Exercise 2

Exercise 2.1

Consider a simple random sample of size \(n\) from the Wald distribution, with density \[\begin{align*} f(y; \nu, \lambda) = \left(\frac{\lambda}{2\pi y^{3}}\right)^{1/2} \exp\left\{ - \frac{\lambda (y-\nu)^2}{2\nu^2y}\right\}\mathrm{I}(y > 0) \end{align*}\] for location \(\nu >0\) and shape \(\tau>0\).

  1. Write down the likelihood and show that it can be written in terms of the sufficient statistics \(\sum_{i=1}^n y_i\) and \(\sum_{i=1} y_i^{-1}\).
  2. Show that the joint prior \[ p(\lambda) \sim \mathsf{Ga}(\alpha, \beta), \quad p(1/\nu \mid \lambda) \sim \mathsf{No}(\mu, \tau^{-1}\lambda^{-1}),\] the product of a gamma and a reciprocal Gaussian, is conjugate for the Wald distribution parameters.
  3. Derive the parameters of the posterior distribution and provide an interpretation of the prior parameters. Hint: write down the posterior parameters as a weighted average of data-dependent quantities and prior parameters.
  4. Derive the marginal posterior \(p(\lambda)\).

Exercise 2.2

Consider the Rayleigh distribution with scale \(\sigma>0\). It’s density is \[f(y; \sigma) = \frac{y}{\sigma^2} \exp\left(-\frac{y^2}{2\sigma^2}\right)\mathrm{I}(x \geq 0).\]

Derive the Fisher information matrix and use it to obtain Jeffrey’s prior for \(\sigma\). Determine whether the prior is proper.

Exercise 2.3

Consider a binomial model with an unknown probability of successes \(\theta \in [0,1]\) model. Suppose your prior guess for \(\theta\) has mean \(0.1\) and standard deviation \(0.2\)

Using moment matching, return values for the parameters of the conjugate beta prior corresponding to your opinion.

Plot the resulting beta prior and compare it with a truncated Gaussian distribution on the unit interval with location \(\mu=0.1\) and scale \(\sigma=0.2\).1

Exercise 2.4

Replicate the analysis of Example 2.6 (Should you phrase your headline as a question?) of the course notes using the upworthy_question data from the hecbayes package.

Footnotes

  1. Note that the parameters of the truncated Gaussian distribution do not correspond to moments!↩︎