What is the difference between credible and confidence interval?
Credible intervals capture our current uncertainty in the location of the parameter values and thus can be interpreted as probabilistic statement about the parameter. In contrast, confidence intervals capture the uncertainty about the interval we have obtained (i.e., whether it contains the true value or not).
What is a credible set?
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. The generalisation to multivariate problems is the credible region.
Can a credible interval be computed from a posterior distribution?
In some cases the credible interval can be computed numerically based on a known distribution, but it’s more common to generate a credible interval by sampling from the posterior distribution and then to compute quantiles of the samples.
Which is a better summary of the posterior distribution?
In this case it seems that a highest posterior density region is a better summary of the distribution than the equal-tailed confidence interval. This (imagined) example also demonstrates why it is dangerous to try to reduce the posterior distribution to single summary statistics, such as the mean or the mode of the posterior distribution.
When to use an ad hoc posterior distribution?
We may for example have an ad hoc estimate of the region of the parameter space where the true parameter value lies with 95% certainty. Then we just have to find a prior distribution whose 95% credible interval agrees with this estimate. But usually credible intervals are examined after observing the data.
Is the posterior density of the credible interval symmetric?
However, unless the posterior distribution is unimodal and symmetric, there are point outsed of the equal-tailed credible interval having a higher posterior density than some points of the interval. If we want to choose the credible interval so that this not happen, we can do it by using the highest posterior density criterion for choosing it.