We first assume that \(y = \mathbf{x}^T \mathbf{w} + \epsilon\) where \(\epsilon \sim \mathcal{N}(0, \sigma^2)\). Second, we assume that weights \(\mathbf{w} \sim \mathcal{N}(\mathbf{0}, \tau^2 \mathbf{I})\). Using our posterior distribution \(\mathbf{w} \mid \mathbf{y}, \mathbf{X}\), we can get an understanding of \(\mathbf{w}_{\text{MAP}}\):
\[\ f(\mathbf{w} \mid \mathbf{y}, \mathbf{X}) \propto f(\mathbf{y} \mid \mathbf{w}, \mathbf{X}) f(\mathbf{w}) \implies \mathbf{w}_{\text{MAP}} = \underset{\mathbf{w}}{\text{argmax}} \ [f(\mathbf{y} \mid \mathbf{w}, \mathbf{X}) f(\mathbf{w})]\] \[\implies \mathbf{w}_{\text{MAP}} = \underset{\mathbf{w}}{\text{argmax}} \ [\log f(\mathbf{y} \mid \mathbf{w}, \mathbf{X}) + \log f(\mathbf{w})]\]Given that \(y_i \mid \mathbf{w}, \mathbf{x_i} \sim \mathcal{N}(\mathbf{x_i}^T \mathbf{w}, \sigma^2)\), we can give a nice understanding of \(\log f(\mathbf{y} \mid \mathbf{w}, \mathbf{X})\):
\[\log f(\mathbf{y} \mid \mathbf{w}, \mathbf{X}) = \sum_{i = 1}^n \log f(y_i \mid \mathbf{w}, \mathbf{x_i}) = \sum_{i = 1}^n - \log(\sqrt{2\pi \sigma^2}) - \frac{1}{2\sigma^2} (\mathbf{y_i} - \mathbf{x_i}^T \mathbf{w})^2 = -\frac{1}{2\sigma^2}\sum_{i = 1}^n (\mathbf{y_i} - \mathbf{x_i}^T \mathbf{w})^2 + \text{const}\]where the constant is w.r.t to \(\mathbf{w}\). For the log-density of our prior \(\log f(\mathbf{w})\) we have:
\[\log f(\mathbf{w}) = \log[\exp(-\frac{1}{2} (\mathbf{w} - 0)^T (\tau^2 \mathbf{I})^{-1} (\mathbf{w} - 0) )] + \text{const} = -\frac{1}{2 \tau^2} \mathbf{w}^T \mathbf{w}\]where \(\mathbf{w}^T \mathbf{w}\) is just the square of the L2 norm. Putting this together we have:
\[\mathbf{w}_{\text{MAP}} = \underset{\mathbf{w}}{\text{argmax}} \ [-\frac{1}{2\sigma^2}\sum_{i = 1}^n (\mathbf{y_i} - \mathbf{x_i}^T \mathbf{w})^2 -\frac{1}{2 \tau^2} \mathbf{w}^T \mathbf{w}] = \underset{\mathbf{w}}{\text{argmin}} \ [\sum_{i = 1}^n (\mathbf{y_i} - \mathbf{x_i}^T \mathbf{w})^2 + \frac{\sigma^2}{\tau^2} \mathbf{w}^T \mathbf{w}]\]Thus, we can conclude that MAP for weights under a Gaussian prior follows the objective as L2/Ridge Regression (albeit with a tuned \(\lambda\) resembling the \(\frac{\sigma^2}{\tau^2}\)) terms.