Uncertainty measure let x be a random variable taking on a nite number m of di erent values x1. In several applications, it is desirable to have a functional purely of the conditional. Entropy and mutual information 1 introduction imagine two people alice and bob living in toronto and boston respectively. The remainder of the paper is organized as follo ws. Functional properties of minimum meansquare error and. More interesting, this framework leads to an estimator with some nice properties. I ix 1 ix 2 if px 1 mutual information entropy mutual information dr. In probability theory and information theory, the mutual information mi of two random variables is a measure of the mutual dependence between the two variables. In this chapter, we have looked at a variety of topics that will be useful to us when we study various lossy compression techniques, including distortion and its measurement, some new concepts from information theory, average mutual information and its connection to the rate of a compression scheme, and the. The properties manager serves the bishop by overseeing projects involving diocesan and church properties and works with pastors and others in coordinating these projects. The conditional entropy is a measure of how much uncertainty remains about the random variable.
Y has the distinction of retaining its fundamental significance as a measure of discrete information since it is actually the limit of the discrete mutual information of partitions of x. Y pxxpy y g 2 the units of mi are sometimes referred to as nats. The conditional mutual information is a measure of how much uncertainty is shared by x. Our method, called maximum likelihood mutual information mlmi, has several at tractive properties, e. We will use the dataprocessing property of mutual information to be proved shortly, see. Of the available information theoretic based measures, we advocate the normalized information distance nid as a general measure of choice, for it possesses concurrently several important properties, such as being both a metric and a normalized measure, admitting an exact analytical adjustedforchance form, and using the nominal 0,1 range. This puts i a, b in contrast to the more commonly used measures, such as pearson correlation or euclidean distance, which quantify linear. Examples are entropy, mutual information, conditional entropy, conditional information, and relative entropy discrimination, kullbackleibler information, along with the limiting normalized versions of these quantities such as entropy rate and information rate. Properties of mutual information and introduction to channel capacity 25. Of interest for us is, that the mutual information is zero if and only if the measurements on the systems a and b are statistically independent.
Consider again the situation given in examples 51 and 52. Indeed, mutual information is now routinely computed on continuous data in many realworld applications e. The new proof of the epi presented in this paper is based on a convexity inequality for mutual information under the variance preserving transformation 4. In this lecture we step back and introduce the concepts of relative entropy and mutual information that. Average mutual information an overview sciencedirect. Its impact has been crucial to the success of the voyager missions to deep space. Entropy let x be a discrete random variable with alphabet xand probability mass function px px prfx xg. Unlike r 2, the mutual information values i of the underlying relationships in fig. More specifically, it quantifies the amount of information in units such as shannons, commonly called bits obtained about one random variable through observing the other random variable.
Khalid sayood, in introduction to data compression fourth edition, 2012. This is a consequence of the selfequitability of mutual information. It gives their definitions in terms of prob abilities, and a few. This is an important primitive in many learning problems including conditional independence testing, graphical model inference, causal strength estimation and timeseries problems. Average mutual information an overview sciencedirect topics. Entropy and mutual information department of electrical and.
Mar 04, 2014 indeed, mutual information is now routinely computed on continuous data in many realworld applications e. It is a dimensionless quantity with generally units of bits, and can be thought of as the reduction in uncertainty about one random variable given knowledge of another. Properties of mutual information and introduction to channel capacity tutorial of information theory and coding course by prof s. This document is an introduction to entropy and mutual information for discrete random variables. Mutual information is a distance between two probability distributions. Essentially it is a variation of the excess entropy so that it.
It was originally proposed by claude shannon in 1948 to find fundamental limits on signal processing and communication operations such as data compression, in a landmark paper titled a mathematical theory of communication. It is related to wellknown complexity measures like excess entropy or statistical complexity. Some of the properties of the discrete random variables carry over to the continuous case, but some do not. Estimation of entropy and mutual information uc berkeley statistics. Information theory is a mathematical approach to the study of coding of information along with the quantification, storage, and communication of information. Harvard seas es250 information theory 2 relative entropy and mutual information 2. Digital communication information theory tutorialspoint. Although he established no connections with mutual information, unwittingly sibsons measure corresponds to replacing in 15 the conditional relative entropy by the conditional r. September 25, 2008 abstract the objective of this note is to report some potentially useful mutual information inequalities. Renyi entropy, mutual information, and fluctuation properties. Application of information theory, lecture 2 joint. Equitability, mutual information, and the maximal information. Notice that alices actions give information about the weather in toronto.
This strategy bears a striking resemblance to regularization methods employed in abstract statistical inference grenander, 1981, generally known. Y in this section, we show that sibsons choice for mutual information of order satis. Learnedmiller department of computer science university of massachusetts, amherst amherst, ma 01003 september 16, 20 abstract this document is an introduction to entropy and mutual information for discrete random variables. This measure is used to evaluate different clusters given by the agglomerative. Given a tripartite probability distribution pxy z, this information measure is, however, hard to compute in general since a minimization has to be made over all. Entropy, relative entropy, and mutual information 1. On some properties of the mutual information between. Properties of mutual information and introduction to channel. Information theory in computer science october 6, 2010 lecture 3 lecturer.
Abstractwhile most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information, shannons entropy power inequality epi seems to be an exception. Unfortunately, shannon did not derive this formula, and rather just assumed it was the correct continuous analogue of discrete. English letter in a le, last digit of dowjones index, result of coin tossing, password with probability p1. The lipschitz continuity of mmse enables us to gauge the gap between the gaussian channel capacity and the mutual information achieved by a given input by comput. Information is the source of a communication system, whether it is analog or digital. Mutual information analysis is a generic sidechannel dis. Apart from the vi which possesses a fairly comprehensive characterization, less is known about the mutual information and various forms of the socalled normalized mutual information strehl and ghosh, 2002. Alice toronto goes jogging whenever it is not snowing heavily. In section 2 we give a brief revie w of mutual information from. Since p 1, p 2, p 3, and p 4 are 18, 14, 12, and 18, respectively, we. Lecture 1 introduction 2 handouts lecture outline goals and mechanics of the class notation entropy.
Pdf properties of persistent mutual information and. Our properties management department can assist you with questions and about diocesan and parish properties and insurance coverage. Isit2007, nice, france, june 24 june 29, 2007 of mutual information. Nov 4, 2014 iftach haitner tau application of information theory, lecture 2 nov 4, 2014 1 26. Relative entropy and mutual information relationship between entropy and mutual information chain rules for entropy, relative entropy and mutual information 2 inequalities in information theory jensen inequality and its consequences log sum inequality and its applications dataprocessing inequality su. Though the abstract suggestion that mutual information plays a role in object recognition may not be new, to date no concrete representations or ef. Prasang upadhyaya 1 introduction in the previous lecture we looked at the application of entropy to derive inequalities that involved counting.
A simple proof of the entropypower inequality via properties of mutual information olivier rioul dept. Mutual information rate of nonstationary statistical. Approximating mutual information by maximum likelihood. Differential entropy also referred to as continuous entropy is a concept in information theory that began as an attempt by shannon to extend the idea of shannon entropy, a measure of average surprisal of a random variable, to continuous probability distributions. Yao xie, ece587, information theory, duke university.
Downloadswingle2012 mutual information, and fluctuation properties of fermi liquids. Thus, differential entropy does not share all properties of discrete entropy. A new measure for conditional mutual information and its properties. A simple proof of the entropypower inequality via properties. The persistent mutual information pmi is a complexity measure for stochastic processes. Estimators and properties arman rahimzamani and sreeram kannan department of electrical engineering, university of washington, seattle october 16, 2017 abstract the conditional mutual information ix. Oct 11, 2018 mutual information is one of many quantities that measures how much one random variables tells us about another. Functional properties of minimum meansquare error and mutual. Information theory georgia institute of technology. Property, shannons source coding theorem and shannons channel coding theorem. If we consider an event, there are three conditions of occurrence. Information theoretic measures for clusterings comparison. The mutual information rate mir is proposed in information theory, which has been introduced into signal processing society, such as blind source separation, stochastic and deterministic separation, waveform design, blind deconvolution of the seismic data, measuring the interdependence of the neuroscience signals. Yz measures the average information that x and y contain about each other given z.
Quantum conditional mutual information and approximate. A property of the intrinsic mutual information crypto. Calculation of channel capacity for different information channel 26. Correlation is a linear distance between two random variables you can have a mutual information between any two probabilities defined for a set of symbols, while you cannot have a correlation between symbols that cannot naturally be mapped into a rn space. Properties of mutual information and introduction to. Pdf mutual information to assess structural properties. I we want to quantify the information provided by each possible outcome. We present some new results on the nonparametric estimation of entropy and mutual information. A new measure for conditional mutual information and its.