**[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., $$ \hat \theta_{bayes} = \int \theta \, p\left( \theta |D \right) \,\mathrm{d}{\theta} $$ Proof that the Bayes estimate minimizes the expected mean-squared error, i.e., proof that $$ \hat \theta_{bayes} = \arg\min_{\hat \theta} \int_\theta (\hat \theta -\theta)^2 p \left( \theta |D \right) \,\mathrm{d}{\theta} $$

**[4]**(###) We make $N$ IID observations $D=\{x_1 \dots x_N\}$ and assume the following model $$ x_k = A + \epsilon_k $$ 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).

**[5]**(##) We consider the coin toss example from the notebook and use a conjugate prior for a Bernoulli likelihood function.

(a) Derive the Maximum Likelihood estimate.

(b) Derive the MAP estimate.

(c) Do these two estimates ever coincide (if so under what circumstances)?

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