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Watanabe–Akaike Information Criterion
In statistics, the widely applicable information criterion (WAIC), also known as Watanabe–Akaike information criterion, is the generalized version of the Akaike information criterion (AIC) onto singular statistical models. Widely applicable Bayesian information criterion (WBIC) is the generalized version of Bayesian information criterion (BIC) onto singular statistical models. WBIC is the average log likelihood function over the posterior distribution with the inverse temperature > 1/log ''n'' where ''n'' is the sample size. Both WAIC and WBIC can be numerically calculated without any information about a true distribution. See also *Akaike information criterion *Bayesian information criterion *Deviance information criterion *Hannan–Quinn information criterion *Shibata information criterion Shibata may refer to: Places * Shibata, Miyagi, a town in Miyagi Prefecture * Shibata District, Miyagi, a district in Miyagi Prefecture * Shibata, Niigata, a city in Niigata ... [...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] |
Akaike Information Criterion
The Akaike information criterion (AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection. AIC is founded on information theory. When a statistical model is used to represent the process that generated the data, the representation will almost never be exact; so some information will be lost by using the model to represent the process. AIC estimates the relative amount of information lost by a given model: the less information a model loses, the higher the quality of that model. In estimating the amount of information lost by a model, AIC deals with the trade-off between the goodness of fit of the model and the simplicity of the model. In other words, AIC deals with both the risk of overfitting and the risk of underfitting. The Akaike information criterion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Statistical Model
In mathematical statistics, the Fisher information (sometimes simply called 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. In Bayesian statistics, the asymptotic distribution of the posterior mode depends on the Fisher information and not on the prior (according to the Bernstein–von Mises theorem, which was anticipated by Laplace for exponential families). The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized by the statistician Ronald Fisher (following some initial results by Francis Ysidro Edgeworth). The Fisher information is also used in the calculation of the Jeffreys prior, which is used in Bayesian statistics. The Fisher information matrix is used to calculate the covariance matrices associate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Information Criterion
In statistics, the Bayesian information criterion (BIC) or Schwarz information criterion (also SIC, SBC, SBIC) is a criterion for model selection among a finite set of models; models with lower BIC are generally preferred. It is based, in part, on the likelihood function and it is closely related to the Akaike information criterion (AIC). When fitting models, it is possible to increase the likelihood by adding parameters, but doing so may result in overfitting. Both BIC and AIC attempt to resolve this problem by introducing a penalty term for the number of parameters in the model; the penalty term is larger in BIC than in AIC for sample sizes greater than 7. The BIC was developed by Gideon E. Schwarz and published in a 1978 paper, where he gave a Bayesian argument for adopting it. Definition The BIC is formally defined as : \mathrm = k\ln(n) - 2\ln(\widehat L). \ where *\hat L = the maximized value of the likelihood function of the model M, i.e. \hat L=p(x\mid\widehat\theta, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Log Likelihood
The likelihood function (often simply called the likelihood) represents the probability of Realization (probability), random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a Sample (statistics), given sample, the likelihood function indicates which parameter values are more ''likely'' than others, in the sense that they would have made the observed data more probable. Consequently, the likelihood is often written as \mathcal(\theta\mid X) instead of P(X \mid \theta), to emphasize that it is to be understood as a function of the parameters \theta instead of the random variable X. In maximum likelihood estimation, the arg max of the likelihood function serves as a Point estimation, point estimate for \theta, while local curvature (approximated by the likelihood's Hessian matrix) indicates the estimate's Accuracy and precision, precision. Meanwhile in Bayesian statistics, parameter estimates are derived from the converse o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Posterior 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 (HPD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Temperature
In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:\beta = \frac (where is the temperature and is Boltzmann constant).J. Meixner (1975) "Coldness and Temperature", ''Archive for Rational Mechanics and Analysis'' 57:3, 281-29abstract It was originally introduced in 1971 (as "coldness function") by , one of the proponents of the rational thermodynamics school of thought, based on earlier proposals for a "reciprocal temperature" function.Day, W.A. and Gurtin, Morton E. (1969) "On the symmetry of the conductivity tensor and other restrictions in the nonlinear theory of heat conduction", ''Archive for Rational Mechanics and Analysis'' 33:1, 26-32 (Springer-Verlag)abstractJ. Castle, W. Emmenish, R. Henkes, R. Miller, and J. Rayne (1965) Science by Degrees: ''Temperature from Zero to Zero'' (Westinghouse Search Book Series, Walker and Company, New York). Thermodynamic beta has units reciprocal to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sample Size
Sample size determination is the act of choosing the number of observations or Replication (statistics), replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make statistical inference, inferences about a statistical population, population from a sample. In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power. In complicated studies there may be several different sample sizes: for example, in a stratified sampling, stratified survey sampling, survey there would be different sizes for each stratum. In a census, data is sought for an entire population, hence the intended sample size is equal to the population. In experimental design, where a study may be divided into different treatment groups, there may be different sample sizes for each group. Sample sizes may be chosen in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
True Probability Distribution
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" ( Herman Adèr quoting Kenneth Bollen). All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference. Introduction Informally, a statistical model can be thought of as a statistical assumption (or set of statistical assumptions) with a certain property: that the assumption allows us to calculate the probability of any event. As an example, consider a pair of ordinary six-side ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deviance Information Criterion
The deviance information criterion (DIC) is a hierarchical modeling generalization of the Akaike information criterion (AIC). It is particularly useful in Bayesian model selection problems where the posterior distributions of the models have been obtained by Markov chain Monte Carlo (MCMC) simulation. DIC is an asymptotic approximation as the sample size becomes large, like AIC. It is only valid when the posterior distribution is approximately multivariate normal. Definition Define the deviance as D(\theta)=-2 \log(p(y, \theta))+C\, , where y are the data, \theta are the unknown parameters of the model and p(y, \theta) is the likelihood function. C is a constant that cancels out in all calculations that compare different models, and which therefore does not need to be known. There are two calculations in common usage for the effective number of parameters of the model. The first, as described in , is p_D=\overline-D(\bar), where \bar is the expectation of \theta. The second, as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hannan–Quinn Information Criterion
In statistics, the Hannan–Quinn information criterion (HQC) is a criterion for model selection. It is an alternative to Akaike information criterion (AIC) and Bayesian information criterion (BIC). It is given as : \mathrm = -2 L_ + 2 k \ln(\ln(n)), \ where ''L_'' is the log-likelihood, ''k'' is the number of parameters, and ''n'' is the number of observations. Burnham & Anderson (2002, p. 287) say that HQC, "while often cited, seems to have seen little use in practice". They also note that HQC, like BIC, but unlike AIC, is not an estimator of Kullback–Leibler divergence. Claeskens & Hjort (2008, ch. 4) note that HQC, like BIC, but unlike AIC, is not asymptotically efficient; however, it misses the optimal estimation rate by a very small \ln(\ln(n)) factor. They further point out that whatever method is being used for fine-tuning the criterion will be more important in practice than the term \ln(\ln(n)), since this latter number is small even for very large n; h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shibata Information Criterion , Train Coupler
{{disambiguation, geo ...
Shibata may refer to: Places * Shibata, Miyagi, a town in Miyagi Prefecture * Shibata District, Miyagi, a district in Miyagi Prefecture * Shibata, Niigata, a city in Niigata Prefecture ** Shibata Station (Niigata), a railway station in Niigata Prefecture * Shibata Station (Aichi), a railway station in Aichi Prefecture Other uses * Shibata (surname), a Japanese surname *Shibata clan, Japanese clan originating in the 12th century *Shibata coupler A coupling (or a coupler) is a mechanism typically placed at each end of a railway vehicle that connects them together to form a train. A variety of coupler types have been developed over the course of railway history. Key issues in their desig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
