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Distribution Learning Theory
The distributional learning theory or learning of probability distribution is a framework in computational learning theory. It has been proposed from Michael Kearns, Yishay Mansour, Dana Ron, Ronitt Rubinfeld, Robert Schapire and Linda Sellie in 1994 M. Kearns, Y. Mansour, D. Ron, R. Rubinfeld, R. Schapire, L. Sellie ''On the Learnability of Discrete Distributions''. ACM Symposium on Theory of Computing, 199/ref> and it was inspired from the PAC-learning, PAC-framework introduced by Leslie Valiant. In this framework the input is a number of samples drawn from a distribution that belongs to a specific class of distributions. The goal is to find an efficient algorithm that, based on these samples, determines with high probability the distribution from which the samples have been drawn. Because of its generality, this framework has been used in a large variety of different fields like machine learning, approximation algorithms, applied probability and statistics. This article explai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Learning Theory
In computer science, computational learning theory (or just learning theory) is a subfield of artificial intelligence devoted to studying the design and analysis of machine learning algorithms. Overview Theoretical results in machine learning mainly deal with a type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could be whether or not the mushrooms are edible. The algorithm takes these previously labeled samples and uses them to induce a classifier. This classifier is a function that assigns labels to samples, including samples that have not been seen previously by the algorithm. The goal of the supervised learning algorithm is to optimize some measure of performance such as minimizing the number of mistakes made on new samples. In addition to performance bounds, computational learning theory studies the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), 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 statistical survey, surveys and experimental design, 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 sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Variation
In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval 'a'', ''b''⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation ''x'' ↦ ''f''(''x''), for ''x'' ∈ 'a'', ''b'' Functions whose total variation is finite are called functions of bounded variation. Historical note The concept of total variation for functions of one real variable was first introduced by Camille Jordan in the paper . He used the new concept in order to prove a convergence theorem for Fourier series of discontinuous periodic functions whose variation is bounded. The extension of the concept to functions of more than one variable however is not simple for various reasons. Definitions Total variation for functions of one real variable Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gautam Kamath
Gautama, Gautam or Gotama may refer to: Ancient sages and philosophers * Akṣapāda Gautama, a Hindu sage and founder of the Nyaya school of Hindu philosophy: see Nyāya Sūtras * Indrabhuti Gautama, chief disciple of Mahavira * Gautama Buddha, the founder of Buddhism * Gautama Maharishi, one of the Saptarshis in Hinduism who authored hymns in Mandala 1 of the Rigveda Clans * Gautam Brahmins, a sub-group of Hindu Brahmins in India * Gautam Rajputs, a sub-clan of Rajputs found in North India Etymology * Gautam (Etymology) Nepali name *Gautam (Nepali name) Gautam as first name * Gautam Adani, Indian industrialist * Gautam Bhatia (architect) * Gautam Bhatia (lawyer) * Gautam Choudhury, Indian musician * Gautam Gambhir, Indian cricketer * Gautam Gulati, Indian actor * Gautham Krishn, South Indian actor * Gautam Navlakha Gautam Navlakha () is an Indian self-proclaimed human rights activist and journalist. He has written on left-wing extremism and is a critic of army and state ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constantinos Daskalakis
Constantinos Daskalakis (; born 29 April 1981) is a Greek theoretical computer scientist. He is a professor at MIT's Electrical Engineering and Computer Science department and a member of the MIT Computer Science and Artificial Intelligence Laboratory. He was awarded the Rolf Nevanlinna Prize and the Grace Murray Hopper Award in 2018. Early life and education Daskalakis was born in Athens on 29 April 1981. His grandparents originated from Crete, where he summered as a child. He has a younger brother, Nikolaos. When Daskalakis was in third grade, his father bought an Amstrad CPC, which Daskalakis stayed up all night attempting to learn how it worked.A Poet of Computation Who Uncovers Distant Truths ''Quanta Magazine'' He attended [...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 random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Probability Distribution
In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value x of X as a parameter. When both X and Y are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable. If the conditional distribution of Y given X is a continuous distribution, then its probability density function is known as the conditional density function. The properties of a conditional distribution, such as the moments, are often referred to by corresponding names such as the conditional mean and conditional variance. Mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Learning Theory
Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data. Statistical learning theory has led to successful applications in fields such as computer vision, speech recognition, and bioinformatics. Introduction The goals of learning are understanding and prediction. Learning falls into many categories, including supervised learning, unsupervised learning, online learning, and reinforcement learning. From the perspective of statistical learning theory, supervised learning is best understood. Supervised learning involves learning from a training set of data. Every point in the training is an input-output pair, where the input maps to an output. The learning problem consists of inferring the function that maps between the input and the output, such that the learned function can be used to predict t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov–Smirnov Test
In statistics, the Kolmogorov–Smirnov test (K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution (one-sample K–S test), or to compare two samples (two-sample K–S test). In essence, the test answers the question "What is the probability that this collection of samples could have been drawn from that probability distribution?" or, in the second case, "What is the probability that these two sets of samples were drawn from the same (but unknown) probability distribution?". It is named after Andrey Kolmogorov and Nikolai Smirnov. The Kolmogorov–Smirnov statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples. The null distributio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Variation Distance Of Probability Measures
In probability theory, the total variation distance is a distance measure for probability distributions. It is an example of a statistical distance metric, and is sometimes called the statistical distance, statistical difference or variational distance. Definition Consider a measurable space (\Omega, \mathcal) and probability measures P and Q defined on (\Omega, \mathcal). The total variation distance between P and Q is defined as: :\delta(P,Q)=\sup_\left, P(A)-Q(A)\. Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event. Properties Relation to other distances The total variation distance is related to the Kullback–Leibler divergence by Pinsker’s inequality: :\delta(P,Q) \le \sqrt. One also has the following inequality, due to Bretagnolle and Huber (see, also, Tsybakov), which has the advantage of providing a non-vacuous bound even when D_(P\parallel Q)>2: :\delta(P,Q) \le \s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Probability
Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains. Scope Much research involving probability is done under the auspices of applied probability. However, while such research is motivated (to some degree) by applied problems, it is usually the mathematical aspects of the problems that are of most interest to researchers (as is typical of applied mathematics in general). Applied probabilists are particularly concerned with the application of stochastic processes, and probability more generally, to the natural, applied and social sciences, including biology, physics (including astronomy), chemistry, medicine, computer science and information technology, and economics. Another area of interest is in engineering: particularly in areas of uncertainty, risk management, probabilistic design, and Quality assurance. See also *Areas of application: **Ruin theory **Statistical physics **Stoichiometry and modelli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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