added 2 definitions

Author: JoramSoch

D/dir-data.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-10-22 05:06:00
+
+title: "Dirichlet-distributed data"
+chapter: "Statistical Models"
+section: "Probability data"
+topic: "Dirichlet-distributed data"
+definition: "Definition"
+
+sources:
+
+def_id: "D104"
+shortcut: "dir-data"
+username: "JoramSoch"
+---
+
+
+**Definition:** Dirichlet-distributed data are defined as a set of vectors of proportions $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$ where
+
+$$ \label{eq:dir-def}
+\begin{split}
+y_i &= [y_{i1}, \ldots, y_{ik}], \\
+y_{ij} &\in [0,1] \quad \text{and} \\
+\sum_{j=1}^k y_{ij} &= 1
+\end{split}
+$$
+
+for all $i = 1,\ldots,n$ (and $j = 1,\ldots,k$) and each $y_i$ is independent and identically distributed according to a [Dirichlet distribution](/D/dir) with concentration parameters $\alpha = [\alpha_1, \ldots, \alpha_k]$:
+
+$$ \label{eq:dir-data}
+y_i \sim \mathrm{Dir}(\alpha), \quad i = 1, \ldots, n \; .
+$$

D/prob-exc.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-10-22 04:36:00
+
+title: "Exceedance probability"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Probability"
+definition: "Exceedance probability"
+
+sources:
+  - authors: "Stephan KE, Penny WD, Daunizeau J, Moran RJ, Friston KJ"
+    year: 2009
+    title: "Bayesian model selection for group studies"
+    in: "NeuroImage"
+    pages: "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"
+  - authors: "Soch J, Allefeld C"
+    year: 2016
+    title: "Exceedance Probabilities for the Dirichlet Distribution"
+    in: "arXiv stat.AP"
+    pages: "1611.01439"
+    url: "https://arxiv.org/abs/1611.01439"
+
+def_id: "D103"
+shortcut: "prob-exc"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let $X = \left\lbrace X_1, \ldots, X_n \right\rbrace$ be a set of $n$ [random variables](/D/rvar) which the [joint probability distribution](/D/dist-joint) $p(X) = p(X_1, \ldots, X_n)$. Then, the exceedance probability for random variable $X_i$ is the [probability](/D/prob) that $X_i$ is larger than all other random variables $X_j, \; j \neq i$:
+
+$$ \label{eq:EP}
+\begin{split}
+\varphi(X_i) &= \mathrm{Pr}\left( \forall j \in \left\lbrace 1, \ldots, n | j \neq i \right\rbrace: \, X_i > X_j \right) \\
+&= \mathrm{Pr}\left( \bigwedge_{j \neq i} X_i > X_j \right) \\
+&= \mathrm{Pr}\left( X_i = \mathrm{max}(\left\lbrace X_1, \ldots, X_n \right\rbrace) \right) \\
+&= \int_{X_i = \mathrm{max}(X)} p(X) \, \mathrm{d}X \; .
+\end{split}
+$$