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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
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 )} \; .
**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
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
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).
**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).
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