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Observed Information
In statistics, the observed information, or observed Fisher information, is the negative of the second derivative (the Hessian matrix) of the "log-likelihood" (the logarithm of the likelihood function). It is a sample-based version of the Fisher information. Definition Suppose we observe random variables X_1,\ldots,X_n, independent and identically distributed with density ''f''(''X''; θ), where θ is a (possibly unknown) vector. Then the log-likelihood of the parameters \theta given the data X_1,\ldots,X_n is :\ell(\theta , X_1,\ldots,X_n) = \sum_^n \log f(X_i, \theta) . We define the observed information matrix at \theta^ as :\mathcal(\theta^*) = - \left. \nabla \nabla^ \ell(\theta) \_ ::= - \left. \left( \begin \tfrac & \tfrac & \cdots & \tfrac \\ \tfrac & \tfrac & \cdots & \tfrac \\ \vdots & \vdots & \ddots & \vdots \\ \tfrac & \tfrac & \cdots & \tfrac \\ \end \right) \ell(\theta) \_ Since the inverse of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "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. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Rubin
Donald Bruce Rubin (born December 22, 1943) is an Emeritus Professor of Statistics at Harvard University, where he chaired the department of Statistics for 13 years. He also works at Tsinghua University in China and at Temple University in Philadelphia. He is most well known for the Rubin causal model, a set of methods designed for causal inference with observational data, and for his methods for dealing with missing data. In 1977 he was elected as a Fellow of the American Statistical Association. Biography Rubin was born in Washington, D.C. into a Jewish family of lawyers. As an undergraduate Rubin attended the accelerated Princeton University PhD program where he was one of a cohort of 20 students mentored by the physicist John Wheeler (the intention of the program was to confer degrees within 5 years of freshman matriculation). He switched to psychology and graduated in 1965. He began graduate school in psychology at Harvard with a National Science Foundation fellowsh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fisher Information Metric
In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, ''i.e.'', a smooth manifold whose points are probability distributions. It can be used to calculate the distance between probability distributions. The metric is interesting in several aspects. By Chentsov’s theorem, the Fisher information metric on statistical models is the only Riemannian metric (up to rescaling) that is invariant under sufficient statistics. It can also be understood to be the infinitesimal form of the relative entropy (''i.e.'', the Kullback–Leibler divergence); specifically, it is the Hessian matrix, Hessian of the divergence. Alternately, it can be understood as the metric induced by the flat space Euclidean metric, after appropriate changes of variable. When extended to complex projective Hilbert space, it becomes the Fubini–Study metric; when written in terms of mixed state (physics), mixed states, it is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fisher Information Matrix
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that models ''X''. Formally, it is the variance of the score, or the expected value of the observed information. The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized and explored by the statistician Sir Ronald Fisher (following some initial results by Francis Ysidro Edgeworth). The Fisher information matrix is used to calculate the covariance matrices associated with maximum-likelihood estimates. It can also be used in the formulation of test statistics, such as the Wald test. In Bayesian statistics, the Fisher information plays a role in the derivation of non-informative prior distributions according to Jeffreys' rule. It also appears as the large-sample covariance of the posterior distribution, provided that the prior is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Squared Error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the true value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the ''empirical'' risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution). The MSE is a measure of the quality of an estimator. As it is derived from the square of Euclidean distance, it is always a positive value that decreases as the erro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Normality
In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the limiting distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing approximations to the cumulative distribution functions of statistical estimators. Definition A sequence of distributions corresponds to a sequence of random variables ''Zi'' for ''i'' = 1, 2, ..., I . In the simplest case, an asymptotic distribution exists if the probability distribution of ''Zi'' converges to a probability distribution (the asymptotic distribution) as ''i'' increases: see convergence in distribution. A special case of an asymptotic distribution is when the sequence of random variables is always zero or ''Zi'' = 0 as ''i'' approaches infinity. Here the asymptotic distribution is a degenerate distribution, corresponding to the value zero. However, the most usual sense in which the term asymptotic distribution is used arises ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biometrika
''Biometrika'' is a peer-reviewed scientific journal published by Oxford University Press for the Biometrika Trust. The editor-in-chief is Paul Fearnhead (Lancaster University). The principal focus of this journal is theoretical statistics. It was established in 1901 and originally appeared quarterly. It changed to three issues per year in 1977 but returned to quarterly publication in 1992. History ''Biometrika'' was established in 1901 by Francis Galton, Karl Pearson, and Raphael Weldon to promote the study of biometrics. The history of ''Biometrika'' is covered by Cox (2001). The name of the journal was chosen by Pearson, but Francis Edgeworth insisted that it be spelt with a "k" and not a "c". Since the 1930s, it has been a journal for statistical theory and methodology. Galton's role in the journal was essentially that of a patron and the journal was run by Pearson and Weldon and after Weldon's death in 1906 by Pearson alone until he died in 1936. In the early days, the Ameri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David V
David V ( ka, დავით V, tr; 1113 — 1155), of the Bagrationi dynasty, was the king ('' mepe'') of the Kingdom of Georgia from 1154 until his death in 1155. Life David was born around 1113 and was the eldest son of Prince Demetrius and grandson of King David IV the Builder who was reigning at that time. In the 1140s, King Demetrius I had quarreled and disinherited David and chosen his youngest son, Prince George, as heir apparent. Why they quarreled is unknown: perhaps over David's personal defects. probably, for the Abuletisdze family and the status of the city of Ani. Those who had supported Prince Vakhtang during an attempted coup against Demetrius I now opposed Demetrius' unprecedented disinheritance of David and approved the surrender of Ani to Muslim rule. Vasak Artsruni and his brother, who negotiated Saltuk's release, were active supporters of David. A first coup attempt failed around 1150, but in 1154 David's coup against his father succeeded. Demet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bradley Efron
Bradley Efron (; born May 24, 1938) is an American statistician. Efron has been president of the American Statistical Association (2004) and of the Institute of Mathematical Statistics (1987–1988).Cochran, J. (1 September 2015), "ASA Leaders Reminisce: Brad Efron", ''Amstat News''. He is a past editor (for theory and methods) of the ''Journal of the American Statistical Association'', and he is the founding editor of the '' Annals of Applied Statistics''. Efron is also the recipient of many awards (see below). Efron is especially known for proposing the bootstrap resampling technique, which has had a major impact in the field of statistics and virtually every area of statistical application. The bootstrap was one of the first computer-intensive statistical techniques, replacing traditional algebraic derivations with data-based computer simulations. Life and career Efron was born in St. Paul, Minnesota in May 1938, the son of Russian Jewish immigrants Esther and Miles Ef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean, mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would expect to get in reality. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by Integral, integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Posterior Probability
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior probability contains everything there is to know about an uncertain proposition (such as a scientific hypothesis, or parameter values), given prior knowledge and a mathematical model describing the observations available at a particular time. After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating. In the context of Bayesian statistics, the posterior probability distribution usually describes the epistemic uncertainty about statistical parameters conditional on a collection of observed data. From a given posterior distribution, various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest posterior density int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
