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Pitman–Yor Process
In probability theory, a Pitman–Yor process denoted PY(''d'', ''θ'', ''G''0), is a stochastic process whose sample path is a probability distribution. A random sample from this process is an infinite discrete probability distribution, consisting of an infinite set of atoms drawn from ''G''0, with weights drawn from a two-parameter Poisson–Dirichlet distribution. The process is named after Jim Pitman and Marc Yor. The parameters governing the Pitman–Yor process are: 0 ≤ ''d'' −''d'' and a base distribution ''G''0 over a probability space ''X''. When ''d'' = 0, it becomes the Dirichlet process. The discount parameter gives the Pitman–Yor process more flexibility over tail behavior than the Dirichlet process, which has exponential tails. This makes Pitman–Yor process useful for modeling data with power-law tails (e.g., word frequencies in natural language). The exchangeable random partition induced by ... [...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 probab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a rando ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Pitman
Jim or JIM may refer to: * Jim (given name), a given name * Jim, a diminutive form of the given name James (given name), James * Jim, a short form of the given name Jimmy (given name), Jimmy * OPCW-UN Joint Investigative Mechanism * Jim (comics), ''Jim'' (comics), a series by Jim Woodring * Jim (album), ''Jim'' (album), by soul artist Jamie Lidell * Jim (Huckleberry Finn), Jim (''Huckleberry Finn''), a character in Mark Twain's novel * Jim (TV channel), in Finland * JIM (Flemish TV channel) * JIM suit, for atmospheric diving * Jim River, in North and South Dakota, United States * Jim, the nickname of Yelkanum Seclamatan (died April 1911), Native American chief * ''Journal of Internal Medicine'' * Juan Ignacio Martínez (born 1964), Spanish footballer, commonly known as JIM * Jim (horse), milk wagon horse used to produce serum containing diphtheria antitoxin * Jim (song), "Jim" (song), a 1941 song. * JIM, Jiangxi Isuzu Motors, a joint venture between Isuzu and Jiangling Motors Corpo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marc Yor
Marc Yor (24 July 1949 – 9 January 2014) was a French mathematician well known for his work on stochastic processes, especially properties of semimartingales, Brownian motion and other Lévy processes, the Bessel processes, and their applications to mathematical finance. Background Yor was a professor at the Paris VI University in Paris, France, from 1981 until his death in 2014. He was a recipient of several awards, including the Humboldt Prize, the Montyon Prize The Montyon Prize (french: Prix Montyon) is a series of prizes awarded annually by the French Academy of Sciences and the Académie française. They are endowed by the French benefactor Baron de Montyon. History Prior to the start of the French R ...,Official biography at the French Academy website and was awarded the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Process
In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular distribution. As an example, a bag of 100 real-world dice is a ''random probability mass function (random pmf)'' - to sample this random pmf you put your hand in the bag and draw out a die, that is, you draw a pmf. A bag of dice manufactured using a crude process 100 years ago will likely have probabilities that deviate wildly from the uniform pmf, whereas a bag of state-of-the-art dice used by Las Vegas casinos may have barely perceptible i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power-law
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four. Empirical examples The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words in most languages, frequencies of family names, the species richness in clades of organisms, the sizes of power outages, volcanic eruptions, human judgments of stimulus intensity and many other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gibbs Type Random Partition
Gibbs or GIBBS is a surname and acronym. It may refer to: People * Gibbs (surname) Places * Gibbs (crater), on the Moon * Gibbs, Missouri, US * Gibbs, Tennessee, US * Gibbs Island (South Shetland Islands), Antarctica * 2937 Gibbs, an asteroid Science Mathematics and statistics * Gibbs phenomenon * Gibbs' inequality * Gibbs sampling Physics * Gibbs phase rule * Gibbs free energy * Gibbs entropy * Gibbs paradox * Gibbs–Helmholtz equation * Gibbs algorithm * Gibbs state * Gibbs-Marangoni effect * Gibbs phenomenon, an MRI artifact Organisations * Gibbs & Cox naval architecture firm * Gothenburg International Bioscience Business School * Gibbs College, several US locations * Gibbs Technologies, developer and manufacturer of amphibious vehicles * Gibbs High School (other), several schools of this name exist * Antony Gibbs & Sons, British trading company, established in London in 1802 Other uses * Gibbs SR, former name of the toothpaste Mentadent * Gibbs Stadium, Spar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Measure
In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. Definition Random measures can be defined as transition kernels or as random elements. Both definitions are equivalent. For the definitions, let E be a separable complete metric space and let \mathcal E be its Borel \sigma -algebra. (The most common example of a separable complete metric space is \R^n ) As a transition kernel A random measure \zeta is a ( a.s.) locally finite transition kernel from a (abstract) probability space (\Omega, \mathcal A, P) to (E, \mathcal E) . Being a transition kernel means that *For any fixed B \in \mathcal \mathcal E , the mapping : \omega \mapsto \zeta(\omega,B) :is measurable from (\Omega, \mathcal A) to (E, \mathcal E) *For every fixed \omega \in \Omega , the mapping : B \mapsto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chinese Restaurant Process
In probability theory, the Chinese restaurant process is a discrete-time stochastic process, analogous to seating customers at tables in a restaurant. Imagine a restaurant with an infinite number of circular tables, each with infinite capacity. Customer 1 sits at the first table. The next customer either sits at the same table as customer 1, or the next table. This continues, with each customer choosing to either sit at an occupied table with a probability proportional to the number of customers already there (i.e., they are more likely to sit at a table with many customers than few), or an unoccupied table. At time ''n'', the ''n'' customers have been partitioned among ''m'' ≤ ''n'' tables (or blocks of the partition). The results of this process are exchangeable, meaning the order in which the customers sit does not affect the probability of the final distribution. This property greatly simplifies a number of problems in population genetics, linguistic analysis, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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