ndent of <span class="arithmatex">\(\beta\)</span>. Therefore, if we let <span class="arithmatex">\(\lambda = \frac{\sigma^2}{\tau}\)</span>, then maximizing over <span class="arithmatex">\(\beta\)</span> for <span class="arithmatex">\(P(\beta|\textbf{y})\)</span> is equivalent to </p>
<div class="arithmatex">\[\begin{equation}
    \min_{\beta} (\textbf{y}-\textbf{X}\beta)^T(\textbf{y}-\textbf{X}\beta) + \lambda \beta^T\beta,\nonumber
\end{equation}\]</div>
<p>which is essentially solving a ridge regression