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Where <math>Z = \frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}}</math> is the closed form for the [[partition function]]. The parameters of this distribution are <math>\mu</math> and <math>\Sigma</math>. <math>\mu</math> is the vector of [[mean values]] of each variable, and <math>\Sigma^{-1}</math>, the inverse of the [[covariance matrix]], also known as the [[precision matrix]]. Precision matrix contains the pairwise dependencies between the variables. A zero value in <math>\Sigma^{-1}</math> means that conditioned on the values of the other variables, the two corresponding variable are independent of each other.
To learn the graph structure as a multivariate Gaussian graphical model, we can use either [[L-1 regularization]], or [[neghborhood selection]] algorithms. These algorithms simultaneously learn a graph structure and the edge strength of the connected nodes. An edge strength corresponds to
Once the model is learned, we can repeat the same step as in the discrete case, to get the density functions at each node, and use analytical form to calculate the Free energy. Here, the [[partition function]] already has a [[closed form]], so the [[inference]], at least for the Gaussian Graphical Models is trivial. If analytical form of the partition function was not available, we can use [[particle filtering]] or [[expectation propagation]] to approximate the Z, and then perform the inference and calculate free energy.
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