this is for holding javascript data
Xavier Holt added begin_align_L_mathbf_w__.tex
over 8 years ago
Commit id: 1bb7d9a3a293ee01b1780bef4a40d03854a908fb
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\begin{align}
L(\mathbf{w} | X, Y) \propto P(Y | X, W) P(\mathbf{w})
\end{align}
We then have to decide a prior distribution for our weights. In pursuit of generalisability, we adopt the framework that values should be at or near zero with high probability. The most widely used weight-distribution in logistic regression and in deed statistics more broadly is of course Guassian.
\subsubsection{Guassian Prior: $\mathbf{w} \sim \mathcal{N}(\mathbf{0}, \text{diag}(\boldsymbol{\sigma}))$}
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 matrix $\Sigma = \text{diag}(\boldsymbol{\sigma}) = \text{diag}(\sigma_1, \sigma_2, \dots)$.
$p(\mathbf{w} | \boldsymbol{\sigma}) = $
