Credible interval

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 90%, then $35 \le t \le 45$ is a 90% 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 90% confidence interval of 35–45 means that with a large number of repeated samples, 90% 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 % [resp. 90 %] 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 Pr(x | μ) = f(x − μ) ), 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 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

1. ^ Edwards, W., Lindman, H., Savage, L.J. (1963) "Bayesian statistical inference in statistical research". Psychological Research, 70, 193-242
2. ^ Lee, P.M. (1997) Bayesian Statistics: An Introduction, Arnold. ISBN 0-340-67785-6
3. ^ O'Hagan, A. (1994) Kendall's Advance Theory of Statistics, Vol 2B, Bayesian Inference, Section 2.51. Arnold, ISBN 0-340-52922-9
4. ^ Antelman, G. (1997) Elementary Bayesian Statistics (Madansky, A. & McCulloch, R. eds.). Cheltenham, UK: Edward Elgar ISBN 978-1-85898-504-6
5. ^ ^{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