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A Gaussian Bayesian network specifies a linear normal conditional model at each node of a Bayesian network: $X_j\mid X_{\mathrm{pa}(j)}\sim N({\alpha }_j+{\gamma }_j^T{X}_{\mathrm{pa}(j)},{\sigma }_j^2)$. Independent local errors and an acyclic ordering generate a joint multivariate normal distribution. Positive conditional variances give a nonsingular joint multivariate normal distribution. The graph constrains regression coefficients and hence conditional independence.

An ordinal categorical variable has categories with a meaningful order. Replacing its levels by numerical scores imposes a quantitative spacing and can replace several unrelated level effects by one regression coefficient when a linear trend is scientifically reasonable.

Use the normal linear modelHere $Y$ is the response random vector, $X$ is the known design matrix, $\beta \in {\mathbb{R}}^p$ contains the unknown regression coefficients, and ${\sigma }^2>0$ is the common error variance. Conditional on $X$, the errors have normal distributions and are independent random variables. Require $\mathrm{rank}X=p\le n$ for identifiability of $\beta$ and invertibility of $X^{T}X$. Usually $n>p$ is needed to estimate the error variance from the regression residuals. An intercept, when included, is represented by a column of ones in $X$.