This article will be permanently flagged as inappropriate and made unaccessible to everyone. Are you certain this article is inappropriate? Excessive Violence Sexual Content Political / Social
Email Address:
Article Id: WHEBN0003408427 Reproduction Date:
In Bayesian statistics, a credible interval is an interval in the domain of a posterior probability distribution or predictive distribution used for interval estimation.^{[1]} The generalisation to multivariate problems is the credible region. Credible intervals are analogous to confidence intervals in frequentist statistics,^{[2]} although they differ on a philosophical basis;^{[3]} Bayesian intervals treat their bounds as fixed and the estimated parameter as a random variable, whereas frequentist confidence intervals treat their bounds as random variables and the parameter as a fixed value.
For example, in an experiment that determines the uncertainty distribution of parameter t, if the probability that t lies between 35 and 45 is 0.95, then 35 \le t \le 45 is a 95% credible interval.
Credible intervals are not unique on a posterior distribution. Methods for defining a suitable credible interval include:
It is possible to frame the choice of a credible interval within decision theory and, in that context, an optimal interval will always be a highest probability density set.^{[4]}
A frequentist 95% confidence interval means that with a large number of repeated samples, 95% of such calculated confidence intervals would include the true value of the parameter. The probability that the parameter is inside the given interval (say, 35–45) is either 0 or 1 (the non-random unknown parameter is either there or not). In frequentist terms, the parameter is fixed (cannot be considered to have a distribution of possible values) and the confidence interval is random (as it depends on the random sample). Antelman (1997, p. 375) summarizes a [95%] confidence interval as "... one interval generated by a procedure that will give correct intervals 95% of the time".^{[5]}
In general, Bayesian credible intervals do not coincide with frequentist confidence intervals for two reasons:
For the case of a single parameter and data that can be summarised in a single sufficient statistic, it can be shown that the credible interval and the confidence interval will coincide if the unknown parameter is a location parameter (i.e. the forward probability function has the form \mathrm{Pr}(x|\mu) = f(x - \mu) ), with a prior that is a uniform flat distribution;^{[6]} and also if the unknown parameter is a scale parameter (i.e. the forward probability function has the form \mathrm{Pr}(x|s) = f(x/s) ), with a Jeffreys' prior \mathrm{Pr}(s|I) \;\propto\; 1/s ^{[6]} — the latter following because taking the logarithm of such a scale parameter turns it into a location parameter with a uniform distribution. But these are distinctly special (albeit important) cases; in general no such equivalence can be made.
Probability theory, Regression analysis, Mathematics, Observational study, Calculus
Statistics, Probability interpretations, Bayesian inference, Bayesian probability, Hyperparameter
Statistics, Nonparametric regression, Robust regression, Least squares, Ordinary least squares
Statistics, Science, Regression analysis, Bayes' theorem, Bayes' rule
Statistics, Canada, English language, French language, Peer review
Christian democracy, Socialism, Politics, Anarchism, Capitalism
Regression analysis, Bayesian statistics, Normal distribution, Seymour Geisser, Statistical inference
Statistics, Bayesian inference, Regression analysis, Probability distribution, Sampling (statistics)