Definition: Exceedance probability
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The Book of Statistical Proofs ▷
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Exceedance probability
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Metadata: ID: D103 | shortcut: prob-exc | author: JoramSoch | date: 2020-10-22, 04:36.
Definition: Let $X = \left\lbrace X_1, \ldots, X_n \right\rbrace$ be a set of $n$ random variables which the joint probability distribution $p(X) = p(X_1, \ldots, X_n)$. Then, the exceedance probability for random variable $X_i$ is the probability that $X_i$ is larger than all other random variables $X_j, \; j \neq i$:
\[\label{eq:EP} \begin{split} \varphi(X_i) &= \mathrm{Pr}\left( \forall j \in \left\lbrace 1, \ldots, n | j \neq i \right\rbrace: \, X_i > X_j \right) \\ &= \mathrm{Pr}\left( \bigwedge_{j \neq i} X_i > X_j \right) \\ &= \mathrm{Pr}\left( X_i = \mathrm{max}(\left\lbrace X_1, \ldots, X_n \right\rbrace) \right) \\ &= \int_{X_i = \mathrm{max}(X)} p(X) \, \mathrm{d}X \; . \end{split}\]- Stephan KE, Penny WD, Daunizeau J, Moran RJ, Friston KJ (2009): "Bayesian model selection for group studies"; in: NeuroImage, vol. 46, pp. 1004–1017, eq. 16; URL: https://www.sciencedirect.com/science/article/abs/pii/S1053811909002638; DOI: 10.1016/j.neuroimage.2009.03.025.
- Soch J, Allefeld C (2016): "Exceedance Probabilities for the Dirichlet Distribution"; in: arXiv stat.AP, 1611.01439; URL: https://arxiv.org/abs/1611.01439.
Metadata: ID: D103 | shortcut: prob-exc | author: JoramSoch | date: 2020-10-22, 04:36.