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Topic: Kolmogorov axioms

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	Andrey Kolmogorov
		Kolmogorov was one of the broadest of this century's mathematicians.
		Kolmogorov went on to study the motion of the planets and turbulent fluid flows, later publishing two papers in 1941 on turbulence that even today are of fundamental importance.
		Kolmogorov's notion of complexity is a measure of randomness, one that is closely related to Claude Shannon's entropy rate of an information source.
	www.exploratorium.edu /complexity/CompLexicon/kolmogorov.html (330 words)

	Kolmogorov biography
		Kolmogorov's mother had been on a journey from the Crimea back to her home in Tunoshna near Yaroslavl and it was in the home of his maternal grandfather in Tunoshna that Kolmogorov spent his youth.
		Kolmogorov was appointed a professor at Moscow University in 1931.
		If Kolmogorov made a major contribution to Hilbert's sixth problem, he completely solved Hilbert's Thirteenth Problem in 1957 when he showed that Hilbert was wrong in asking for a proof that there exist continuous functions of three variables which could not be represented by continuous functions of two variables.
	www-gap.dcs.st-and.ac.uk /~history/Biographies/Kolmogorov.html (2094 words)

	Andrey Kolmogorov - Wikipedia, the free encyclopedia
		Andrey Nikolaevich Kolmogorov (Russian: Андре́й Никола́евич Колмого́ров) (April 25, 1903 - October 20, 1987) was a Soviet mathematician who made major advances in the fields of probability theory and topology.
		Kolmogorov quickly gained a reputation for wide-ranging erudition; as an undergraduate at Moscow State University in the early 1920s, he published his first research paper — not in any branch of mathematics but on landholding practices in fifteenth and sixteenth century Novgorod.
		The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography.
	en.wikipedia.org /wiki/Andrey_Kolmogorov (627 words)

	Andrey Kolmogorov Summary
		Kolmogorov was born in the town of Tambov in central Russia on April 25, 1903.
		Kolmogorov's father, a Russian agriculturist named Nikolai Kataev, was killed in World War I, and his mother, Mariya Kolmogorova—who was not married to Nikolai—died giving birth to their son in the town of Tambov on April 25, 1903.
		Kolmogorov, however, remained modest to the end, and, in line with his belief that a mathematician could no longer conduct valuable research after the age of 60, he retired in 1963, spending 20 years teaching high school.
	www.bookrags.com /Andrey_Kolmogorov (2461 words)

	Probability axioms
		The probability $P$ of some event $E$ (denoted $P(E)$ ) is defined with respect to a "universe" or sample space $S$ of all possible elementary events in such a way that $P$ must satisfy the Kolmogorov axioms.
		Alternatively, a probability can be interpreted as a measure on a sigma-algebra of subsets of the sample space, those subsets being the events, such that the measure of the whole set equals 1.
		These axioms are known as the Kolmogorov axioms, after Andrey Kolmogorov who developed them.
	www.ebroadcast.com.au /lookup/encyclopedia/ax/Axioms_of_probability.html (464 words)

	Andrei Nikolaevich Kolmogorov
		Kolmogorov's 1941 theory is presented in a novel fashion with emphasis on symmetries (including scaling transformations) which are broken by the mechanisms producing the turbulence and restored by the chaotic character of the cascade to small scales.
		Kolmogorov was Dean of the MechMath Faculty (1954-1958) and Head of the Mathematics Division (1954-1956 and 1978-till his death in 1987).
		Kolmogorov was President of the Moscow Mathematical Society from 1964 to 1966 and from 1973 to 1985.
	www.kolmogorov.com /Kolmogorov.html (4215 words)

	Interpretations of Probability (Stanford Encyclopedia of Philosophy)
		The non-negativity and normalization axioms are largely matters of convention, although it is non-trivial that probability functions take at least the two values 0 and 1, and that they have a maximal value (unlike various other measures, such as length, volume, and so on, which are unbounded).
		Kolmogorov comments that infinite probability spaces are idealized models of real random processes, and that he limits himself arbitrarily to only those models that satisfy countable additivity.
		This axiom is the cornerstone of the assimilation of probability theory to measure theory.
	plato.stanford.edu /entries/probability-interpret (15152 words)

	[No title]
		Kolmogorov complexity is a modern notion of randomness dealing with the quantity of information in individual objects; that is, pointwise randomness rather than average randomness as produced by a random source.
		It was proposed by A.N. Kolmogorov in 1965 to quantify the randomness of individual objects in an objective and absolute manner.
		`Kolmogorov' complexity was earlier proposed by Ray Solomonoff in a Zator Technical Report in 1960 and in a long paper in Information and Control in 1964.
	homepages.cwi.nl /~paulv/kolmogorov.html (824 words)

	20th WCP: Coherence and Epistemic Rationality
		The claim that degrees of confidence should satisfy the probability axioms is most often defended by appealing to the so-called Dutch Book argument, which was first presented by Ramsey in his famous paper "Truth and Probability".
		If rationality requires satisfying the probability axioms and the theorem is true, then we must assign it probability one now, but this would involve being fully confident without sufficient evidence.
		What this seems to show is not that failure to satisfy the probability axioms involves making a logical error in reasoning from the evidence, but rather that degrees of confidence are not always taken as fair betting quotients.
	www.bu.edu /wcp/Papers/TKno/TKnoVine.htm (3166 words)

	Classical Probability Theory and Learning from Experience
		These new axioms make it easier to join probability theory to propositional logic than Kolmogorov's axioms and are more elementary and simpler than his in several respects - as shall be shown.
		I have now proved all of Kolmogorov's axioms for the finite case: A1 follows from T3; A2 is AxA; and A3 is T8.
		The advantage and use of axioms is that one can use them to prove the theorems one needs - and having given a valid proof one knows that any objection against the theorem must be directed against the axioms, for the theorem was proved to follow from them.
	www.xs4all.nl /~maartens/logic/CPTandLFE.htm (4177 words)

	Probability axioms (Site not responding. Last check: 2007-11-06)
		The probability P of some event E, denoted P(E), is defined with respect to a "universe", or sample space Ω, of all possible elementary events in such a way that P must satisfy the Kolmogorov axioms.
		Alternatively, a probability can be interpreted as a measure on a σ-algebra of subsets of the sample space, those subsets being the events, such that the measure of the whole set equals 1.
		The following three axioms are known as the Kolmogorov axioms, after Andrey Kolmogorov who developed them.
	www.tocatch.info /en/Axioms_of_probability.htm (548 words)

	Week 2
		Axiom 1: The probability of an event is a nonnegative real number; that is, P(A) for any subset A of S.
		The interpretation of a continuous probability function is complicated by the fact that the Kolmogorov axioms need to be modified.
		Axiom 3: If A is any event defined on S, f f is a continuous probability function, f(y) is not the probability that the outcome of the experiment is y.
	isolatium.uhh.hawaii.edu /m321/w2/probfun.html (535 words)

	Harald Cramér, Mathematical Methods of Statistics
		A superior definition was however provided by the polymathic Andrei Kolmogorov in 1933; it is the one used by Cramér, and by all other competent authorities.
		Namely, we define a probability space as a measurable space in which the measure satisfies certain requirements, the ``axioms of probability theory'' --- for instance, the measure of the entire space must be one.
		Kolmogorov's axioms exhaust the meaning of purely mathematical probability.
	cscs.umich.edu /~crshalizi/reviews/cramer-on-math-stat (1151 words)

	Kolmogorov space information - Search.com (Site not responding. Last check: 2007-11-06)
		axiom fits in with the rest of the separation axioms.
		Topological spaces X and Y are Kolmogorov equivalent when their Kolmogorov quotients are homeomorphic.
		(R) usually denotes the Kolmogorov quotient, the set of equivalence classes of square integrable functions which differ on sets of measure zero, rather than simply the vector space of square integrable functions which the notation suggests.
	c10-ss-1-lb.cnet.com /reference/Kolmogorov_space (1195 words)

	University of South Carolina: CSCE 582 {=STAT 582} Lecture Log
		Three historically important interpretations in which the axioms of Kolmogorov are true (i.e., three models of them): classical, frequentist, and subjective.
		Detailed presentation of the classical approach, including proof of the three properties that are the axioms of Kolmogorov.
		Definition of conditional probability as an additional axiom of probability, which is true in the three major models of probability (classical, frequentist, subjective).
	www.cse.sc.edu /~mgv/csce582f03/log/index.html (2357 words)

	Probability
		It involves the argument of the long run and is compatible with the experience that the relative frequency seems to stabilize when the number of trials is increased.
		The most important advantage may be the target of the relation: the Kolmogorov axioms for probability and the whole theory build upon this formalization of probability.
		However, within the Kolmogorov axioms, randomness is not associated with single sequences and will not be a suitable criterion for selecting some sequences in favor of others: such a selection can only be made with respect to an actual application.
	random.mat.sbg.ac.at /~ste/dipl/node6.html (1157 words)

	Kolmogorov Complexity and Solomonoff Induction (Site not responding. Last check: 2007-11-06)
		Kolmogorov defined the complexity of a string x as the length of its shortest description p on a universal Turing machine U (K(x)=min{l(p):U(p)=x}).
		The Kolmogorov Complexity mailing list is a moderated mailing list intended for people in Computer Sciences, Statistics, Mathematics, and other areas or disciplines with interests in Kolmogorov Complexity and Solomonoff Induction.
		A gentle tour of Kolmogorov complexity and its applications, presented by Chris Hillman at UIUC, April 1999, in PostScript.
	www.hutter1.de /kolmo.htm (1308 words)

	Probability Measure
		Axioms 1 and 2 are really just a matter of convention; we choose to measure the probability of an event with a number between 0 and 1 (as opposed, say, to a number between −5 and 7).
		In all these cases, the size of a set that is composed of countably many disjoint pieces is the sum of the sizes of the pieces.
		The countable additivity axiom holds because of a basic property of integrals, which we will assume.
	www.math.uah.edu /statold/prob/Probability.xhtml (3005 words)

	Appendix 2 of Eells (Site not responding. Last check: 2007-11-06)
		These three conditions are the probability axioms, also called "the Kolmogorov axioms" (for Kolmogorov 1933).
		A function Pr that satisfies the axioms, relative to an algebra B, is said to be a probability function on B -- that is, with "domain" B (that is, the set of propositions of B) and range the closed interval [0,1].
		Various interpretations of probability (such as frequency, degree of belief, and partial logical entailment interpretations) are discussed in Chapter 1; here, the focus is on the formal calculus.
	philosophy.wisc.edu /sober/eells-pr.html (1104 words)

	Antimeta: Sets of Worlds
		It's a consequence of any form of the axioms of probability that any tautologically false proposition must get probability zero, but depending on which form of the axioms one takes, this won't have to be the case even for first-order logically false propositions, much less metaphysically impossible ones.
		The reason I say "depending on which form of the axioms one takes" is because there are several different forms that are generally considered to be equivalent.
		For instance, Kolmogorov says that probability (1) is a non-negative function on sets of possible worlds (2) that assigns the value 1 to the complete set (3) and is (finitely or countably) additive on disjoint sets of worlds.
	www.ocf.berkeley.edu /~easwaran/blog/2006/03/sets_of_worlds.html (1838 words)

	Learn more about Andrey Nikolaevich Kolmogorov in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		Learn more about Andrey Nikolaevich Kolmogorov in the online encyclopedia.
		Hint: Play with putting spaces before and after your words to see the different results you get.
		Andrey Nikolaevich Kolmogorov (April 25, 1903 - October 20, 1987) was a Russian mathematician who made major advances in the fields of probability theory and topology.
	www.onlineencyclopedia.org /a/an/andrey_nikolaevich_kolmogorov.html (198 words)

	Probability - Kolmogorov
		The (first 3) Kolmogorov axioms and the Bayesian axioms can be derived from each other (NO!), and we sometimes blur the distinction.
		The Bayesian approach always states probablilites in terms of what is already known.
		Kolmogorov from Bayes: (1) P(SS) = P(SS) + P(S)> = 0 since P(is in [0,1].
	www.quantum.bowmain.com /probability.htm (361 words)

	Reference.com/Encyclopedia/Probability axioms
		, of all possible elementary events in such a way that P must satisfy the Kolmogorov axioms.
		In the case that the sample space is finite or countably infinite, a probability function can also be defined by its values on the elementary events
		We have an underlying set Ω, a sigma-algebra F of subsets of Ω, and a function P assigning real numbers to members of F.
	www.reference.com /browse/wiki/Kolmogorov_axioms (602 words)

	Information
		You suggest that even if BT is derivable from the standard probability axioms one could construct an alternate system which takes BT as an axiom and drops one or more of the other standard axioms.
		Using Bayes' Theorem as a starting point "and then add[ing] other axioms as seem necessary" still sounds weird to me. I think that my worry is that the expressions appearing in BT require more fundamental axioms to define their meaning.
		It hadn't been clear in my mind that the "givens" are not only axioms and "sentence generation mechanisms" and "inference rules" (my terms, do they bear some relation to your vocabulary?) but also (in all cases?) "atomic propositions" and arbitrary "value" assignments.
	serendip.brynmawr.edu /local/scisoc/information/baker.html (7627 words)

	[MLUG - DISCUSSION] Kolmogorov's Laws of Probability (was statistical inference) (Site not responding. Last check: 2007-11-06)
		By the way, my suspicion is that Bell's inequalities, and their failure, is the proof that Kolmogorov's axioms don't apply.
		I really don't understand it at all, but as best I can tell the do some experiment with two electrons having spin up and spin down, which correspond to having a cow or a horse in the field.
		Well I tried reading "no hidden variables" on wikopedia, and this is what I thought I got from it.
	mlug.missouri.edu /pipermail/discussion/2006-June/021107.html (588 words)

	Probability 'Cepts
		Sometimes mathematicians will start with a proposition or domain of mathematical reasoning they wish to establish and work backwards to identify the primitives and axioms that would be sufficient to deduce the proposition or establish the domain.
		Andrey Kolmogorov, first proposed the axioms that are the starting point for the logical system that is modern probability theory.
		This axiom says that the probability of all the events in the sample space is one.
	www.uri.edu /cels/cpla/marsh/cpl526/probabilityCepts.htm (1340 words)

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