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Topic: Pairwise independence

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Statistical independence

Normally distributed and uncorrelated does not imply independent

Conditional expectation

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	Statistical independence - Wikipedia, the free encyclopedia
		Similarly, two random variables are independent if the conditional probability distribution of either given the observed value of the other is the same as if the other's value had not been observed.
		If any two of a collection of random variables are independent, they may nonetheless fail to be mutually independent; this is called pairwise independence.
		Independence can be seen as a special kind of conditional independence, since probability can be seen as a kind of conditional probability given no events.
	en.wikipedia.org /wiki/Statistical_independence (903 words)

	The cardinal-weighted pairwise comparison method (Site not responding. Last check: 2007-11-07)
		It is based on Condorcet’s method of pairwise comparison, but in addition to asking voters to rank the candidates in order of preference, this method also asks them to rate the candidates, for example on a scale from 0 to 100.
		Cardinal pairwise minimizes this barrier to entry, in that the entry of a new candidate is extremely unlikely to affect the result in retrograde to the will of his would-be supporters.
		In this respect, the cardinal pairwise version of the counterstrategy is preferable, in that it does not alter the direction of any pairwise defeats, and therefore will not interfere with the identification of a Condorcet winner.
	fc.antioch.edu /~james_green-armytage/cwp13.htm (4474 words)

	Statistical independence (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-11-07)
		In probability theory, to say that two events are independent intuitively means that knowing whether or not one of them occurs makes it neither more probable nor less probable that the other occurs.
		Similarly, when we assert that two random variables are independent, we intuitively mean that knowing something about the value of one of them does not yield any information about the value of the other.
		Two random variables X and Y are independent iff for any numbers a and b the events [''X'' ≤ a] (the event of X being less than or equal to a) and [''Y'' ≤ b] are independent events as defined above.
	statistical-independence.iqnaut.net.cob-web.org:8888 (933 words)

	Pairwise independence (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-11-07)
		In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent.
		Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not independent.
		X, Y, and Z are not independent (since the values of any two determine the value of the third).
	publicliterature.org.cob-web.org:8888 /en/wikipedia/p/pa/pairwise_independence.html (127 words)

	Independence (Site not responding. Last check: 2007-11-07)
		Independence is frequently invoked as a modeling assumption, and moreover, probability itself is based on the idea of independent replications of the experiment.
		General independence of a collection of events is much stronger than mere pairwise independence of the events in the collection.
		is a vector of independent indicator random variables that specifies the states of all of the components.
	www.math.uah.edu.cob-web.org:8888 /stat/prob/Independence.xhtml (2736 words)

	Statistical independence (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-11-07)
		In probability theory, when we assert that two events are independent, we intuitively mean that knowing whether or not one of them occurred makes it neither more probable nor less probable that the other occurred.
		Another typical example of two independent variables is given by repeating an experiment: roll a die twice, let X be the number you get the first time, and Y the number you get the second time.
		If we have more than two events, then pairwise independence is insufficient to capture the intuitive sense of independence.
	www.fact-index.com.cob-web.org:8888 /s/st/statistical_independence_1.html (333 words)

	Borel-Cantelli Lemma
		Note that no assumption of independence is required.
		are independent and the sum of their probabilities diverges to infinity, then the probability that infinitely many of them occur is 1.
		(The assumption of independence can be weakened to pairwise independence, but in that case the proof is more difficult.)
	www.ebroadcast.com.au /lookup/encyclopedia/bo/Borel-Cantelli_Lemma.html (226 words)

	Randomized Methods in Computation (lecture summaries)
		Motivated by the problem of approximating the average of a function, we present stronger bounds than Chebyshev's inequality: moving from pairwise independent sampling to $2k$-wise independent sampling and then to totally independent sampling makes the the error probability exponentially vanishing (in the amount of independence).
		In the case of totally independent random variables the probability space is at least of the size $S^m$, where $m$ is the number of variables in the sequence and $S$ is the sample space for an individual variable.
		In contrast, we present a construction for pairwise independent sample space of the size $\max(S,m)^2$ and its extension to the $k$-wise independent case.
	www.wisdom.weizmann.ac.il /~oded/rnd-sum.html (1261 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		It has been shown that pairwise independence is equivalent to mutual independence in this class.
		When the infinitely divisible variables contain no normal component (in particular, when they are discrete), the cumulant of order (2,2) can be used as a measure of pairwise dependence; when a normal component is present, the appropriate measure also involves the covariance.
		Results for testing independence of infinitely divisible random variables are discussed.
	www.uic.edu /classes/idsc/ids594/my_abstracts/abstr20.html (110 words)

	Analysis Seminar Abstracts (Site not responding. Last check: 2007-11-07)
		The Borel-Cantelli Lemma gives the probability that a sequence of events occurs infinitely often: if the sum of the probabilities is finite, the probability is 0, if it diverges and the events are totally independent, the probability is 1.
		Total independence has been weakened significantly to ‘pairwise independence’ and further, at the cost of sacrificing probability 1 to positive probability, to ‘pairwise quasi-independence’.
		This has a natural interpretation in the theory of Diophantine approximation and the general Borel-Cantelli Lemma has led to new proofs and generalisations of Khintchine's theorem on the size of sets of real points close to the rationals.
	www.maths.gla.ac.uk /~lw/abstracts.html (450 words)

	Themis' corner of the Internet: Research
		Maximum entropy reconstruction from a number of aggregates is performed based on the assumption that the aggregates of interest are pairwise independent.
		Any data value that violates the pairwise independence assumption, will induce a larger reconstruction error than one that does not, and this data value is potentially interesting.
		The main advantage of our approach is that, contrary to previous work in the area, the deviations are reported without any need for parameter setting or fine tuning, and without any prior knowledge of the domain.
	www.cs.toronto.edu /~themis/researchstatementp.html (904 words)

	Statistics Glossary - Probability (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-11-07)
		Two events are independent if the occurrence of one of the events gives us no information about whether or not the other event will occur; that is, the events have no influence on each other.
		a) A and B are independent; A and C are independent and B and C are independent (pairwise independence);
		The Geometric distribution is related to the Binomial distribution in that both are based on independent trials in which the probability of success is constant and equal to p.
	www.cas.lancs.ac.uk.cob-web.org:8888 /glossary_v1.1/prob.html (3540 words)

	[No title]
		The method is based on the remarkable fact that in the class of infinitely divisible distributions, as in the Gaussian family, mutual independence reduces to pairwise independence, but the measure of dependence is the cosquariance rather than the covariance.
		they are pairwise independent, and they are pairwise independent iff.
		they are pairwise independent, and they are pairwise indep.
	www.uic.edu /classes/idsc/ids594/research/BBC/talks/IDtalk.doc (1513 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Min-wise independence is a recently introduced notion of limited independence, similar in spirit to pairwise independence.
		Here we show that approximate min-wise independence allows similar uses, by presenting a derandomization of the RNC algorithm for approximate set cover due to S. Rajagopalan and V. Vazirani.
		The multi-cover case leads us to discuss the concept of k-minima-wise independence, a natural counterpart to k-wise independence.
	www.eecs.harvard.edu /~michaelm/NEWWORK/abstracts/minwise-derand.html (90 words)

	mutual& pairwise independence (Site not responding. Last check: 2007-11-07)
		For Random events we consider the notions of a mutual independence and pairwise independence.
		Question: Are there sets of random events which are pairwise independent but not mutually independent?
		CONCLUSION: Pairwise independence of a given set of random events does not imply that these events are mutually independent.
	www.mrs.umn.edu /~sungurea/introstat/stat2611/independence.html (131 words)

	Melissa Cline: Recent Projects
		Objective: Analyze the information content of pairwise amino acid contacts, types of amino acids that tend to be close in protein structures.
		Tested sets of pairwise contacts for statistical independence, using chi-squared and G-tests.
		Developed method to estimate the expected mutual information of two quantities, given sample size and assuming pairwise independence.
	www.bioinfocus.com /projects.html (465 words)

	hw2sol (Site not responding. Last check: 2007-11-07)
		are determined by distinct independent rolls, and are therefore independent.
		Therefore, the triple does not factor, and we have only pairwise independence, not independence.
		These are independent events (so the probabilities multiply).
	www.engineering.usu.edu /classes/ece/6010/hw2sol/hw2sol.html (713 words)

	Proceedings of the American Mathematical Society (Site not responding. Last check: 2007-11-07)
		Convergence of sequences of pairwise independent random variables
		-algebra of a sequence of pairwise independent random variables may not be trivial, we have discovered that if such a sequence converges in probability or almost everywhere, then the limit has to be a constant.
		Robertson, J. and Womack, J. A pairwise independent stationary stochastic process.
	www.mathaware.org /proc/2004-132-04/S0002-9939-03-07236-8/home.html (165 words)

	Problem Set #1 (Site not responding. Last check: 2007-11-07)
		In class, we mentioned the defintion of pairwise independence as well, but didn't write it on the board.
		Give an example of a collection of events which are pairwise independent, but not mutually independent.
		Prove that no protocol exists which allows Alice to communicate M to Bob in such a way that Eve obtains no information about M. (To say that Eve learns nothing means that the probability of Alice saying S is independent of the message.
	www.cs.colorado.edu /~jrblack/class/csci7000/s03/hw/hw1.html (328 words)

	Pairwise Independence and Derandomization \| EECS at UC Berkeley (Site not responding. Last check: 2007-11-07)
		This set of notes gives several applications of the following paradigm.
		The first part is to design a probabilistic algorithm described by a sequence of random variables so that the analysis is valid assuming limited independence between the random variables.
		The second part is the design of a small probability space for the random variables such that they are somewhat independent of each other.
	www.eecs.berkeley.edu /Pubs/TechRpts/1995/5484.html (147 words)

	[No title]
		Recall that pairwise independence does not guarantee that a set of vector is independent, so a set of dependent vectors could have subsets that are linearly independent (Problem 3, C2M4).
		For the linear system below, do the following: (a) Express this system in the form , and use Matsolve to solve this matrix-vector equation.
		What set of vectors have you thereby shown is linearly independent or linearly dependent?
	www.dean.usma.edu /MATH/courses/ma371/MA371/Quiz02.doc (510 words)

	Pairwise independence - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-11-07)
		X, Y, and Z are not independent, since any of them is just the mod 2 sum of the other two, and so is completely determined by the other two.
		That is as far from independence as random variables can get.
		This page was last modified 06:22, 4 October 2006.
	en.wikipedia.org.cob-web.org:8888 /wiki/Pairwise_independence (147 words)

	Past Tjalling C. Koopmans Prizes, 1985-1999
		This paper proposes a diagnostic test for the adequacy of time series models, allowing for rather general model classes, involving possibly nonlinear parametric functions of past information that include both conditional heterogeneity and conditional duration specifications.
		The test is based on the estimated noise process, relying on the joint and marginal characteristic functions of pairs of noise variables at different time distances to test for pairwise independence.
		The test employs the (generalized) spectrum of these quantities and therefore does not require moment conditions, and applies under a suitable mixing condition.
	korora.econ.yale.edu /et/award/tck-past.htm (1018 words)

	Pairwise Independence and Derandomization - Luby, Wigderson (ResearchIndex) (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-11-07)
		Pairwise Independence and Derandomization - Luby, Wigderson (ResearchIndex) (via CobWeb/3.1 planetlab1.tamu.edu)
		This document uses CoBlitz to cache paper downloads.
		Luby and A. Wigderson, "Pairwise independence and derandomization," Tech.
	citeseer.ist.psu.edu.cob-web.org:8888 /luby95pairwise.html (455 words)

	[No title]
		Developed it into a notion of Conditional Independence relative to a function f which takes values into an Abelian Group If(.,..).
		Intersection ¡ I çÿ çÿ=Iª»±ó ¨Conclusions (2)¨Axioms 1-4 imply the equivalence of the Global, Local and Pairwise Markov Properties for our notion conditional independence relation If(.,..)) based on the result from [Pearl and Paz '87].
		We proved a natural generalization of the Hammersley-Clifford which allows us to factorize the function f over the cliques of an associated Markov Network which reflects the Conditional Independencies of subsets of variables with respect to f.
	www.cs.iastate.edu /~silvescu/papers/idfab/idfab.ppt (652 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Lectures 3,4 (Jan 14,16) -- Pairwise independence and maximal independent set, lecture notes.
		Related randomized algorithms courses with lecture notes on the web:
		Lecture notes from Luby and Wigderson on pairwise independence.
	www.cc.gatech.edu /~vigoda/RandAlgs_Course (395 words)

	Collection of Lecture Notes, Survey Papers, etc (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-11-07)
		Michael Luby & Avi Wigderson: Pairwise Independence and Derandomization
		Michael Luby & Avi Wigderson : Pairwise Independence and Derandomization
		Lecture Notes (in german) to the course Effiziente Algorithmen, Spring 1995, Technischen Universität München, scribed by Jens Ernst
	wwwcs.uni-paderborn.de.cob-web.org:8888 /fachbereich/AG/agmadh/WWW/english/scripts.html (2934 words)

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