Bayesian Statistics

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8.5 Exercises on Chapter 8

8.1 $\color {blue}\star $ Inside the files 8_mh_random_walk_case.ipynb and 8_mh_random_walk_case.Rmd you will find code that generates the first two plots in Example 8.2.3.

This question investigates some modifications to the code from Example 8.2.3.
- (a) In each case (i)-(iv) listed below, start from the original code and make the change(s) listed. Then, note any differences between the two new plots and the two plots in Example 8.2.3. Give a brief explanation for any differences you notice.
  - (i) Change the proposal distribution from $Q=Y+\Normal (0,1)$ to $Q=Y+\Normal (0,10)$.
  - (ii) Change the initial location from $0$ to $10$.
  - (iii) Change the initial location from $0$ to $20$.
  - (iv) Change the initial location from $0$ to $20$ and change the proposal distribution from $Q=Y+\Normal (0,1)$ to $Q=Y+\Normal (0,10)$.
- (b) For (i)-(iv) in part (a), in which cases does it look like the MH algorithm has converged after 100 steps?
- (c) Change the target distribution from $\Cauchy (0,1)$ to $\Exp (1)$. You should now notice that the sequence $(y_m)$ generated by the MH algorithm never goes below zero. Why has this occurred?
8.2 $\color {blue}\star \,\star $ A Cauchy distribution $M_\theta =\Cauchy (0,\theta )$ with parameter $\theta \in (0,\infty )$ is used to model the errors made by a financial model, which forecasts the price in dollars of a particular asset. By using historical data it is possible to compare the predictions of the model to the prices that were later observed. Using ten independent time periods, the following errors were observed

\[(-11.93,\; 23.26,\; -13.17,\; 38.25,\; -0.36).\]

It is decided to use a prior density

\[f(\theta ) \propto \begin {cases} \frac {1}{\theta ^2} & \text { for }\theta \in (1,\infty ) \\ 0 & \text { otherwise.} \end {cases} \]
- (a) Find the posterior density of $\theta |x$. Is this a proper or improper density function?
- (b) Write an MCMC algorithm that produces samples from $\theta |x$ and plot a histogram of $100$ samples.
  
  Hint: Re-use some of your code from Exercise 8.1.
8.3 $\color {blue}\star $ Look at the right hand column of the reference sheet ‘Bayesian Models and Related Formulae’ in Appendix A. For each item listed there, identify which Section, Lemma, equation, or other part of Chapter 7 it comes from.

Do the same for the both columns of the reference sheet ‘Some Useful Algorithms’, with Chapter 8.
8.4
- (a) $\color {blue}\star \,\star $ Convince yourself that, within the MH algorithm, the choice of proposal distribution $Q=Y$ would result in $\alpha =1$ i.e. all proposals are accepted. This observation is not helpful in practice – why not?
- (b) A related question: what happens if we try to use the Gibbs sampler, as described in Section 8.4, with $d=1$?
8.5 $\color {blue}\star \,\star $ Write Definition 8.4.3 in Bayesian shorthand.
8.6 $\color {blue}\star \star \star $ In this question we prove the final claim in Example 8.2.4.
- (a) Let $X$ be a continuous random variable with p.d.f. $f_X$. Show that $X\eqd -X$ if and only if $f_X(x)=f_{X}(-x)$.
- (b) Let $Y,Q$ and $Z$ be continuous random variables with range $\R $. Suppose that $Q=Y+Z$, where $Z$ is independent of $(Y,Q)$ and $f_Z(z)=f_Z(-z)$ for all $z\in \R $. Show that
  
  \[f_{Q|_{\{Y=y\}}}(\tilde {y})=f_{Q|_{\{Y=\tilde {y}\}}}(y)\]
  
  for all $y,\wt {y}\in \R $, assuming that both are well defined.