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‎D/cvlme.md‎

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---
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layout: definition
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-11-19 04:55:00
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title: "Cross-validated log model evidence"
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chapter: "Model Selection"
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section: "Bayesian model selection"
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topic: "Log model evidence"
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definition: "Cross-validated log model evidence"
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sources:
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- authors: "Soch J, Allefeld C, Haynes JD"
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year: 2016
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title: "How to avoid mismodelling in GLM-based fMRI data analysis: cross-validated Bayesian model selection"
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in: "NeuroImage"
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pages: "vol. 141, pp. 469-489, eqs. 13-15"
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url: "https://www.sciencedirect.com/science/article/pii/S1053811916303615"
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doi: "10.1016/j.neuroimage.2016.07.047"
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- authors: "Soch J, Meyer AP, Allefeld C, Haynes JD"
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year: 2017
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title: "How to improve parameter estimates in GLM-based fMRI data analysis: cross-validated Bayesian model averaging"
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in: "NeuroImage"
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pages: "vol. 158, pp. 186-195, eq. 6"
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url: "https://www.sciencedirect.com/science/article/pii/S105381191730527X"
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doi: "10.1016/j.neuroimage.2017.06.056"
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- authors: "Soch J, Allefeld C"
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year: 2018
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title: "MACS – a new SPM toolbox for model assessment, comparison and selection"
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in: "Journal of Neuroscience Methods"
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pages: "vol. 306, pp. 19-31, eqs. 14-15"
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url: "https://www.sciencedirect.com/science/article/pii/S0165027018301468"
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doi: "10.1016/j.jneumeth.2018.05.017"
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- authors: "Soch J"
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year: 2018
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title: "cvBMS and cvBMA: filling in the gaps"
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in: "arXiv stat.ME"
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pages: "arXiv:1807.01585"
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url: "https://arxiv.org/abs/1807.01585"
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def_id: "D111"
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shortcut: "cvlme"
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username: "JoramSoch"
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---
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**Definition:** Let there be a [data set](/D/data) $y$ with mutually exclusive and collectively exhaustive subsets $y_1, \ldots, y_S$. Assume a [generative model](/D/gm) $m$ with model parameters $\theta$ implying a [likelihood function](/D/lf) $p(y \vert \theta, m)$ and a [non-informative](/D/prior-inf) [prior density](/D/prior) $p(\theta \vert m)$.
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Then, the cross-validated log model evidence of $m$ is given by
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$$ \label{eq:cvLME}
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\mathrm{cvLME}(m) = \sum_{i=1}^{S} \log \int p( y_i \vert \theta, m ) \, p( \theta \vert y_{\neg i}, m ) \, \mathrm{d}\theta
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$$
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where $y_{\neg i} = \bigcup_{j \neq i} y_j$ is the union of all data subsets except $y_i$ and $p( \theta \vert y_{\neg i}, m )$ is the [posterior distribution](/D/post) obtained from $y_{\neg i}$ when using the [prior distribution](/D/prior) $p(\theta \vert m)$:
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$$ \label{eq:post}
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p( \theta \vert y_{\neg i}, m ) = \frac{p( y_{\neg i} \vert \theta, m ) \, p(\theta \vert m)}{p( y_{\neg i} \vert m )} \; .
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$$

‎D/ind-cond.md‎

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---
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layout: definition
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-11-19 05:40:00
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title: "Conditional independence"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Probability"
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definition: "Conditional independence"
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sources:
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- authors: "Wikipedia"
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year: 2020
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title: "Conditional independence"
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in: "Wikipedia, the free encyclopedia"
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pages: "retrieved on 2020-11-19"
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url: "https://en.wikipedia.org/wiki/Conditional_independence#Conditional_independence_of_random_variables"
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def_id: "D112"
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shortcut: "ind-cond"
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username: "JoramSoch"
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---
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**Definition:** Generally speaking, [random variables](/D/rvar) are conditionally independent given another random variable, if they are [statistically independent](/D/ind) in their [conditional probability distributions](/D/dist-cond) given this random variable.
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<br>
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1) A set of [discrete random variables](/D/rvar-disc) $X_1, \ldots, X_n$ with possible values $\mathcal{X}_1, \ldots, \mathcal{X}_n$ is called conditionally independent given the random variable $Y$ with possible values $\mathcal{Y}$, if
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$$ \label{eq:disc-ind}
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p(X_1 = x_1, \ldots, X_n = x_n|Y = y) = \prod_{i=1}^{n} p(X_i = x_i|Y = y) \quad \text{for all} \; x_i \in \mathcal{X}_i \quad \text{and all} \; y \in \mathcal{Y}
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$$
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where $p(x_1, \ldots, x_n \vert y)$ are the [joint (conditional) probabilities](/D/prob-joint) of $X_1, \ldots, X_n$ given $Y$ and $p(x_i)$ are the [marginal (conditional) probabilities](/D/prob-marg) of $X_i$ given $Y$.
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<br>
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2) A set of [random variables](/D/rvar) $X_1, \ldots, X_n$ with possible values $\mathcal{X}_1, \ldots, \mathcal{X}_n$ is called conditionally independent given the random variable $Y$ with possible values $\mathcal{Y}$, if
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$$ \label{eq:cond-ind-F}
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F_{X_1,\ldots,X_n|Y=y}(x_1,\ldots,x_n) = \prod_{i=1}^{n} F_{X_i|Y=y}(x_i) \quad \text{for all} \; x_i \in \mathcal{X}_i \quad \text{and all} \; y \in \mathcal{Y}
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$$
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or equivalently, if the [probability densities](/D/pdf) exist, if
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$$ \label{eq:cont-ind-f}
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f_{X_1,\ldots,X_n|Y=y}(x_1,\ldots,x_n) = \prod_{i=1}^{n} f_{X_i|Y=y}(x_i) \quad \text{for all} \; x_i \in \mathcal{X}_i \quad \text{and all} \; y \in \mathcal{Y}
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$$
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where $F$ are the [joint (conditional)](/D/dist-joint) or [marginal (conditional)](/D/dist-marg) [cumulative distribution functions](/D/cdf) and $f$ are the respective [probability density functions](/D/pdf).

‎D/reve.md‎

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---
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layout: definition
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-11-19 04:33:00
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title: "Random event"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Random variables"
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definition: "Random event"
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sources:
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- authors: "Wikipedia"
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year: 2020
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title: "Event (probability theory)"
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in: "Wikipedia, the free encyclopedia"
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pages: "retrieved on 2020-11-19"
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url: "https://en.wikipedia.org/wiki/Event_(probability_theory)"
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def_id: "D110"
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shortcut: "reve"
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username: "JoramSoch"
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---
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**Definition:** A random event $E$ is the outcome of a [random experiment](/D/rexp) which can be described by a statement that is either true or false.
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* If the statement is true, the event is said to take place, denoted as $E$.
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* If the statement is false, the complement of $E$ occurs, denoted as $\overline{E}$.
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In other words, a random event is a [random variable](/D/rvar) with two possible values (true and false, or 1 and 0). A [random experiment](/D/rexp) with two possible outcomes is called a [Bernoulli trial](/D/bern).

‎D/rexp.md‎

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---
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layout: definition
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-11-19 04:10:00
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title: "Random experiment"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Random variables"
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definition: "Random experiment"
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sources:
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- authors: "Wikipedia"
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year: 2020
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title: "Experiment (probability theory)"
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in: "Wikipedia, the free encyclopedia"
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pages: "retrieved on 2020-11-19"
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url: "https://en.wikipedia.org/wiki/Experiment_(probability_theory)"
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def_id: "D109"
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shortcut: "rexp"
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username: "JoramSoch"
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---
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**Definition:** A random experiment is any repeatable procedure that [results in one](/D/rvar) out of a well-defined set of possible outcomes.
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* The set of possible outcomes is called sample space.
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* A set of zero or more outcomes is called a [random event](/D/reve).
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* A function that maps from events to probabilities is called a [probability function](/D/pmf).
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Together, sample space, event space and probability function characterize a random experiment.

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