Credible interval

Credible intervals are typically used to characterize posterior probability distributions or predictive probability distributions. Their generalization to disconnected or multivariate sets is called credible set or credible region.

Credible intervals are a Bayesian analog to confidence intervals in frequentist statistics. The two concepts arise from different philosophies: 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.

Definitions

Credible sets are not unique, as any given probability distribution has an infinite number of ${\displaystyle \gamma }$-credible sets, i.e. sets of probability ${\displaystyle \gamma }$. For example, in the univariate case, there are multiple definitions for a suitable interval or set:

• The smallest credible interval (SCI), sometimes also called the highest density interval. This interval necessarily contains the median whenever ${\displaystyle \gamma \ge 0.5}$. When the distribution is unimodal, this interval also contains the mode.
• The smallest credible set (SCS), sometimes also called the highest density region. For a multimodal distribution, this is not necessarily an interval as it can be disconnected. This set always contains the mode.
• A quantile-based credible interval, which is computed by taking the inter-quantile interval ${\displaystyle [q_{\delta },q_{\delta +\gamma }]}$ for some predefined ${\displaystyle \delta \in [0,1-\gamma ]}$. For instance, the median credible interval (MCI) of probability ${\displaystyle \gamma }$ is the interval where the probability of being below the interval is as likely as being above it, that is to say the interval ${\displaystyle [q_{(1-\gamma )/2},q_{(1+\gamma )/2}]}$. It is sometimes also called the equal-tailed interval, and it always contains the median. Other quantile-based credible intervals can be defined, such as the lowest credible interval (LCI) which is ${\displaystyle [q_0,q_{\gamma }]}$, or the highest credible interval (HCI) which is ${\displaystyle [q_{1-\gamma },q_1]}$. These intervals may be more suited for bounded variables.

One may also define an interval for which the mean is the central point, assuming that the mean exists.

${\displaystyle \gamma }$-Smallest Credible Sets (${\displaystyle \gamma }$-SCS) can easily be generalized to the multivariate case, and are bounded by probability density contour lines. They always contain the mode, but not necessarily the mean, the coordinate-wise median, nor the geometric median.

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