Credibility interval

In statistics  , a credibility interval , also known as the credibility interval , is the Bayesian counterpart to the confidence interval in frequentist statistics . Bayesian interval estimators are derived from the posterior distribution . To distinguish them from the confidence intervals that have a different interpretation, they are called belief intervals. The confidence interval says that the unknown parameter is likely to be in this interval. ${\displaystyle \vartheta }$ ${\displaystyle \gamma }$

definition

For a fixedly specified , a credibility interval for the credibility level (also: a credibility interval ) is defined by two real numbers and which ones ${\displaystyle \gamma \in (0,1)}$ ${\displaystyle \gamma \cdot 100\,\%}$ ${\displaystyle \vartheta }$ ${\displaystyle \gamma }$ ${\displaystyle \gamma }$ ${\displaystyle t_{l}}$ ${\displaystyle t_{u}}$

${\displaystyle \int _{t_{l}}^{t_{u}}f(\vartheta \mid x)\,\mathrm {d} \vartheta =\gamma }$

fulfill. Here is . Represents the a posteriori distribution The easiest way to complete a credibility interval is to construct than that - quantile and as the choose quantile of the a posteriori distribution. In order to calculate such confidence intervals, one has to calculate the quantiles of the posterior distribution. ${\displaystyle f(\vartheta \mid x)}$ ${\displaystyle t_{l}}$ ${\displaystyle (1-\gamma )/2}$ ${\displaystyle t_{u}}$ ${\displaystyle (1+\gamma )/2}$

interpretation

Since the unknown parameter is a random variable , it can be said that there is a confidence interval with probability . In contrast to this interpretation, a confidence interval says that if the random experiment is repeated in an identical manner, then a confidence interval will cover the unknown parameter in all cases. ${\displaystyle \vartheta \mid x}$ ${\displaystyle \vartheta \mid x}$ ${\displaystyle \gamma \cdot 100\,\%}$ ${\displaystyle \gamma }$ ${\displaystyle \gamma \cdot 100\,\%}$ ${\displaystyle \vartheta }$ ${\displaystyle \gamma \cdot 100\,\%}$

Individual evidence

1. ^ Leonhard Held and Daniel Sabanés Bové: Applied Statistical Inference: Likelihood and Bayes. Springer Heidelberg New York Dordrecht London (2014). ISBN 978-3-642-37886-7 , p. 172.
2. ^ Leonhard Held and Daniel Sabanés Bové: Applied Statistical Inference: Likelihood and Bayes. Springer Heidelberg New York Dordrecht London (2014). ISBN 978-3-642-37886-7 , p. 172.