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Scaled Inverse Chi-squared Distribution
The scaled inverse chi-squared distribution \psi \, \mbox \chi^2(\nu), where \psi is the scale parameter, equals the univariate inverse Wishart distribution \mathcal^(\psi,\nu) with degrees of freedom \nu. This family of scaled inverse chi-squared distributions is linked to the inverse-chi-squared distribution and to the chi-squared distribution: If X \sim \psi \, \mbox \chi^2(\nu) then X/\psi \sim \mbox \chi^2(\nu) as well as \psi/X \sim \chi^2(\nu) and 1/X \sim \psi^\chi^2(\nu) . Instead of \psi, the scaled inverse chi-squared distribution is however most frequently parametrized by the scale parameter \tau^2 = \psi/\nu and the distribution \nu \tau^2 \, \mbox \chi^2(\nu) is denoted by \mbox\chi^2(\nu, \tau^2). In terms of \tau^2 the above relations can be written as follows: If X \sim \mbox\chi^2(\nu, \tau^2) then \frac \sim \mbox \chi^2(\nu) as well as \frac \sim \chi^2(\nu) and 1/X \sim \frac\chi^2(\nu) . This family of scaled inverse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scaled Inverse Chi Squared
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Scaled may mean: * Scaled Composites (often abbreviated as Scaled), formerly the Rutan Aircraft Factory * Scaled Aviation Industries of Lahore, Pakistan, a Light Sports Aircraft Manufacturer * Something which has undergone a scale transformation ** Scale model#Scales ** Scaling (geometry) See also *Scale (other) Scale or scales may refer to: Mathematics * Scale (descriptive set theory), an object defined on a set of points * Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original * Scale factor, a number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regularized Gamma Function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity. Definition The upper incomplete gamma function is defined as: \Gamma(s,x) = \int_x^ t^\,e^\, dt , whereas the lower incomplete gamma function is defined as: \gamma(s,x) = \int_0^x t^\,e^\, dt . In both cases is a complex parameter, such that the real part of is positive. Properties By integration by parts we find the recurrence relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse-gamma Distribution
In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution. Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution, if an uninformative prior is used, and as an analytically tractable conjugate prior, if an informative prior is required. It is common among some Bayesians to consider an alternative parametrization of the normal distribution in terms of the precision, defined as the reciprocal of the variance, which allows the gamma distribution to be used directly as a conjugate prior. Other Bayesians prefer to parametrize the inverse gamma distribution differently, as a scaled inverse chi-squared distribution. Characteriza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse-chi-squared Distribution
In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distributionBernardo, J.M.; Smith, A.F.M. (1993) ''Bayesian Theory'', Wiley (pages 119, 431) ) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution. It is used in Bayesian inference as conjugate prior for the variance of the normal distribution. Definition The inverse chi-squared distribution (or inverted-chi-square distribution ) is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-squared distribution. If X follows a chi-squared distribution with \nu degrees of freedom then 1/X follows the inverse chi-squared distribution with \nu degrees of freedom. The probability density function of the inverse chi-squared distribution is given by : f(x; \nu) = \frac\,x^ e^ In the above x>0 and \nu is the degrees of freedom parameter. Further, \Gamma is the gamm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeffreys Prior
In Bayesian statistics, the Jeffreys prior is a non-informative prior distribution for a parameter space. Named after Sir Harold Jeffreys, its density function is proportional to the square root of the determinant of the Fisher information matrix: p\left( \theta \right) \propto \left, I (\theta) \^ .\, It has the key feature that it is invariant under a change of coordinates for the parameter vector \theta. That is, the relative probability assigned to a volume of a probability space using a Jeffreys prior will be the same regardless of the parameterization used to define the Jeffreys prior. This makes it of special interest for use with ''scale parameters''. As a concrete example, a Bernoulli distribution can be parameterized by the probability of occurrence , or by the odds . A uniform prior on one of these is not the same as a uniform prior on the other, even accounting for reparameterization in the usual way, but the Jeffreys prior on one reparameterizes to the Jeffreys p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Probability
In probability theory, conditional probability is a measure of the probability of an Event (probability theory), event occurring, given that another event (by assumption, presumption, assertion or evidence) is already known to have occurred. This particular method relies on event A occurring with some sort of relationship with another event B. In this situation, the event A can be analyzed by a conditional probability with respect to B. If the event of interest is and the event is known or assumed to have occurred, "the conditional probability of given ", or "the probability of under the condition ", is usually written as or occasionally . This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the "given" one happening (how many times A occurs rather than not assuming B has occurred): P(A \mid B) = \frac. For example, the probability that any given person has a cough on any given day ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Likelihood Function
A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the joint probability distribution of the random variable that (presumably) generated the observations. When evaluated on the actual data points, it becomes a function solely of the model parameters. In maximum likelihood estimation, the argument that maximizes the likelihood function serves as a point estimate for the unknown parameter, while the Fisher information (often approximated by the likelihood's Hessian matrix at the maximum) gives an indication of the estimate's precision. In contrast, in Bayesian statistics, the estimate of interest is the ''converse'' of the likelihood, the so-called posterior probability of the parameter given the observed data, which is calculated via Bayes' rule. Definition The likelihood function, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prior Distribution
A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable. In Bayesian statistics, Bayes' rule prescribes how to update the prior with new information to obtain the posterior probability distribution, which is the conditional distribution of the uncertain quantity given new data. Historically, the choice of priors was often constrained to a conjugate family of a given likelihood function, so that it would result in a tractable posterior of the same family. The widespread availability of Markov chain Monte Carlo methods, however, has made this less of a concern. There are many ways to constru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Posterior Probability Distribution
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 interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayes' Theorem
Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting Conditional probability, conditional probabilities, allowing one to find the probability of a cause given its effect. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to someone of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the person is typical of the population as a whole. Based on Bayes' law, both the prevalence of a disease in a given population and the error rate of an infectious disease test must be taken into account to evaluate the meaning of a positive test result and avoid the ''base-rate fallacy''. One of Bayes' theorem's many applications is Bayesian inference, an approach to statistical inference, where it is used to invert the probability of Realization (probability), observations given a model configuration (i.e., th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digamma Function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing and Concave function, strictly concave on (0,\infty), and it Asymptotic analysis, asymptotically behaves as :\psi(z) \sim \ln - \frac, for complex numbers with large modulus (, z, \rightarrow\infty) in the Circular sector, sector , \arg z, 0. The digamma function is often denoted as \psi_0(x), \psi^(x) or (the uppercase form of the archaic Greek consonant digamma meaning Gamma, double-gamma). Gamma. Relation to harmonic numbers The gamma function obeys the equation :\Gamma(z+1)=z\Gamma(z). \, Taking the logarithm on both sides and using the functional equation property of the log-gamma function gives: :\log \Gamma(z+1)=\log(z)+\log \Gamma(z), Differentiating both sides with respect to gives: :\psi(z+1)=\psi(z)+\frac Since the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method
In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a real-valued function , its derivative , and an initial guess for a root of . If satisfies certain assumptions and the initial guess is close, then x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the x-intercept of the tangent of the graph of at : that is, the improved guess, , is the unique root of the linear approximation of at the initial guess, . The process is repeated as x_ = x_n - \frac until a sufficiently precise value is reached. The number of correct digits roughly doubles with each step. This algorithm is first in the class of Householder's methods, and was succeeded by Halley's method. The method can also be extended t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
