(a) Derive the log-likelihood of the parameters for these data.
(b) Derive the maximum likelihood estimates for the mean $\mu$ and variance $\Sigma$ by setting the derivative of the log-likelihood to zero.
[2] (#) Shortly explain why the Gaussian distribution is often preferred as a prior distribution over other distributions with the same support?
[3] (###) We make $N$ IID observations $D=\{x_1 \dots x_N\}$ and assume the following model
We assume that $\sigma$ has a known value and are interested in deriving an estimator for $A$ .
(a) Derive the Bayesian (posterior) estimate $p(A|D)$.
(b) (##) Derive the Maximum Likelihood estimate for $A$.
(c) Derive the MAP estimates for $A$.
(d) Now assume that we do not know the variance of the noise term? Describe the procedure for Bayesian estimation of both $A$ and $\sigma^2$ (No need to fully work out to closed-form estimates).
$x \sim \mathcal{N}(\mu_x,\sigma_x^2)$ and $y \sim \mathcal{N}(\mu_y,\sigma_y^2)$, what is the PDF for $z = A\cdot(x -y) + b$?