|
Lindley's Paradox
Lindley's paradox is a counterintuitive situation in statistics in which the Bayesian and frequentist approaches to a hypothesis testing problem give different results for certain choices of the prior distribution. The problem of the disagreement between the two approaches was discussed in Harold Jeffreys' 1939 textbook; it became known as Lindley's paradox after Dennis Lindley called the disagreement a paradox in a 1957 paper. Although referred to as a ''paradox'', the differing results from the Bayesian and frequentist approaches can be explained as using them to answer fundamentally different questions, rather than actual disagreement between the two methods. Nevertheless, for a large class of priors the differences between the frequentist and Bayesian approach are caused by keeping the significance level fixed: as even Lindley recognized, "the theory does not justify the practice of keeping the significance level fixed" and even "some computations by Prof. Pearson in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dennis Lindley
Dennis Victor Lindley (25 July 1923 – 14 December 2013) was an English statistician, decision theorist and leading advocate of Bayesian statistics. Biography Lindley grew up in the south-west London suburb of Surbiton. He was an only child and his father was a local building contractor. Lindley recalled (to Adrian Smith) that the family had "little culture" and that both his parents were "proud of the fact that they had never read a book". The school Lindley attended, Tiffin School, introduced him to "ordinary cultural activities"."Lindley Prize – Dennis Lindley" International Society for Bayesian Analysis, ''accessed 31 October 2023'' From there Lindley went t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counterintuitive
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites". In logic, many paradoxes exist that are known to be invalid arguments, yet are nevertheless valuable in promoting critical thinking, while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself and showed that attempts to found set theory on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-value
In null-hypothesis significance testing, the ''p''-value is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. A very small ''p''-value means that such an extreme observed outcome would be very unlikely ''under the null hypothesis''. Even though reporting ''p''-values of statistical tests is common practice in academic publications of many quantitative fields, misinterpretation and misuse of p-values is widespread and has been a major topic in mathematics and metascience. In 2016, the American Statistical Association (ASA) made a formal statement that "''p''-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone" and that "a ''p''-value, or statistical significance, does not measure the size of an effect or the importance of a result" or "evidence regarding a model or hypothesis". That ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Hypothesis Testing
A statistical hypothesis test is a method of statistical inference used to decide whether the data provide sufficient evidence to reject a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. Then a decision is made, either by comparing the test statistic to a Critical value (statistics), critical value or equivalently by evaluating a p-value, ''p''-value computed from the test statistic. Roughly 100 list of statistical tests, specialized statistical tests are in use and noteworthy. History While hypothesis testing was popularized early in the 20th century, early forms were used in the 1700s. The first use is credited to John Arbuthnot (1710), followed by Pierre-Simon Laplace (1770s), in analyzing the human sex ratio at birth; see . Choice of null hypothesis Paul Meehl has argued that the epistemological importance of the choice of null hypothesis has gone largely unacknowledged. When the null hypothesis is predicted by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Statistical Association
The ''Journal of the American Statistical Association'' is a quarterly peer-reviewed scientific journal published by Taylor & Francis on behalf of the American Statistical Association. It covers work primarily focused on the application of statistics, statistical theory and methods in economic, social, physical, engineering, and health sciences. The journal also includes reviews of books which are relevant to the field. The journal was established in 1888 as the ''Publications of the American Statistical Association''. It was renamed ''Quarterly Publications of the American Statistical Association'' in 1912, obtaining its current title in 1922. Reception According to the ''Journal Citation Reports ''Journal Citation Reports'' (''JCR'') is an annual publication by Clarivate. It has been integrated with the Web of Science and is accessed from the Web of Science Core Collection. It provides information about academic journals in the natur ...'', the journal has a 2023 impac ... [...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] |
Maximum Likelihood
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, ML ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delta Function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as \delta (x) = \begin 0, & x \neq 0 \\ , & x = 0 \end such that \int_^ \delta(x) dx=1. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions. The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1. The mathematical rigor of the delta function wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Effect Size
In statistics, an effect size is a value measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the value of one parameter for a hypothetical population, or to the equation that operationalizes how statistics or parameters lead to the effect size value. Examples of effect sizes include the correlation between two variables, the regression coefficient in a regression, the mean difference, or the risk of a particular event (such as a heart attack) happening. Effect sizes are a complement tool for statistical hypothesis testing, and play an important role in power analyses to assess the sample size required for new experiments. Effect size are fundamental in meta-analyses which aim to provide the combined effect size based on data from multiple studies. The cluster of data-analysis methods concerning effect sizes is referred to as estima ... [...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] |
Illustration
An illustration is a decoration, interpretation, or visual explanation of a text, concept, or process, designed for integration in print and digitally published media, such as posters, flyers, magazines, books, teaching materials, animations, video games and films. An illustration is typically created by an illustrator. Digital illustrations are often used to make websites and apps more user-friendly, such as the use of emojis to accompany digital type. Illustration also means providing an example; either in writing or in picture form. The origin of the word "illustration" is late Middle English (in the sense ‘illumination; spiritual or intellectual enlightenment’): via Old French from Latin">-4; we might wonder whether there's a point at which it's appropriate to talk of the beginnings of French, that is, when it wa ... from Latin ''illustratio''(n-), from the verb ''illustrare''. Illustration styles Contemporary illustration uses a wide range of styles and techniq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Sure Hypothesis Testing
In statistics, almost sure hypothesis testing or a.s. hypothesis testing utilizes almost sure convergence in order to determine the validity of a statistical hypothesis with probability one. This is to say that whenever the null hypothesis is true, then an a.s. hypothesis test will fail to reject the null hypothesis w.p. 1 for all sufficiently large samples. Similarly, whenever the alternative hypothesis is true, then an a.s. hypothesis test will reject the null hypothesis with probability one, for all sufficiently large samples. Along similar lines, an a.s. confidence interval eventually contains the parameter of interest with probability 1. Dembo and Peres (1994) proved the existence of almost sure hypothesis tests. Description For simplicity, assume we have a sequence of independent and identically distributed normal random variables, \textstyle x_i \sim N(\mu,1), with mean \textstyle \mu , and unit variance. Suppose that nature or simulation has chosen the true mean to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
