Bayesian Statistics

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2.5 Exercises on Chapter 2

You can find formulae for named distributions in Appendix A.

2.1 $\color {blue}\star $ This exercise provides template code for drawing several sketches of distributions, which you will find helpful in many later exercises.

Use a computer package of your choice to complete the following questions. You will need the file 2_dist_sketching.ipynb if you use Python, or 2_dist_sketching.Rmd if you use R.
- (a) You will find code that produces a sketch of the probability density functions of the $\Exp (3)$ and $\Exp (5)$ distributions. Modify this code to produce a sketch of the probability density functions of the $\Gam (4,5)$ and $\Gam (6,7)$ distributions.
- (b) You will find code that produces a sketch of the $\Geo (\frac 12)$ distribution. Modify this code to produce a sketch of the $\Bin (10,\frac 23)$ distribution.
- (c) In this question we look at distributions in the form of equation (2.4).
  
  You will find code that produces a sketch of the discrete distribution with p.m.f.
  
  \[\P [X=x]=\int _0^1 \P [\Bin (10,p)=x]f_{\Beta (2,3)}(p)\,dp\]
  
  for $x\in \{0,1,\ldots ,10\}$. Note that this distribution is the sampling distribution associated to a discrete Bayesian model with model family $\Bin (10,p)$ and prior $P\sim \Beta (2,3)$.
  
  Modify this code to produce a sketch of the discrete distribution with p.m.f.
  
  \[\P [X=x]=\int _0^1 \P [\Geo (p)=x]f_{\Beta (\frac 12,\frac 12)}(p)\,dp\]
  
  for $x\in \{0,1,\ldots ,\}$.
2.2 $\color {blue}\star \,\star $ Let $(X,P)$ be a discrete Bayesian model with model family $M_p\sim \Geo (p)$ where $p\in [0,1]$.

We regard $M_p$ as a model for the number of times an experiment fails before the experiment is successful. The probability of success on each try is $p\in [0,1]$, which is an unknown parameter. We assume that the experiments are independent of each other.
- (a) Write down the probability mass function $\P [M_p=n]$, and the range of this model.
- (b) Take a prior $P\sim \Unif ([0,1])$. Given the single data point $x=5$, show that the posterior distribution is given by $P|_{\{X=x\}}\sim \Beta (2,5)$.
- (c) Use a computer package of your choice to sketch the p.d.f. of this distribution, alongside the prior distribution.
- (d) A second Bayesian update is made using a new data point, $x=9$. Find the new posterior distribution and add it to your sketch from (c).
- (e) Write down the p.m.f. of the predictive distribution, after the second update. Use a computer package of your choice to sketch it.
2.3 $\color {blue}\star \,\star $ Let $(X,\Lambda )$ be a discrete Bayesian model with model family $M_\lambda \sim \Poisson (\lambda )$, where $\lambda \in (0,\infty )$. Take the prior to be $\Lambda \sim \Exp (5)$. Find the distribution of the posterior $\Lambda |_{\{x=7\}}$ and write down the p.m.f. of the predictive distribution.