# Credible interval

In Bayesian statistics, a **credible interval** is an interval within which an unobserved parameter value falls with a particular probability. It is an interval in the domain of a posterior probability distribution or a predictive distribution.^{[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. Also, Bayesian credible intervals use (and indeed, require) knowledge of the situation-specific prior distribution, while the frequentist confidence intervals do not.

For example, in an experiment that determines the distribution of possible values of the parameter , if the subjective probability that lies between 35 and 45 is 0.95, then is a 95% credible interval.

## Choosing a credible intervalEdit

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 (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.

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]}

## Contrasts with confidence intervalEdit

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 can be quite different from 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 ), 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 ), with a Jeffreys' prior ^{[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.

## ReferencesEdit

**^**Edwards, Ward, Lindman, Harold, Savage, Leonard J. (1963) "Bayesian statistical inference in psychological research".*Psychological Review*,**70**, 193-242**^**Lee, P.M. (1997)*Bayesian Statistics: An Introduction*, Arnold. ISBN 0-340-67785-6**^**"Frequentism and Bayesianism".**^**O'Hagan, A. (1994)*Kendall's Advanced Theory of Statistics, Vol 2B, Bayesian Inference*, Section 2.51. Arnold, ISBN 0-340-52922-9- ^
^{a}^{b}Jaynes, E. T. (1976). "Confidence Intervals vs Bayesian Intervals", in*Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science*, (W. L. Harper and C. A. Hooker, eds.), Dordrecht: D. Reidel, pp. 175*et seq*

## Further readingEdit

- Morey, R. D.; Hoekstra, R.; Rouder, J. N.; Lee, M. D.; Wagenmakers, E.-J. (2016). "The fallacy of placing confidence in confidence intervals".
*Psychonomic Bulletin & Review*.**23**(1): 103–123. doi:10.3758/s13423-015-0947-8. PMC 4742505. PMID 26450628.