Why can we suppose $\epsilon \sim \mathcal{N}(0,\sigma^2)$? Suppose we want a regression of a function $f(x)$. Suppose $r = f(x) + \epsilon$. Why can we suppose that $\epsilon  \sim \mathcal{N}(0,\sigma^2)$? What is the advantage of such supposition?
 A: This assumption has two reasons.
First, it is reasonable to assume a normal distribution for the error. We make this assumption because if you have many random variables that are influencing the error independently and additively the distribution of the resulting random variable follows the normal distribution.
Another advantage of this assumption is rooted in parameter estimation for linear regression. If we assume that $f(\mathbf{x}) = \mathbf{w}^T\mathbf{x}$, we can rewrite the residual of observation $i$ as
$$\varepsilon_i = r_i - f(\mathbf{x}_i).$$
If $\varepsilon \sim \mathcal{N}(0, \sigma^2)$, then we know that for observation $i$ we have
$$p(\varepsilon_i) = \dfrac{1}{\sqrt{2\pi \sigma^2}}\exp\left[-0.5(\varepsilon_i - 0)^2/\sigma^2\right]$$
Swichting from $\varepsilon_i$ to $\mathbf{x}_i$ and $r_i$ will result in
$$p(\mathbf{x}_i,r_i|\mathbf{w}) = \dfrac{1}{\sqrt{2\pi \sigma^2}}\exp\left[-0.5(r_i - \mathbf{w}^T\mathbf{x}_i)^2/\sigma^2\right].$$
If we assume that our errors are independent, then we can express the likelihood for observing the data $\mathcal{D}=\{(\mathbf{x}_1,r_i),\ldots,(\mathbf{x}_N, r_N) \}$ as
$$L(\mathcal{D}|\mathbf{w}) = \prod_{n=1}^N\dfrac{1}{\sqrt{2\pi \sigma^2}}\exp\left[-0.5(r_n - \mathbf{w}^T\mathbf{x}_n)^2/\sigma^2\right].$$
The log-likelihood of this expression is given as
$$\log L(\mathcal{D}|\mathbf{w}) = \log \left[\dfrac{1}{\sqrt{2\pi \sigma^2}}\right]^N -\dfrac{1}{2\sigma^2}\sum_{n=1}^N\left[r_n - \mathbf{w}^T\mathbf{x}_n\right]^2.$$
If we want to maximize the log-likelihood (maximizes the likelihood of observing the data $\mathcal{D}$) we need to minimize (note the negative sign of the sum)
$$\sum_{n=1}^N\left[r_n - \mathbf{w}^T\mathbf{x}_n\right]^2.$$
But this is the sum of squared errors that we minimize in the standard case of multiple linear regression.
