this is for holding javascript data
Xavier Holt edited subsection_Prior_for_Weights_In__.tex
over 8 years ago
Commit id: 6a6c10091cb02868ffbe03f9b54095becda7e6e7
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\subsubsection{Guassian Prior: $\mathbf{w} \sim \mathcal{N}_{d\times d}(\mathbf{0}, \text{diag}(\boldsymbol{\sigma}))$}
Clearly having a zero mean vector is desirable. Additionally, we have opted to assume that the weights are heteroscedastic but independent of one another. That is, the covariance matrix is a diagonal $d\times d$
covariance matrix $\Sigma = \text{diag}(\boldsymbol{\sigma}) = \text{diag}(\sigma_1, \sigma_2, \dots \sigma_d)$. Consequently,
$$p(\mathbf{w} | \boldsymbol{\sigma}) = (2\pi )^{-\frac{d}{2}} | \Sigma | ^{-\frac{1}{2}} \exp(-\frac{1}{2} \mathbf{w^T} \Sigma ^{-1} \mathbf{w})$$