# Credible interval

Short description: Concept in Bayesian statistics

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 a Bayesian analog to confidence intervals in frequentist statistics.[2] The two concepts arise from different philosophies:[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 $\displaystyle{ \mu }$, if the subjective probability that $\displaystyle{ \mu }$ lies between 35 and 45 is 0.95, then $\displaystyle{ 35 \le \mu \le 45 }$ 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 (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, a smallest interval will always be a highest probability density set. It is bounded by the contour of the density.[4]

Credible intervals can also be estimated through the use of simulation techniques such as Markov chain Monte Carlo.[5]

## Contrasts with confidence 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.

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 coincide if the unknown parameter is a location parameter (i.e. the forward probability function has the form $\displaystyle{ \mathrm{Pr}(x|\mu) = f(x - \mu) }$), 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 $\displaystyle{ \mathrm{Pr}(x|s) = f(x/s) }$), with a Jeffreys' prior   $\displaystyle{ \mathrm{Pr}(s|I) \;\propto\; 1/s }$ [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.

## References

1. Edwards, Ward; Lindman, Harold; Savage, Leonard J. (1963). "Bayesian statistical inference in psychological research". Psychological Review 70 (3): 193-242. doi:10.1037/h0044139.
2. Lee, P.M. (1997) Bayesian Statistics: An Introduction, Arnold. ISBN:0-340-67785-6
3. O'Hagan, A. (1994) Kendall's Advanced Theory of Statistics, Vol 2B, Bayesian Inference, Section 2.51. Arnold, ISBN:0-340-52922-9
4. Chen, Ming-Hui; Shao, Qi-Man (1 March 1999). "Monte Carlo Estimation of Bayesian Credible and HPD Intervals". Journal of Computational and Graphical Statistics 8 (1): 69–92. doi:10.1080/10618600.1999.10474802.
5. 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