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Credible interval
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intervals
//-->{{short description|Concept in Bayesian statistics}}(File:Highest posterior density interval.svg|thumb|350px|The highest-density 90% credible interval of a probability distribution){{Bayesian statistics}}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 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 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.JOURNAL, Edwards, Ward, Lindman, Harold, Savage, Leonard J., 1963, Bayesian statistical inference in psychological research, Psychological Review, 70, 3, 193-242, 10.1037/h0044139, The generalisation to multivariate problems is the credible region. Credible intervals are a Bayesian analog to confidence intervals in frequentist statistics.Lee, P.M. (1997) Bayesian Statistics: An Introduction, Arnold. {{ISBN|0-340-67785-6}} The two concepts arise from different philosophies:WEB, VanderPlas, Jake, Frequentism and Bayesianism III: Confidence, Credibility, and why Frequentism and Science do not Mix {{!, Pythonic Perambulations |url=https://jakevdp.github.io/blog/2014/06/12/frequentism-and-bayesianism-3-confidence-credibility/ |website=jakevdp.github.io}} 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. Also, Bayesian credible intervals use (and indeed, require) knowledge of the situation-specific prior distribution, while the frequentist confidence intervals do not.Choosing a credible interval
Credible intervals are not unique; any given posterior probability distribution has an infinite number of 95% credible intervals. There are therefore multiple methods for defining a suitable credible interval:- Choosing the narrowest interval. For a unimodal distribution, this interval will include the mode (the maximum a posteriori). This is sometimes called the highest posterior density interval (HPDI).
- Choosing the interval where the probability of being below the interval is as likely as being above it. This interval will include the median. This is sometimes called the equal-tailed interval.
- Assuming that the mean exists, choosing the interval for which the mean is the central point.
Contrasts with confidence interval{{anchor|Confidence interval}}
{{see also|Confidence interval#Credible interval}}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. 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).Bayesian credible intervals differ from frequentist confidence intervals by two major aspects:- credible intervals are intervals whose values have a (posterior) probability density, representing the plausibility that the parameter has those values, whereas confidence intervals regard the population parameter as fixed and therefore not the object of probability. Within confidence intervals, confidence refers to the randomness of the very confidence interval under repeated trials, whereas credible intervals analyse the uncertainty of the target parameter given the data at hand.
- credible intervals and confidence intervals treat nuisance parameters in radically different ways.
References
{{Reflist}}Further reading
- JOURNAL, Morey, R. D., Hoekstra, R., Rouder, J. N., Lee, M. D., Wagenmakers, E.-J., The fallacy of placing confidence in confidence intervals, Psychonomic Bulletin & Review, 23, 1, 2016, 103â123, 10.3758/s13423-015-0947-8, 26450628, 4742505,
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