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Interval Estimation
In statistics, interval estimation is the use of sample data to estimate an '' interval'' of possible values of a parameter of interest. This is in contrast to point estimation, which gives a single value. The most prevalent forms of interval estimation are '' confidence intervals'' (a frequentist method) and ''credible intervals'' (a Bayesian method). Less common forms include '' likelihood intervals,'' '' fiducial intervals,'' ''tolerance intervals,'' and ''prediction intervals''. For a non-statistical method, interval estimates can be deduced from fuzzy logic. Types Confidence intervals Confidence intervals are used to estimate the parameter of interest from a sampled data set, commonly the mean or standard deviation. A confidence interval states there is a 100γ% confidence that the parameter of interest is within a lower and upper bound. A common misconception of confidence intervals is 100γ% of the data set fits within or above/below the bounds, this is referred ... [...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] |
Variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who almost single-handedly created the foundations for modern statistical science" and "the single most important figure in 20th century statistics". In genetics, Fisher was the one to most comprehensively combine the ideas of Gregor Mendel and Charles Darwin, as his work used mathematics to combine Mendelian genetics and natural selection; this contributed to the revival of Darwinism in the early 20th-century revision of the theory of evolution known as the Modern synthesis (20th century), modern synthesis. For his contributions to biology, Richard Dawkins declared Fisher to be the greatest of Darwin's successors. He is also considered one of the founding fathers of Neo-Darwinism. According to statistician Jeffrey T. Leek, Fisher is the most in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Inference
Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. In machine learning, the term ''inference'' is sometimes used instead to mean "make a prediction, by evaluating an already trained model"; in this context inferring properties of the model is referred to as ''training'' or ''learning'' (rather than ''inference''), and using a model for prediction is referred to as ''inference'' (instead of ''prediction''); se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiducial Inference
Fiducial inference is one of a number of different types of statistical inference. These are rules, intended for general application, by which conclusions can be drawn from samples of data. In modern statistical practice, attempts to work with fiducial inference have fallen out of fashion in favour of frequentist inference, Bayesian inference and decision theory. However, fiducial inference is important in the history of statistics since its development led to the parallel development of concepts and tools in theoretical statistics that are widely used. Some current research in statistical methodology is either explicitly linked to fiducial inference or is closely connected to it. Background The general approach of fiducial inference was proposed by Ronald Fisher. Here "fiducial" comes from the Latin for faith. Fiducial inference can be interpreted as an attempt to perform inverse probability without calling on prior probability distributions. Fiducial inference quickly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highest Posterior Density Interval
In Bayesian statistics, a credible interval is an interval used to characterize a probability distribution. It is defined such that an unobserved parameter value has a particular probability \gamma to fall within it. For example, in an experiment that determines the distribution of possible values of the parameter \mu, if the probability that \mu lies between 35 and 45 is \gamma=0.95, then 35 \le \mu \le 45 is a 95% credible interval. Credible intervals are typically used to characterize posterior probability distributions or predictive probability distributions. Their generalization to disconnected or multivariate sets is called credible set or credible region. Credible intervals are a Bayesian analog to confidence intervals in frequentist statistics. The two concepts arise from different philosophies: Bayesian intervals treat their bounds as fixed and the estimated parameter as a random variable, whereas frequentist confidence intervals treat their bounds as random varia ... [...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] |
Bayes Factor
The Bayes factor is a ratio of two competing statistical models represented by their evidence, and is used to quantify the support for one model over the other. The models in question can have a common set of parameters, such as a null hypothesis and an alternative, but this is not necessary; for instance, it could also be a non-linear model compared to its linear approximation. The Bayes factor can be thought of as a Bayesian analog to the likelihood-ratio test, although it uses the integrated (i.e., marginal) likelihood rather than the maximized likelihood. As such, both quantities only coincide under simple hypotheses (e.g., two specific parameter values). Also, in contrast with null hypothesis significance testing, Bayes factors support evaluation of evidence ''in favor'' of a null hypothesis, rather than only allowing the null to be rejected or not rejected. Although conceptually simple, the computation of the Bayes factor can be challenging depending on the complexity of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prior Probability
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 const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bootstrapping (statistics)
Bootstrapping is a procedure for estimating the distribution of an estimator by resampling (often with replacement) one's data or a model estimated from the data. Bootstrapping assigns measures of accuracy ( bias, variance, confidence intervals, prediction error, etc.) to sample estimates.software This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods. Bootstrapping estimates the properties of an estimand (such as its ) by measuring those properties when sampling from an approximating distribution. One standard choice for an approximating distributi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Distribution
In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1 (e.g., number of events in a given area or volume). The Poisson distribution is named after French mathematician Siméon Denis Poisson. It plays an important role for discrete-stable distributions. Under a Poisson distribution with the expectation of ''λ'' events in a given interval, the probability of ''k'' events in the same interval is: :\frac . For instance, consider a call center which receives an average of ''λ ='' 3 calls per minute at all times of day. If the calls are independent, receiving one does not change the probability of when the next on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
