Multivariate Gaussian as Bayesian or Markov network

Question

How can I represent a Multivariate Gaussian distribution as a Bayesian network? How can I represent a Multivariate Gaussian distribution as a Markov network/Markov Process?

Answer 1

I'll assume a mean of 0 for simplicity. The density of a multivariate Gaussian is proportional to $\exp(-0.5x^T \Sigma^{-1} x)$. If you define $\Lambda\equiv \Sigma^{-1}$, that gives a characterization of the density as a pairwise Markov random field. To be more specific, the quadratic form $x^T \Lambda x$ can be written as $\sum_{i,j} x_ix_j\lambda_{ij}$, so the density is proportional to a product of potential functions $\prod_{i,j} \phi_{i,j}(x_i, x_j)$. Each of these potential functions depends on only two coordinates, which makes this representation "Markovian" in the sense that it relies only on local/pairwise interactions.