[1] (#) (a) Explain shortly the relation between machine learning and Bayes rule.
(b) How are Maximum a Posteriori (MAP) and Maximum Likelihood (ML) estimation related to Bayes rule and machine learning?
[2] (#) What are the four stages of the Bayesian design approach?
[3] (##) The Bayes estimate is a summary of a posterior distribution by a delta distribution on its mean, i.e.,
Proof that the Bayes estimate minimizes the expected mean-squared error, i.e., proof that $$ \hat \theta_{bayes} = \arg\min_{\hat \theta} \int_\theta (\hat \theta -\theta)^2 p \left( \theta |D \right) \,\mathrm{d}{\theta} $$
where $\epsilon_k = \mathcal{N}(\epsilon_k | 0,\sigma^2)$ with known $\sigma^2=1$. We are interested in deriving an estimator for $A$.
(a) Make a reasonable assumption for a prior on $A$ and derive a Bayesian (posterior) estimate.
(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).