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I have difficulties in understanding the way of calculation of covariance matrix in Gaussian Bayesian Nets (from conditional to joint):

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The last formula is about to calculates covariance between parent Xi with child Y, right? then what is sigma for? is k standing for all parents of Y or parents of Xi itself?

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The second part of the theorem describes the nature of the joint distribution of $(X_1, \dots, X_k, Y)$, namely, that it is a multivariate normal distribution. Since the means of these random variables are already specified and the covariance matrix $\Sigma$ gives the covariance between all pairs $(X_i, X_j)$, all that remains is to compute the covariance of each $X_i$ with $Y$, for $i = 1, \dots, k$. The theorem says that this is a $\beta$-weighted linear combination of the $i$th row of the covariance matrix.

So to answer your three questions directly:

  • Yes, the formula gives the covariance between each parent node $X_i$ and the child $Y$.

  • $\Sigma$ is the covariance matrix of $\mathbf{X}$.

  • $k$ is the number of parents of $Y$.

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