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Bernstein–von Mises Theorem
In Bayesian inference, the Bernstein–von Mises theorem provides the basis for using Bayesian credible sets for confidence statements in parametric models. It states that under some conditions, a posterior distribution converges in total variation distance to a multivariate normal distribution centered at the maximum likelihood estimator \widehat_n with covariance matrix given by n^\mathcal(\theta_0)^ , where \theta_0 is the true population parameter and \mathcal(\theta_0) is the Fisher information matrix at the true population parameter value: :, , P(\theta, x_1,\dots x_n) - \mathcal(_n, n^\mathcal(\theta_0)^), , _ \xrightarrow = 0 The Bernstein–von Mises theorem links Bayesian inference with frequentist inference. It assumes there is some true probabilistic process that generates the observations, as in frequentism, and then studies the quality of Bayesian methods of recovering that process, and making uncertainty statements about that process. In particular, it states tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Inference
Bayesian inference ( or ) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence, and update it as more information becomes available. Fundamentally, Bayesian inference uses a prior distribution to estimate posterior probabilities. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability". Introduction to Bayes' rule Formal explanation Bayesian inference derives the posterior probability as a consequence of two antecedents: a prior probability and a "likelihood function" derive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametric Model
In statistics, a parametric model or parametric family or finite-dimensional model is a particular class of statistical models. Specifically, a parametric model is a family of probability distributions that has a finite number of parameters. Definition A statistical model is a collection of probability distributions on some sample space. We assume that the collection, , is indexed by some set . The set is called the parameter set or, more commonly, the parameter space. For each , let denote the corresponding member of the collection; so is a cumulative distribution function. Then a statistical model can be written as : \mathcal = \big\. The model is a parametric model if for some positive integer . When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions: : \mathcal = \big\. Examples * The Poisson family of distributions is parametrized by a single number : : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Variation Distance Of Probability Measures
In probability theory, the total variation distance is a statistical distance between probability distributions, 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)\. This is the largest absolute difference between the probabilities that the two probability distributions assign to the same event. Properties The total variation distance is an ''f''-divergence and an integral probability metric. 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 ), which has the advantage of providing a non-vacuous bound even when \textstyle D_(P\pa ... [...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] |
Frequentist Inference
Frequentist inference is a type of statistical inference based in frequentist probability, which treats “probability” in equivalent terms to “frequency” and draws conclusions from sample-data by means of emphasizing the frequency or proportion of findings in the data. Frequentist inference underlies frequentist statistics, in which the well-established methodologies of statistical hypothesis testing and confidence intervals are founded. History of frequentist statistics Frequentism is based on the presumption that statistics represent probabilistic frequencies. This view was primarily developed by Ronald Fisher and the team of Jerzy Neyman and Egon Pearson. Ronald Fisher contributed to frequentist statistics by developing the frequentist concept of "significance testing", which is the study of the significance of a measure of a statistic when compared to the hypothesis. Neyman-Pearson extended Fisher's ideas to apply to multiple hypotheses. They posed that the rati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Likelihood Estimator
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference. If the likelihood function is differentiable, the derivative test for finding maxima can be applied. In some cases, the first-order conditions of the likelihood function can be solved analytically; for instance, the ordinary least squares estimator for a linear regression model maximizes the likelihood when the random errors are assumed to have normal distributions with the same variance. From the perspective of Bayesian inference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Von Mises
Richard Martin Edler von Mises (; 19 April 1883 – 14 July 1953) was an Austrian scientist and mathematician who worked on solid mechanics, fluid mechanics, aerodynamics, aeronautics, statistics and probability theory. He held the position of Gordon McKay Professor of Aerodynamics and Applied Mathematics at Harvard University. He described his work in his own words shortly before his death as: Although best known for his mathematical work, von Mises also contributed to the philosophy of science as a neo-positivist and empiricist, following the line of Ernst Mach. Historians of the Vienna Circle of logical empiricism recognize a "first phase" from 1907 through 1914 with Philipp Frank, Hans Hahn, and Otto Neurath. His older brother, Ludwig von Mises, held an opposite point of view with respect to positivism and epistemology. His brother developed ''praxeology'', an ''a priori'' view. During his time in Istanbul, Mises maintained close contact with Philipp Frank, a lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph L
Joseph is a common male name, derived from the Hebrew (). "Joseph" is used, along with "Josef (given name), Josef", mostly in English, French and partially German languages. This spelling is also found as a variant in the languages of the modern-day Nordic countries. In Portuguese language, Portuguese and Spanish language, Spanish, the name is "José". In Arabic, including in the Quran, the name is spelled , . In Kurdish language, Kurdish (''Kurdî''), the name is , Persian language, Persian, the name is , and in Turkish language, Turkish it is . In Pashto the name is spelled ''Esaf'' (ايسپ) and in Malayalam it is spelled ''Ousep'' (ഔസേപ്പ്). In Tamil language, Tamil, it is spelled as ''Yosepu'' (யோசேப்பு). The name has enjoyed significant popularity in its many forms in numerous countries, and ''Joseph'' was one of the two names, along with ''Robert'', to have remained in the top 10 boys' names list in the US from 1925 to 1972. It is especiall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a . A probability space consists of three elements:Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press. # A '' sample space'', \Omega, which is the set of all possible outcomes of a random process under consideration. # An event space, \mathcal, which is a set of events, where an event is a subset of outcomes in the sample space. # A '' probability function'', P, which assigns, to each event in the event space, a probability, which is a number between 0 and 1 (inclusive). In order to provide a model of probability, these elements must satisfy probability axioms. In the example of the throw of a standard die, # The sample space \Omega is typically the set \ where each element in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucien Le Cam
Lucien Marie Le Cam (November 18, 1924 – April 25, 2000) was a mathematician and statistician. Biography Le Cam was born November 18, 1924, in Croze, France. His parents were farmers, and unable to afford higher education for him; his father died when he was 13. After graduating from a Catholic school in 1942, he began studying at a seminary in Limoges, but immediately quit upon learning that he would not be allowed to study chemistry there. Instead he continued his studies at a lycée, which did not teach chemistry but did teach mathematics. In May 1944 he joined an underground group, and then went into hiding, returning to his school the following November but soon afterwards moving to Paris, where he began studying at the University of Paris. He graduated in 1945 with the degree '' licence ès sciences''.. Le Cam then worked for a hydroelectric utility for five years, while meeting at the University of Paris for a weekly seminar in statistics. In 1950, he was invited to beco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorraine Schwartz (statistician)
Lorraine Schwartz is an American bespoke high jewellery designer. She has designed jewelry for a host of famous female celebrities including Beyoncé, Jennifer Lopez and Kim Kardashian. In 2009, Angelina Jolie wore to the Oscars a pair of emerald earrings designed by Schwartz, valued at $2.5 million. Career Schwartz states that in 2002 Halle Berry wore her jewellery on the red carpet. Since, she has designed for celebrities such as Kelly Rowland, Beyoncé Knowles, Jennifer Lopez and Cate Blanchett. For ESPN The Magazine, she made a "30-carat, diamond-and-ruby logo" for their 10th anniversary baseball caps. She was also involved in the "Art for Life" gala in 2007, which was led by Russell Simmons and his ex-wife, Kimora Lee Simmons. In the 2006 Beyoncé song, " Upgrade U", and the 2023 song, " My House", Lorraine Schwartz is mentioned. Kim Kardashian wears a 15-carat diamond engagement ring by Schwartz, given to her by her now ex-husband, Kanye West Ye ( ; born Kanye Oma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David A
David (; , "beloved one") was a king of ancient Israel and Judah and the third king of the United Monarchy, according to the Hebrew Bible and Old Testament. The Tel Dan stele, an Aramaic-inscribed stone erected by a king of Aram-Damascus in the late 9th/early 8th centuries BCE to commemorate a victory over two enemy kings, contains the phrase (), which is translated as " House of David" by most scholars. The Mesha Stele, erected by King Mesha of Moab in the 9th century BCE, may also refer to the "House of David", although this is disputed. According to Jewish works such as the '' Seder Olam Rabbah'', '' Seder Olam Zutta'', and '' Sefer ha-Qabbalah'' (all written over a thousand years later), David ascended the throne as the king of Judah in 885 BCE. Apart from this, all that is known of David comes from biblical literature, the historicity of which has been extensively challenged,Writing and Rewriting the Story of Solomon in Ancient Israel; by Isaac Kalimi; page 3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
