added 4 definitions

Author: JoramSoch

D/betabin-data.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-10-20 08:20:00
+
+title: "Beta-binomial data"
+chapter: "Statistical Models"
+section: "Frequency data"
+topic: "Beta-binomial data"
+definition: "Definition"
+
+sources:
+
+def_id: "D178"
+shortcut: "betabin-data"
+username: "JoramSoch"
+---
+
+
+**Definition:** Beta-binomial data are defined as a set of counts $y = \left\lbrace y_1, \ldots, y_N \right\rbrace$ with $y_i \in \mathbb{N}, \; i = 1,\ldots,N$, independent and identically distributed according to a [beta-binomial distribution](/D/betabin) with number of trials $n$ as well as shapes $\alpha$ and $\beta$:
+
+$$ \label{eq:betabin-data}
+y_i \sim \mathrm{BetBin}(n,\alpha,\beta), \quad i = 1, \ldots, N \; .
+$$

D/betabin.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-10-20 08:09:00
+
+title: "Beta-binomial distribution"
+chapter: "Probability Distributions"
+section: "Univariate discrete distributions"
+topic: "Beta-binomial distribution"
+definition: "Definition"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2022
+    title: "Beta-binomial distribution"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2022-10-20"
+    url: "https://en.wikipedia.org/wiki/Beta-binomial_distribution#Motivation_and_derivation"
+
+def_id: "D177"
+shortcut: "betabin"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let $p$ be a [random variable](/D/rvar) following a [beta distribution](/D/beta)
+
+$$ \label{eq:beta}
+p \sim \mathrm{Bet}(\alpha, \beta)
+$$
+
+and let $X$ be a [random variable](/D/rvar) following a [binomial distribution](/D/bin) conditional on $p$
+
+$$ \label{eq:bin}
+X \mid p \sim \mathrm{Bin}(n, p) \; .
+$$
+
+Then, the [marginal distribution](/D/dist-marg) of $X$ is called a beta-binomial distribution
+
+$$ \label{eq:betabin}
+X \sim \mathrm{BetBin}(n, \alpha, \beta)
+$$
+
+with [number of trials](/D/bin) $n$ and [shape parameters](/D/beta) $\alpha$ and $\beta$.

D/fe.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-10-20 09:57:00
+
+title: "Family evidence"
+chapter: "Model Selection"
+section: "Bayesian model selection"
+topic: "Family evidence"
+definition: "Definition"
+
+sources:
+  - authors: "Soch J, Allefeld C"
+    year: 2018
+    title: "MACS – a new SPM toolbox for model assessment, comparison and selection"
+    in: "Journal of Neuroscience Methods"
+    pages: "vol. 306, pp. 19-31, eq. 16"
+    url: "https://www.sciencedirect.com/science/article/pii/S0165027018301468"
+    doi: "10.1016/j.jneumeth.2018.05.017"
+
+def_id: "D180"
+shortcut: "fe"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let $f$ be a family of $M$ [generative models](/D/gm) $m_1, \ldots, m_M$, such that the following statement holds true:
+
+$$ \label{eq:fam}
+f \Leftrightarrow m_1 \vee \ldots \vee m_M \; .
+$$
+
+Then, the family evidence (FE) of $f$ is defined as the [marginal probability](/D/prob-marg) relative to the [model evidences](/D/me) $p(y \vert m_i), conditional only on $f$:
+
+$$ \label{eq:fe}
+\mathrm{FE}(f) = p(y|f) \; .
+$$

D/me.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-10-20 09:43:00
+
+title: "Model evidence"
+chapter: "Model Selection"
+section: "Bayesian model selection"
+topic: "Model evidence"
+definition: "Definition"
+
+sources:
+  - authors: "Penny WD"
+    year: 2012
+    title: "Comparing Dynamic Causal Models using AIC, BIC and Free Energy"
+    in: "NeuroImage"
+    pages: "vol. 59, iss. 2, pp. 319-330, eq. 15"
+    url: "https://www.sciencedirect.com/science/article/pii/S1053811911008160"
+    doi: "10.1016/j.neuroimage.2011.07.039"
+
+def_id: "D179"
+shortcut: "me"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let $m$ be a [generative model](/D/gm) with [likelihood function](/D/lf) $p(y \vert \theta, m)$ and [prior distribution](/D/prior) $p(\theta \vert m)$. Then, the model evidence (ME) of $m$ is defined as the [marginal likelihood](/D/ml) of this model:
+
+$$ \label{eq:ME}
+\mathrm{ME}(m) = p(y|m) \; .
+$$