Credible interval

In Bayesian statistics, a credible interval is an interval in the domain of a posterior probability distribution or predictive 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] 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 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. This is sometimes called the highest posterior density interval.
• 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 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. 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 [95%] confidence interval as "... one interval generated by a procedure that will give correct intervals 95% of the time".^[5]

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;^[6] 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   $\mathrm{Pr}(s|I) \;\propto\; 1/s$ ^[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, 193-242
2. Lee, P.M. (1997) Bayesian Statistics: An Introduction, Arnold. ISBN 0-340-67785-6
3. "Frequentism and Bayesianism".
4. O'Hagan, A. (1994) Kendall's Advanced Theory of Statistics, Vol 2B, Bayesian Inference, Section 2.51. Arnold, ISBN 0-340-52922-9
5. Antelman, G. (1997) Elementary Bayesian Statistics (Madansky, A. & McCulloch, R. eds.). Cheltenham, UK: Edward Elgar ISBN 978-1-85898-504-6
6. 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