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Categorical Probability
In mathematics, the term categorical probability denotes a collection of category theory, category-theoretic approaches to probability theory and related fields such as statistics, information theory and ergodic theory. The earliest ideas in the field were developed independently by William Lawvere, Lawvere and by Nikolai Chentsov, Chentsov, where they defined a version of what we today call the category of Markov kernels, and appeared in 1962 and 1965 respectively.N. N. Chentsov, The categories of mathematical statistics, Dokl. Akad. SSSR 164, 1965. Some of the most widely used structures in the theory are *The category of measurable spaces; *Markov categories such as the category of Markov kernels; *Probability monads such as Giry monad. References *https://ncatlab.org/nlab/show/category-theoretic+approaches+to+probability+theory *https://golem.ph.utexas.edu/category/2024/07/imprecise_probabilities_toward.html#more Further reading * https://ncatlab.org/nlab/show/Giry+mona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure in information theory is entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a die (with six equally likely outcomes). Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. Important sub-fields of information theory include s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Lawvere
Francis William Lawvere (; born February 9, 1937) is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics. Biography Lawvere studied continuum mechanics as an undergraduate with Clifford Truesdell. He learned of category theory while teaching a course on functional analysis for Truesdell, specifically from a problem in John L. Kelley's textbook ''General Topology''. Lawvere found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell and Walter Noll. Truesdell supported Lawvere's application to study further with Samuel Eilenberg, a founder of category theory, at Columbia University in 1960. Before completing the Ph.D. Lawvere spent a year in Berkeley as an informal student of model theory and set theory, following lectures by Alfred Tarski and Dana Scott. In his first teaching position at Reed College he was instructed to devise courses in calculus and abstract algebra from a foundational perspec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolai Chentsov
Nikolai Nikolaevich Chentsov (19 February 1930 – 5 July 1992), written also as Nikolaj Nikolajevič Čencov, or Nikolai Chentsov, N. N. Čencov for short, was a Soviet mathematician who made important contributions to stochastic processes, convergence theory and information geometry. Education and career Chentsov was born in Moscow and showed an early interest in mathematics. In the eighth grade (1944), he joined a school mathematics club for high school students who had just returned from evacuation at the Faculty of Mechanics and Mathematics of the Moscow State University, which was led by Alexander Kronrod and Olga Ladyzhenskaya, who was at that time a graduate student. Chentsov continued his studies in the circle under the guidance of a graduate student Eugene Dynkin, who later became his thesis advisor. In 1947, Chentsov entered the Faculty of Mechanics and Mathematics of Moscow State University and was very actively involved in leading a mathematical club for schoolchil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Markov Kernels
In mathematics, the category of Markov kernels, often denoted Stoch, is the category whose objects are measurable spaces and whose morphisms are Markov kernels. It is analogous to the category of sets and functions, but where the arrows can be interpreted as being stochastic. Several variants of this category are used in the literature. For example, one can use subprobability kernels instead of probability kernels, or more general s-finite kernels. Also, one can take as morphisms equivalence classes of Markov kernels under almost sure equality; see below. Definition Recall that a Markov kernel between measurable spaces (X,\mathcal) and (Y,\mathcal) is an assignment k:X\times\mathcal\to\mathbb which is measurable as a function on X and which is a probability measure on \mathcal. We denote its values by k(B, x) for x\in X and B\in\mathcal, which suggests an interpretation as conditional probability. The category Stoch has: * As objects, measurable spaces; * As morphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Measurable Spaces
In mathematics, the category of measurable spaces, often denoted Meas, is the category whose objects are measurable spaces and whose morphisms are measurable maps. This is a category because the composition of two measurable maps is again measurable, and the identity function is measurable. N.B. Some authors reserve the name Meas for categories whose objects are measure spaces, and denote the category of measurable spaces as Mble, or other notations. Some authors also restrict the category only to particular well-behaved measurable spaces, such as standard Borel spaces. As a concrete category Like many categories, the category Meas is a concrete category, meaning its objects are sets with additional structure (i.e. sigma-algebras) and its morphisms are functions preserving this structure. There is a natural forgetful functor :''U'' : Meas → Set to the category of sets which assigns to each measurable space the underlying set and to each measurable map the underlying function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Categories
Markov (Bulgarian, russian: Марков), Markova, and Markoff are common surnames used in Russia and Bulgaria. Notable people with the name include: Academics *Ivana Markova (born 1938), Czechoslovak-British emeritus professor of psychology at the University of Stirling *John Markoff (sociologist) (born 1942), American professor of sociology and history at the University of Pittsburgh *Konstantin Markov (1905–1980), Soviet geomorphologist and quaternary geologist Mathematics, science, and technology *Alexander V. Markov (1965-), Russian biologist *Andrey Markov (1856–1922), Russian mathematician *Vladimir Andreevich Markov (1871–1897), Russian mathematician, brother of Andrey Markov (Sr.) *Andrey Markov Jr. (1903–1979), Russian mathematician and son of Andrey Markov *John Markoff (born 1949), American journalist of computer industry and technology *Moisey Markov (1908–1994), Russian physicist Performing arts *Albert Markov, Russian American violinist, composer *Alexand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Monad
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giry Monad , from The Phantom of the Opera
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Giry may refer to: People * Arthur Giry (1848–1899), French historian * Louis Giry (1596–1665), French lawyer, translator and writer * Odet-Joseph Giry (1699–1761), French clergyman * Sylvie Giry-Rousset (born 1965), French cross-country skier Places * Giry, Nièvre, France Fictional characters * Madame Giry, from The Phantom of the Opera * Meg Giry Meg is a feminine given name, often a short form of Megatron, Megan, Megumi (Japanese), etc. It may refer to: People * Meg (singer), a Japanese singer * Meg Cabot (born 1967), American author of romantic and paranormal fiction * Meg Burton Cahil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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