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# Credible intervals

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 Title: Credible intervals Author: World Heritage Encyclopedia Language: English Subject: Statistical inference Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Credible intervals

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 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.
• 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 95% confidence interval of 35–45 means that with a large number of repeated samples, 95% 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 % 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 $\mathrm\left\{Pr\right\}\left(x|\mu\right) = f\left(x - \mu\right)$ ), 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 $\mathrm\left\{Pr\right\}\left(x|s\right) = f\left(x/s\right)$ ), with a Jeffreys' prior $\scriptstyle\left\{\mathrm\left\{Pr\right\}\left(s|I\right) \;\propto\; 1/s\right\}$ [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

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