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Stochastic Kernel
In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space. Formal definition Let (X,\mathcal A) and (Y,\mathcal B) be measurable spaces. A ''Markov kernel'' with source (X,\mathcal A) and target (Y,\mathcal B) is a map \kappa : \mathcal B \times X \to ,1/math> with the following properties: # For every (fixed) B \in \mathcal B, the map x \mapsto \kappa(B, x) is \mathcal A-measurable # For every (fixed) x \in X, the map B \mapsto \kappa(B, x) is a probability measure on (Y, \mathcal B) In other words it associates to each point x \in X a probability measure \kappa(dy, x): B \mapsto \kappa(B, x) on (Y,\mathcal B) such that, for every measurable set B\in\mathcal B, the map x\mapsto \kappa(B, x) is measurable with respect to the \sigma-algebra \mathcal A. Examples Simple r ... [...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. Typically these axioms formalise probability in terms of a probability space, which assigns a 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. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counting Measure
In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity \infty if the subset is infinite. The counting measure can be defined on any measurable space (that is, any set X along with a sigma-algebra) but is mostly used on countable sets. In formal notation, we can turn any set X into a measurable space by taking the power set of X as the sigma-algebra \Sigma; that is, all subsets of X are measurable sets. Then the counting measure \mu on this measurable space (X,\Sigma) is the positive measure \Sigma \to ,+\infty/math> defined by \mu(A) = \begin \vert A \vert & \text A \text\\ +\infty & \text A \text \end for all A\in\Sigma, where \vert A\vert denotes the cardinality of the set A. The counting measure on (X,\Sigma) is σ-finite if and only if the space X is countable In mathematics, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur. If the random variable can take on only a finite number of values, the “conditions” are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space. Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted E(X\mid Y) analogously to conditional probability. The function form is either denoted E(X\mid Y=y) or a separate function symbol such as f(y) is introduced with the meaning E(X\mid Y) = f(Y). Examples Example 1: Dice rolling Consider the roll of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Category
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 * A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. '' Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galton Board
The Galton board, also known as the Galton box or quincunx or bean machine, is a device invented by Sir Francis Galton to demonstrate the central limit theorem, in particular that with sufficient sample size the binomial distribution approximates a normal distribution. Among its applications, it afforded insight into regression to the mean or "reversion to mediocrity". Description The Galton board consists of a vertical board with interleaved rows of pegs. Beads are dropped from the top and, when the device is level, bounce either left or right as they hit the pegs. Eventually they are collected into bins at the bottom, where the height of bead columns accumulated in the bins approximate a bell curve. Overlaying Pascal's triangle onto the pins shows the number of different paths that can be taken to get to each bin. Large-scale working models of this device created by Charles and Ray Eames can be seen in the '' Mathematica: A World of Numbers... and Beyond'' exhibits permane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Distribution
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability q = 1-p. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/ yes/true/ one with probability ''p'' and failure/no/ false/zero with probability ''q''. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and ''p'' would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and ''p'' would be the probability of tails). In particular, unfair coins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads H and tails T) in a sample space (e.g., the set \) to a measurable space, often the real numbers (e.g., \ in which 1 corresponding to H and -1 corresponding to T). Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory, a random var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent And Identically Distributed Random Variables
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as ''i.i.d.'', ''iid'', or ''IID''. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Introduction In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent, identically distributed (IID) random variables”. In other words, the terms ''random sample'' and ''IID'' are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to say “IID.” * Identically Distributed means that there are no overall trends–the distribution doesn’t fluctuate and all items in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space ''X'', the collection of all Borel sets on ''X'' forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on ''X'' is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galton–Watson Process
The Galton–Watson process is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. The process models family names as patrilineal (passed from father to son), while offspring are randomly either male or female, and names become extinct if the family name line dies out (holders of the family name die without male descendants). This is an accurate description of Y chromosome transmission in genetics, and the model is thus useful for understanding human Y-chromosome DNA haplogroups. Likewise, since mitochondria are inherited only on the maternal line, the same mathematical formulation describes transmission of mitochondria. The formula is of limited usefulness in understanding actual family name distributions, since in practice family names change for many other reasons, and dying out of name line is only one factor. History There was concern amongst the Victorians that aristocratic surnames were becoming ext ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivalued Function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a codomain associates each in to one or more values in ; it is thus a serial binary relation. Some authors allow a multivalued function to have no value for some inputs (in this case a multivalued function is simply a binary relation). However, in some contexts such as in complex analysis (''X'' = ''Y'' = C), authors prefer to mimic function theory as they extend concepts of the ordinary (single-valued) functions. In this context, an ordinary function is often called a single-valued function to avoid confusion. The term ''multivalued function'' originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function f(z) in some neighbourhood of a point z=a. This is the case fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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