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@jemoka / Jemoka Knowledge Base / wiki/concepts/inference_for_gaussian_models.md
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--- title: "Inference for Gaussian Models" type: concept related: [Covariance] source: https://www.jemoka.com/posts/kbhinference_for_gaussian_models/ confidence: high status: active --- If we know that \(a,b\) are both Gaussian distributions, then we have that: \begin{equation} \mqty[a \\ b] \sim \mathcal{N} \qty(\mqty[\mu_{a} \\mu_{b}], \mqty[A & C \\ C^{T} & B]) \end{equation} whereby: \(A\) is the covariance of each element of \(A\) \(B\) is the covariance of each element of \(B\) \(C\) is the covariance of \(A\) against \(B\) To perform inference: \begin{equation} p(a|b) = \mathcal{N}(a | \mu_{a|B}, \Sigma_{a|b}) \end{equation} wherby: \begin{equation} \mu_{a|b} = \mu_{a} + CB^{-1}(b-\mu_{b}) \end{equation} \begin{equation} \Sigma_{a|b} = A - CB^{-1}C^{T} \end{equation} Its a closed form solution. Tada. We know that \(B\) is positive semidefinite, and that its invertible, from the fact that its a covariance.