deletions | additions       diff --git a/subsection_Prior_for_Weights_In__.tex b/subsection_Prior_for_Weights_In__.tex  index d247a9f..b45eacb 100644  --- a/subsection_Prior_for_Weights_In__.tex  +++ b/subsection_Prior_for_Weights_In__.tex 
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It is then readily apparent that, given our diagonal $\Sigma$ we have $\Sigma^{-1} = \text{diag}(\sigma_1^{-1}, \dots, \sigma_d^{-1})$ and similarly $|\Sigma^{-1}| = |\text{diag}(\sigma_1^{-1}, \dots, \sigma_d^{-1})| = \prod(\sigma_i^{-1})$.  Using this information, it can be shown that finding the MAP estimator of $\mathbf{w}$ is equivalent to solving the maximum likelihood problem with a ridge regularisation parameter \cite{Hoerl_1970}. The success and ubiquity of this approach then speaks to the reasonableness of assuming Guassian priors.\begin{align}  L(\mathbf{w} | X, Y) \propto P(Y | X, W) P(\mathbf{w})  \end{align}