Proof: Cross-validated log model evidence for the univariate Gaussian
Theorem: Let
\[\label{eq:UG} y = \left\lbrace y_1, \ldots, y_n \right\rbrace, \quad y_i \sim \mathcal{N}(\mu, \sigma^2), \quad i = 1, \ldots, n\]be a univariate Gaussian data set with unknown mean $\mu$ and unknown variance $\sigma^2$. Moreover, assume two statistical models, one assuming that $\mu$ is zero (null model), the other imposing a normal distribution as the prior distribution on the mean parameter $\mu$ (alternative) and both imposing a gamma distribtion on the precision parameter $\tau = 1/\sigma^2$:
\[\label{eq:UG-m01} \begin{split} m_0 &: \; y_i \sim \mathcal{N}(\mu, \tau^{-1}), \; \mu = 0, \; \tau \sim \mathrm{Gam}(a_0, b_0) \\ m_1 &: \; y_i \sim \mathcal{N}(\mu, \tau^{-1}), \; \mu|\tau \sim \mathcal{N}(\mu_0, (\tau \lambda_0)^{-1}), \; \tau \sim \mathrm{Gam}(a_0, b_0) \end{split}\]Then, the cross-validated log model evidences of $m_0$ and $m_1$ are
\[\label{eq:UG-cvLME-m01} \begin{split} \mathrm{cvLME}(m_0) = & - \frac{n}{2} \log (2 \pi) + S \cdot \log \Gamma \left( \frac{n}{2} \right) - S \cdot \log \Gamma \left( \frac{1}{2} \frac{S-1}{S} n \right) \\ &- \frac{S \cdot n}{2} \log \left[ \frac{1}{2} \left( y^\mathrm{T} y \right) \right] + \frac{n_1}{2} \sum_{i=1}^S \log \left[ \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} \right) \right] \\ \mathrm{cvLME}(m_1) = & - \frac{n}{2} \log (2 \pi) + \frac{S}{2} \log \left( \frac{S-1}{S} \right) + S \cdot \log \Gamma \left( \frac{n}{2} \right) - S \cdot \log \Gamma \left( \frac{1}{2} \frac{S-1}{S} n \right) \\ &- \frac{S \cdot n}{2} \log \left[ \frac{1}{2} \left( y^\mathrm{T} y - n {\bar{y}}^2 \right) \right] + \frac{n_1}{2} \sum_{i=1}^S \log \left[ \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} - n_1 \bar{y}_1^{(i)} \right) \right] \end{split}\]where $\bar{y}$ is the sample mean, $y_1^{(i)}$ are the training data in the $i$-th cross-validation fold with $n_1$ data points and $S$ is the number of data subsets.
Proof: For evaluation of the cross-validated log model evidences (cvLME), we assume that $n$ data points are divided into $S \mid n$ data subsets without remainder. Then, the number of training data points $n_1$ and test data points $n_2$ are given by
\[\label{eq:CV-n12} \begin{split} n &= n_1 + n_2 \\ n_1 &= \frac{S-1}{S} n \\ n_2 &= \frac{1}{S} n \; , \end{split}\]such that training data $y_1$ and test data $y_2$ in the $i$-th cross-validation fold are
\[\label{eq:CV-y12} \begin{split} y &= \left\lbrace y_1, \ldots, y_n \right\rbrace \\ y_1^{(i)} &= \left\lbrace x \in y \mid x \notin y_2^{(i)} \right\rbrace = y \backslash y_2^{(i)} \\ y_2^{(i)} &= \left\lbrace y_{(i-1) \cdot n_2 + 1}, \ldots, y_{i \cdot n_2} \right\rbrace \; . \end{split}\]
First, we have a look at the alternative $m_1$ assuming $\mu \neq 0$. To begin, the training data $y_1^{(i)}$ are analyzed using a non-informative prior distribution and applying the posterior distribution for the univariate Gaussian:
This results in a posterior characterized by $\mu_n^{(1)}$, $\lambda_n^{(1)}$, $a_n^{(1)}$ and $b_n^{(1)}$. Then, the test data $y_2^{(i)}$ are analyzed using this posterior as an informative prior distribution, again applying the posterior distribution for the univariate Gaussian:
\[\label{eq:UG-m1-y2} \begin{split} \mu_0^{(2)} &= \mu_n^{(1)} = \bar{y}_1^{(i)} \\ \Rightarrow \quad \mu_n^{(2)} &= \frac{\lambda_0^{(2)} \mu_0^{(2)} + n_2 \bar{y}_2^{(i)}}{\lambda_0^{(2)} + n_2} \\ &= \frac{n_1 \bar{y}_1^{(i)} + n_2 \bar{y}_2^{(i)}}{n_1 + n_2} = \frac{n \bar{y}}{n} = \bar{y} \\ \lambda_0^{(2)} &= \lambda_n^{(1)} = n_1 \\ \Rightarrow \quad \lambda_n^{(2)} &= \lambda_0^{(2)} + n_2 = n_1 + n_2 = n \\ a_0^{(2)} &= a_n^{(1)} = \frac{n_1}{2} \\ \Rightarrow \quad a_n^{(2)} &= a_0^{(2)} + \frac{n_2}{2} = \frac{n_1}{2} + \frac{n_2}{2} = \frac{n}{2} \\ b_0^{(2)} &= b_n^{(1)} = \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} - n_1 \bar{y}_1^{(i)} \right) \\ \Rightarrow \quad b_n^{(2)} &= b_0^{(2)} + \frac{1}{2} \left( {y_2^{(i)}}^\mathrm{T} y_2^{(i)} + \lambda_0^{(2)} {\mu_0^{(2)}}^2 - \lambda_n^{(2)} {\mu_n^{(2)}}^2 \right) \\ &= \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} - n_1 \left( \bar{y}_1^{(i)} \right)^2 \right) + \frac{1}{2} \left( {y_2^{(i)}}^\mathrm{T} y_2^{(i)} + n_1 \left( \bar{y}_1^{(i)} \right)^2 - n {\bar{y}}^2 \right) \\ &= \frac{1}{2} \left( y^\mathrm{T} y - n {\bar{y}}^2 \right) \; . \end{split}\]In the test data, we now have a prior characterized by $\mu_0^{(2)}$, $\lambda_0^{(2)}$, $a_0^{(2)}$ and $b_0^{(2)}$ and a posterior characterized $\mu_n^{(2)}$, $\lambda_n^{(2)}$, $a_n^{(2)}$ and $b_n^{(2)}$. Applying the log model evidence for the univariate Gaussian, the out-of-sample log model evidence (oosLME) therefore follows as
\[\label{eq:UG-m1-oosLME} \begin{split} \mathrm{oosLME}_i(m_1) = &- \frac{n_2}{2} \log (2 \pi) + \frac{1}{2} \log \frac{\lambda_0^{(2)}}{\lambda_n^{(2)}} + \log \Gamma \left( a_n^{(2)} \right) - \log \Gamma \left( a_0^{(2)} \right) \\ &+ a_0^{(2)} \log b_0^{(2)} - a_n^{(2)} \log b_n^{(2)} \\ = &- \frac{n_2}{2} \log (2 \pi) + \frac{1}{2} \log \left( \frac{n_1}{n} \right) + \log \Gamma \left( \frac{n}{2} \right) - \log \Gamma \left( \frac{n_1}{2} \right) \\ &+ \frac{n_1}{2} \log \left[ \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} - n_1 \bar{y}_1^{(i)} \right) \right] - \frac{n}{2} \log \left[ \frac{1}{2} \left( y^\mathrm{T} y - n {\bar{y}}^2 \right) \right] \\ = &- \frac{n}{2S} \log (2 \pi) + \frac{1}{2} \log \left( \frac{S-1}{S} \right) + \log \Gamma \left( \frac{n}{2} \right) - \log \Gamma \left( \frac{(S-1)n}{2S} \right) \\ &+ \frac{(S-1)n}{2S} \log \left[ \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} - n_1 \bar{y}_1^{(i)} \right) \right] - \frac{n}{2} \log \left[ \frac{1}{2} \left( y^\mathrm{T} y - n {\bar{y}}^2 \right) \right] \; . \end{split}\]By definition, the cross-validated log model evidence is the sum of out-of-sample log model evidences over cross-validation folds, such that the cvLME of $m_1$ is:
\[\label{eq:UG-m1-cvLME} \begin{split} \mathrm{cvLME}(m_1) = &\sum_{i=1}^S \mathrm{oosLME}_i(m_1) \\ = &\sum_{i=1}^S \left( - \frac{n}{2S} \log (2 \pi) + \frac{1}{2} \log \left( \frac{S-1}{S} \right) + \log \Gamma \left( \frac{n}{2} \right) - \log \Gamma \left( \frac{(S-1)n}{2S} \right) \right. \\ &+ \left. \frac{(S-1)n}{2S} \log \left[ \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} - n_1 \bar{y}_1^{(i)} \right) \right] - \frac{n}{2} \log \left[ \frac{1}{2} \left( y^\mathrm{T} y - n {\bar{y}}^2 \right) \right] \right) \\ = & - \frac{n}{2} \log (2 \pi) + \frac{S}{2} \log \left( \frac{S-1}{S} \right) + S \cdot \log \Gamma \left( \frac{n}{2} \right) - S \cdot \log \Gamma \left( \frac{1}{2} \frac{S-1}{S} n \right) \\ &- \frac{S \cdot n}{2} \log \left[ \frac{1}{2} \left( y^\mathrm{T} y - n {\bar{y}}^2 \right) \right] + \frac{n_1}{2} \sum_{i=1}^S \log \left[ \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} - n_1 \bar{y}_1^{(i)} \right) \right] \; . \end{split}\]
Next, we consider the null model $m_0$ assuming $\mu = 0$. Because this model has no parameter for $\mu$, there are no hyperparameters $\mu_0$ and $\lambda_0$, but only hyperparameters $a_0$ and $b_0$ for the parameter $\tau$. In the first step, the training data $y_1^{(i)}$ are analyzed using a non-informative prior distribution and applying the posterior distribution for the univariate Gaussian:
In the second step, the test data $y_2^{(i)}$ are analyzed using this posterior as an informative prior distribution, again applying the posterior distribution for the univariate Gaussian:
\[\label{eq:UG-m0-y2} \begin{split} a_0^{(2)} &= a_n^{(1)} = \frac{n_1}{2} \\ \Rightarrow \quad a_n^{(2)} &= a_0^{(2)} + \frac{n_2}{2} = \frac{n_1}{2} + \frac{n_2}{2} = \frac{n}{2} \\ b_0^{(2)} &= b_n^{(1)} = \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} \right) \\ \Rightarrow \quad b_n^{(2)} &= b_0^{(2)} + \frac{1}{2} \left( {y_2^{(i)}}^\mathrm{T} y_2^{(i)} \right) = \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} \right) + \frac{1}{2} \left( {y_2^{(i)}}^\mathrm{T} y_2^{(i)} \right) = \frac{1}{2} \left( y^\mathrm{T} y \right) \; . \end{split}\]In the test data, we now have a prior characterized by $a_0^{(2)}$ and $b_0^{(2)}$ and a posterior characterized $a_n^{(2)}$ and $b_n^{(2)}$. Applying the log model evidence for the univariate Gaussian, the out-of-sample log model evidence (oosLME) therefore follows as
\[\label{eq:UG-m0-oosLME} \begin{split} \mathrm{oosLME}_i(m_1) = &- \frac{n_2}{2} \log (2 \pi) + \log \Gamma \left( a_n^{(2)} \right) - \log \Gamma \left( a_0^{(2)} \right) \\ &+ a_0^{(2)} \log b_0^{(2)} - a_n^{(2)} \log b_n^{(2)} \\ = &- \frac{n_2}{2} \log (2 \pi) + \log \Gamma \left( \frac{n}{2} \right) - \log \Gamma \left( \frac{n_1}{2} \right) \\ &+ \frac{n_1}{2} \log \left[ \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} \right) \right] - \frac{n}{2} \log \left[ \frac{1}{2} \left( y^\mathrm{T} y \right) \right] \\ = &- \frac{n}{2S} \log (2 \pi) + \log \Gamma \left( \frac{n}{2} \right) - \log \Gamma \left( \frac{(S-1)n}{2S} \right) \\ &+ \frac{(S-1)n}{2S} \log \left[ \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} \right) \right] - \frac{n}{2} \log \left[ \frac{1}{2} \left( y^\mathrm{T} y \right) \right] \; . \end{split}\]Again, because the cross-validated log model evidence is the sum of out-of-sample log model evidences over cross-validation folds, the cvLME of $m_0$ becomes:
\[\label{eq:UG-m0-cvLME} \begin{split} \mathrm{cvLME}(m_0) = &\sum_{i=1}^S \mathrm{oosLME}_i(m_0) \\ = &\sum_{i=1}^S \left( - \frac{n}{2S} \log (2 \pi) + \log \Gamma \left( \frac{n}{2} \right) - \log \Gamma \left( \frac{(S-1)n}{2S} \right) \right. \\ &+ \left. \frac{(S-1)n}{2S} \log \left[ \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} \right) \right] - \frac{n}{2} \log \left[ \frac{1}{2} \left( y^\mathrm{T} y \right) \right] \right) \\ = & - \frac{n}{2} \log (2 \pi) + S \cdot \log \Gamma \left( \frac{n}{2} \right) - S \cdot \log \Gamma \left( \frac{1}{2} \frac{S-1}{S} n \right) \\ &- \frac{S \cdot n}{2} \log \left[ \frac{1}{2} \left( y^\mathrm{T} y \right) \right] + \frac{n_1}{2} \sum_{i=1}^S \log \left[ \frac{1}{2} \left( {y_1^{(i)}}^\mathrm{T} y_1^{(i)} \right) \right] \; . \end{split}\]Together, \eqref{eq:UG-m0-cvLME} and \eqref{eq:UG-m1-cvLME} conform to the results given in \eqref{eq:UG-cvLME-m01}.
Metadata: ID: P490 | shortcut: ug-cvlme | author: JoramSoch | date: 2025-03-06, 16:43.