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Copy file name to clipboardExpand all lines: D/cvlme.md
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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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**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) $p_{\mathrm{ni}}(\theta \vert m)$.
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Then, the cross-validated log model evidence of $m$ is given by
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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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) $p_{\mathrm{ni}}(\theta \vert m)$:
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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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p( \theta \vert y_{\neg i}, m ) = \frac{p( y_{\neg i} \vert \theta, m ) \, p_{\mathrm{ni}}(\theta \vert m)}{p( y_{\neg i} \vert m )} \; .
Copy file name to clipboardExpand all lines: D/rvar-uni.md
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**Definition:** Let $X$ be a [random variable](/D/rvar) with possible outcomes $\mathcal{X}$. Then,
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* $X$ is called a two-valued random variable or [random event](/D/reve), if $\mathcal{X}$ has exactly two elements, e.g. $\mathcal{X} = \left\lbrace \mathrm{true}, \mathrm{false} \right\rbrace$ or $\mathcal{X} = \left\lbrace 1, 0 \right\rbrace$;
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* $X$ is called a univariate random variable or [random scalar](/D/rvar), if $\mathcal{X}$ is one-dimensional, i.e. (a subset of) the real numbers $\mathbb{R}$;
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* $X$ is called a multivariate random variable or [random vector](/D/rvec), if $\mathcal{X}$ is multi-dimensional, e.g. (a subset of) the $n$-dimensional Euclidean space $\mathbb{R}^n$;
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