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Rate Function
In mathematics — specifically, in large deviations theory — a rate function is a function used to quantify the probabilities of rare events. Such functions are used to formulate large deviation principles. A large deviation principle quantifies the asymptotic probability of rare events for a sequence of probabilities. A rate function is also called a Cramér function, after the Swedish probabilist Harald Cramér. Definitions Rate function An extended real-valued function I: X \to , +\infty/math> defined on a Hausdorff topological space X is said to be a rate function if it is not identically +\infty and is lower semi-continuous ''i.e.'' all the sub-level sets :\ \mbox c \geq 0 are closed in X. If, furthermore, they are compact, then I is said to be a good rate function. A family of probability measures (\mu_)_ on X is said to satisfy the large deviation principle with rate function I: X \to , +\infty) (and rate 1/\delta) if, for every closed set F \subseteq X a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contraction Principle (large Deviations Theory)
In mathematics — specifically, in large deviations theory — the contraction principle is a theorem that states how a large deviation principle on one space "pushes forward" (via the pushforward of a probability measure) to a large deviation principle on another space ''via'' a continuous function. Statement Let ''X'' and ''Y'' be Hausdorff topological spaces and let (''μ''''ε'')''ε''>0 be a family of probability measures on ''X'' that satisfies the large deviation principle with rate function ''I'' : ''X'' → , +∞ Let ''T'' : ''X'' → ''Y'' be a continuous function, and let ''ν''''ε'' = ''T''∗(''μ''''ε'') be the push-forward measure of ''μ''''ε'' by ''T'', i.e., for each measurable set/event ''E'' ⊆ ''Y'', ''ν''''ε''(''E'') = ''μ''''ε''(''T''−1(''E'')). Let :J(y) := \inf \, with the convention that the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fields Institute
The Fields Institute for Research in Mathematical Sciences, commonly known simply as the Fields Institute, is an international centre for scientific research in mathematical sciences. It is an independent non-profit with strong ties to 20 Ontario universities, including the University of Toronto, where it occupies a purpose-built building on the St. George campus. Fields was established in 1992, and was briefly based at the University of Waterloo before relocating to Toronto in 1995. The institute is named after Canadian mathematician John Charles Fields, after whom the Fields Medal is also named. Fields' name was given to the institute in recognition of his contributions to mathematics and his work on behalf of high level mathematical scholarship in Canada. As a centre for mathematical activity, the institute brings together mathematicians from Canada and abroad. It also supports the collaboration between professional mathematicians and researchers in other domains, such as s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moderate Deviation Principle
Moderate is an ideological category which entails centrist views on a liberal-conservative spectrum. It may also designate a rejection of radical or extreme views, especially in regard to politics and religion. Political position Canada At the federal level in Canada as of 2024, there are five active political parties who have seats in the House of Commons, for which most of them have a wide range of goals and political opinions, that differ between each others. Per definition, where "political moderate" is used, in a specific context to being far conservative, the Conservative Party of Canada could be used as a representation. However, we can now see that those beliefs might contain "inverted" or different effects-opinions. If we could measure them from a "political spectrum" point of view, the variations for instance, conservatism, who tend to be defined in the same way toward being resistant with the idea of future changes, is not always the case. In parallel, liberali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extreme Value Theory
Extreme value theory or extreme value analysis (EVA) is the study of extremes in statistical distributions. It is widely used in many disciplines, such as structural engineering, finance, economics, earth sciences, traffic prediction, and Engineering geology, geological engineering. For example, EVA might be used in the field of hydrology to estimate the probability of an unusually large flooding event, such as the 100-year flood. Similarly, for the design of a breakwater (structure), breakwater, a coastal engineer would seek to estimate the 50 year wave and design the structure accordingly. Data analysis Two main approaches exist for practical extreme value analysis. The first method relies on deriving block maxima (minima) series as a preliminary step. In many situations it is customary and convenient to extract the annual maxima (minima), generating an ''annual maxima series'' (AMS). The second method relies on extracting, from a continuous record, the peak values reac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space
In computer science, a state space is a discrete space representing the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the toy problem Vacuum World has a discrete finite state space in which there are a limited set of configurations that the vacuum and dirt can be in. A "counter" system, where states are the natural numbers starting at 1 and are incremented over time has an infinite discrete state space. The angular position of an undamped pendulum is a continuous (and therefore infinite) state space. Definition State spaces are useful in computer science as a simple model of machines. Formally, a state space can be defined as a tuple [''N'', ''A'', ''S'', ''G''] where: * ''N'' is a Set (mathematics), set of states * ''A'' is a set of arcs connecting the states * ''S'' is a nonempty subset of ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a . A probability space consists of three elements:Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press. # A '' sample space'', \Omega, which is the set of all possible outcomes of a random process under consideration. # An event space, \mathcal, which is a set of events, where an event is a subset of outcomes in the sample space. # A '' probability function'', P, which assigns, to each event in the event space, a probability, which is a number between 0 and 1 (inclusive). In order to provide a model of probability, these elements must satisfy probability axioms. In the example of the throw of a standard die, # The sample space \Omega is typically the set \ where each element in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa. The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the th-order cumulant of their sum is equal to the sum of their th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property. Just as for moments, where ''joint moments'' are used for collections of random variables ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Conjugate
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). The convex conjugate is widely used for constructing the dual problem in optimization theory, thus generalizing Lagrangian duality. Definition Let X be a real topological vector space and let X^ be the dual space to X. Denote by :\langle \cdot , \cdot \rangle : X^ \times X \to \mathbb the canonical dual pairing, which is defined by \left\langle x^*, x \right\rangle \mapsto x^* (x). For a function f : X \to \mathbb \cup \ taking values on the extended real number line, its is the function :f^ : X^ \to \mathbb \cup \ whose value at x^* \in X^ is defined to be the supremum: :f^ \left( x^ \right) := \sup \left\, or, equivalently, in terms of the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentially Equivalent Measures
In mathematics, exponential equivalence of measures is how two sequences or families of probability measures are "the same" from the point of view of large deviations theory. Definition Let (M,d) be a metric space and consider two one-parameter families of probability measures on M, say (\mu_\varepsilon)_ and (\nu_\varepsilon)_. These two families are said to be exponentially equivalent if there exist * a one-parameter family of probability spaces (\Omega,\Sigma_\varepsilon,P_\varepsilon)_, * two families of M-valued random variables (Y_\varepsilon)_ and (Z_\varepsilon)_, such that * for each \varepsilon >0, the P_\varepsilon-law (i.e. the push-forward measure) of Y_\varepsilon is \mu_\varepsilon, and the P_\varepsilon-law of Z_\varepsilon is \nu_\varepsilon, * for each \delta >0, "Y_\varepsilon and Z_\varepsilon are further than \delta apart" is a \Sigma_\varepsilon- measurable event, i.e. ::\big\ \in \Sigma_, * for each \delta >0, ::\limsup_\, \varepsilon \log P_\varepsilon \big( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional (mathematics)
In mathematics, a functional is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author). * In linear algebra, it is synonymous with a linear form, which is a linear mapping from a vector space V into its field of scalars (that is, it is an element of the dual space V^*) "Let ''E'' be a free module over a commutative ring ''A''. We view ''A'' as a free module of rank 1 over itself. By the dual module ''E''∨ of ''E'' we shall mean the module Hom(''E'', ''A''). Its elements will be called functionals. Thus a functional on ''E'' is an ''A''-linear map ''f'' : ''E'' → ''A''." * In functional analysis and related fields, it refers to a mapping from a space X into the field of real or complex numbers. "A numerical function ''f''(''x'') defined on a normed linear space ''R'' will be called a ''functional''. A functional ''f''(''x'') is said to be ''linear'' if ''f''(α''x'' + β''y'') = α''f''(''x'') + β ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tilted Large Deviation Principle
In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by exponential tilting, i.e. integration against an exponential functional. It can be seen as an alternative formulation of Varadhan's lemma. Statement of the theorem Let ''X'' be a Polish space (i.e., a separable, completely metrizable topological space), and let (''μ''''ε'')''ε''>0 be a family of probability measures on ''X'' that satisfies the large deviation principle with rate function ''I'' : ''X'' → , +∞ Let ''F'' : ''X'' → R be a continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
