\begin{aligned} \mathbf{w} &= \operatorname*{argmax}_{\mathbf{\mathbf{w}}} P(y_1,\mathbf{x}_1,...,y_n,\mathbf{x}_n|\mathbf{w})\\ &= \operatorname*{argmax}_{\mathbf{\mathbf{w}}} \prod_{i=1}^n P(y_i,\mathbf{x}_i|\mathbf{w}) & \textrm{(Because of independence.)}\\ &= \operatorname*{argmax}_{\mathbf{\mathbf{w}}} \prod_{i=1}^n P(y_i|\mathbf{x}_i,\mathbf{w})P(\mathbf{x}_i|\mathbf{w}) \\ &= \operatorname*{argmax}_{\mathbf{\mathbf{w}}} \prod_{i=1}^n P(y_i|\mathbf{x}_i,\mathbf{w})P(\mathbf{x}_i) & \textrm{($\mathbf{x}_i$ is independent of $\mathbf{w}$)}\\ &= \operatorname*{argmax}_{\mathbf{\mathbf{w}}} \prod_{i=1}^n P(y_i|\mathbf{x}_i,\mathbf{w}) & \textrm{($P(\mathbf{x}_i)$ is a constant - can be dropped.)}\\ &= \operatorname*{argmax}_{\mathbf{\mathbf{w}}} \sum_{i=1}^n \log\left[P(y_i|\mathbf{x}_i,\mathbf{w})\right] \\ &= \operatorname*{argmax}_{\mathbf{\mathbf{w}}} \sum_{i=1}^n \left[ \log\left(\frac{1}{\sqrt{2\pi\sigma^2}}\right) + \log\left(e^{-\frac{(\mathbf{x}_i^\top\mathbf{w}-y_i)^2}{2\sigma^2}}\right)\right] \\ &= \operatorname*{argmax}_{\mathbf{\mathbf{w}}} -\frac{1}{2\sigma^2}\sum_{i=1}^n (\mathbf{x}_i^\top\mathbf{w}-y_i)^2 \\ &= \operatorname*{argmin}_{\mathbf{\mathbf{w}}} \frac{1}{n}\sum_{i=1}^n (\mathbf{x}_i^\top\mathbf{w}-y_i)^2 \\ \end{aligned}
In the last couple of steps, can anyone clarify why $\frac{1}{2\sigma^2}$ is replaced with $\frac{1}{n}$?