The "confidence interval" is a term used by frequentist - I am a frequentist. If we say an estimate has its 90% confidence interval of 35-45, it means that with a large number of repeated samples, 90% of times, the true value of the parameter will fall within the range of 35-45.
The term 'credible interval' is used by Bayesian statisticians and it may also be called 'Bayesian Posterior Interval'. In Bayesian statistics, a credible interval is a posterior probability interval, used for purposes similar to those of confidence intervals in frequentist statistics. Bayesian inference is statistical inference in which probabilities are interpreted not as frequencies or proportions or the like, but rather as degrees of belief. ...
The posterior probability can be calculated by Bayes theorem from the prior probability and the likelihood function. ... In statistics, a confidence interval (CI) is an interval between two numbers, where there is a certain specified level of confidence that a population parameter lies. ... Statistical regularity has motivated the development of the relative frequency concept of probability. ...
For example, a statement such as "following the experiment, a 95% credible interval for the parameter t is 35-45" means that the posterior probability that t lies in the interval from 35 to 45 is 0.9.
A Bayesian credible interval incorporates information from the prior distribution into the estimate, while confidence intervals are based solely on the data.
Like 'confidence interval' vs 'credible interval, there is also 'confidence region' vs 'credible region'.
Here are some links for further reading:
- Credible interval at wikepedia
- Bland and Altman (1998) Bayesians and frequentists. BMJ
- Fisher (1996) Comments on Bayesian and Frequentist Analysis and Interpretation of Clinical Trials. Controlled Clinical Trials
- Credible Interval
- Credible Intervals vs. Confidence Intervals