random variable x with alphabet x is. Also, it arises as answers to speci c operational problems, e. Information theory can be viewed as simply a branch of applied probability theory. Apart from entropy, the mutual information is perhaps the most important concept in shannons theory. Estimation of entropy and mutual information 1195 ducing anything particularly novel, but merely formalizing what statisticians have been doing naturally since well before shannon wrote his papers. Communication theory provides a technique for measuring the joint entropy with respect to the marginal entropies.
The information content of one of these partial messages is a measure of how much uncertainty this resolves for the receiver. Information theory is the natural domain to quantify these operations of information processing, and we have seen a number of measures recently introduced for this purpose, including the wellknown transfer entropy 22, as well as active information storage 23 and predictive information 24, 25. Chain rules for entropy, relative entropy, and mutual information. This document is an introduction to entropy and mutual information for discrete. Motivationinformation entropy compressing information motivation. The defining expression for entropy in the theory of statistical mechanics established by ludwig boltzmann and j. The second notion of information used by shannon was mutual information. Information theory 3 note that entropy is a function of the distribution of x. Entropy and mutual information 1 introduction imagine two people alice and bob living in toronto and boston respectively.
Entropy is measure uncertainty of an random variable in information theory. The relationship between joint entropy, marginal entropy, conditional entropy and mutual information source. In information theory, the conditional entropy or equivocation quantifies the amount of information needed to describe the outcome of a random variable given that the value of another random variable is known. To make this more precise, let a x denote the simple event in which x takes its value. More precisely, the entropy of a system represents the amount. Properties of differential entropy, relative entropy, and mutual information. Binary erasure channel, uniqueness of the entropy function, joint entropy and conditional entropy, relative entropy and. You need to be consistent with how many bins youre using for both entropy and what we talked about with joint entropy.
Information theory, the mathematical theory of communication, has two primary goals. Such a theory is not yet well developed, for several historical reasons. This measure is known as mutual information ia, b and was independently and simultaneously proposed for intermodality medical image registration by researchers in leuven, belgium 18, 19, and mit in the united states 1, 20. The proposed measure is based on the fact of the joint entropy is. The joint entropy of two discrete random variables x and y is merely the entropy of their pairing.
Because of its dependence on ergodic theorems, however, it can also be viewed as a branch of ergodic theory, the theory of invariant transformations and transformations related to invariant transformations. Joint entropy is a measure of the uncertainty associated with a set of variables. E log f x corollary if x 1,x 2,x n are mutually independent, then hx n i1 hx i. Entropy and mutual information department of electrical and. Relation of differential entropy to discrete entropy. Say, the above objective function works good for neural networks, but violate information theory in the sense that we are adding a probability value, a, to a loss value, which is crossentropy. It tells how much information there is in an event. Joint and conditional entropy code information free.
Informationentropyisthelogbase2of thenumberofpossibleoutcomes. X, y, this implies that if x and y are independent, then their joint entropy. This strategy bears a striking resemblance to regularization methods employed in abstract statistical inference grenander, 1981, generally known. Mutual information between ensembles of random variables. In this section, we define joint differential entropy, conditional differential entropy and mutual information. Tatu proposed a new similarity measure that base on joint entropy joint histogram 10. Definition the differential entropy of a continuous random variable x with p. There is a simple relationship between the entropy concept in information theory and the boltzmanngibbs entropy concept in thermodynamics, brie. More clearly stated, information is an increase in uncertainty or entropy. In shannons information theory, a message is a random draw from a probability distribution on messages and entropy gives the data compression source coding limit.
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