World Library  
Flag as Inappropriate
Email this Article

Credible intervals

Article Id: WHEBN0021752285
Reproduction Date:

Title: Credible intervals  
Author: World Heritage Encyclopedia
Language: English
Subject: Statistical inference
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Credible intervals

In Bayesian statistics, a credible interval (or Bayesian confidence interval) is an interval in the domain of a posterior probability 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]

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.

Choosing a credible interval

Credible intervals are not unique on a posterior distribution. Methods for defining a suitable credible interval include:

  • Choosing the narrowest interval, which for a unimodal distribution will involve choosing those values of highest probability density including the mode.
  • Choosing the interval where the probability of being below the interval is as likely as being above it. This interval will include the median.
  • Assuming the mean exists, choosing the interval for which the mean is the central point.

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.[3]

Contrasts with confidence interval

A frequentist 95% confidence interval of 35–45 means that with a large number of repeated samples, 95% of the 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 confidence interval as "... one interval generated by a procedure that will give correct intervals 95 % of the time".[4]

In general, Bayesian credible intervals do not coincide with frequentist confidence intervals for two reasons:

  • credible intervals incorporate problem-specific contextual information from the prior distribution whereas confidence intervals are based only on the data;
  • credible intervals and confidence intervals treat nuisance parameters in radically different ways.

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;[5] 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 \scriptstyle{\mathrm{Pr}(s|I) \;\propto\; 1/s} [5] — 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.

References

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.