Question Sheet Three

Set: Problems class, Thursday 3rd March 2022.

Due in: Problems class, Thursday 10th March 2022. Paper copies may be submitted in the problems class or directly to me in lectures or my office, 4W4.10. Pdf copies may be submitted to the portal available on the Moodle page.

Task: Attempt questions 1-2, 3(d). If you are feeling particularly worthy you may also consider 3(e); questions 3(a)-(c), and 4 are extra questions which may be discussed in the problems class.

Question 1

Let \(X_{1}, \ldots, X_{n}\) be conditionally independent given \(\theta\), so \(f(x \, | \, \theta) = \prod_{i=1}^{n} f(x_{i} \, | \, \theta)\) where \(x = (x_{1}, \ldots, x_{n})\), with each \(X_{i} \, | \, \theta \sim Po(\theta)\). Suppose we judge that \(\theta \sim Gamma(\alpha, \beta)\).

  1. Find the distribution of \(\theta \, | \, x\).

  2. Show that the posterior mean can be written as a weighted average of the prior mean, denoted \(\theta_{0}\), and the maximum likelihood estimate of \(\theta\), \(\overline{x}\).

  3. Let \(Z\) be a future (unobserved) observation. Find the mean and variance of the predictive distribution \(Z \, | \, x\). You may assume that \(X\) and \(Z\) are conditionally independent given \(\theta\) and that \(Z \, | \, \theta \sim Po(\theta)\).

  4. The data in the table below are the number of fatal accidents on scheduled airline flights between 1976 and 1985. \[\begin{eqnarray*} \begin{array}{cccccccccc} 1976 & 1977 & 1978 & 1979 & 1980 & 1981 & 1982 & 1983 & 1984 & 1985 \\ 24 & 25 & 31 & 31 & 22 & 21 & 26 & 20 & 16 & 22 \end{array} \end{eqnarray*}\] Let \(Z\) be the number of fatal accidents in 1986 and \(X_{i}\) the number of fatal accidents in 1975\(+ i\). Adopting the model given above, with \(\alpha = 101\) and \(\beta = 5\) and assuming that \(Z \, | \, x\) may be approximated by a \(N(E(Z \, | \, X), Var(Z \, | \, X))\), find an approximate 95% prediction interval for the number of accidents in 1986, that is an interval \((z_{L}, z_{U})\) such that \(P(z_{L} < Z < z_{U} \, | \, x) = 0.95\).

Question 2

Let \(X_{1}, \ldots, X_{n}\) be a finitely exchangeable sequence of random quantities and consider any \(i \neq j \in \{1, \ldots, n\}\).

  1. Explain why the marginal distribution of any \(X_{i}\) does not depend upon \(i\) so that \(E(X_{i})\) and \(Var(X_{i})\) do not depend upon \(i\).

  2. Explain why the joint distribution of any \(X_{i}\), \(X_{j}\) does not depend upon either \(i\) or \(j\) so that \(Cov(X_{i}, X_{j})\) does not depend on either \(i\) or \(j\).

  3. Let \(Y = \sum_{i=1}^{n} X_{i}\). By considering \(Var(Y) \geq 0\), or otherwise, show that \[\begin{eqnarray*} Corr(X_{i}, X_{j}) & = & \frac{Cov(X_{i}, X_{j})}{\sqrt{Var(X_{i})Var(X_{j})}} \ \geq \ \frac{-1}{n-1}. \end{eqnarray*}\]

Question 3

Suppose \(X \, | \, \mu \sim N(\mu, \sigma^{2})\) and \(Y \, | \, \mu, \delta \sim N(\mu + \delta, \sigma^{2})\), where \(\sigma^{2}\) is known and \(X\) and \(Y\) are conditionally independent given \(\mu\) and \(\delta\).

  1. Find the joint distribution of \(X\) and \(Y\) given \(\mu\) and \(\delta\).

  2. Consider the improper noninformative joint prior distribution \[\begin{eqnarray*} f(\mu, \delta) & \propto & 1. \end{eqnarray*}\] Find, up to a constant of proportionality, the joint posterior distribution of \(\mu\) and \(\delta\) given \(x\) and \(y\). Are \(\mu \, | \, x, y\) and \(\delta \, | \, x, y\) independent?

  3. Find the marginal posterior distribution \(f(\delta \, | \, x, y)\).

  4. Find the marginal posterior distribution \(f(\mu \, | \, x, y)\).

  5. Consider a future observation \(Z\) where \(Z \, | \, \mu, \delta \sim N(\mu - \delta, \sigma^{2})\) and \(Z\) is conditionally independent of \(X\) and \(Y\) given \(\mu\) and \(\delta\). Find the predictive distribution \(f(z \, | \, x, y)\).
    [Hint: Use the distributions \(f(z \, | \, \mu, \delta)\) and \(f(\mu, \delta \, | \, x,y)\). You will need a double integral to integrate out both \(\mu\) and \(\delta\).]

Question 4

Let \(X_{1}, \ldots, X_{n}\) be exchangeable so that the \(X_{i}\) are conditionally independent given a parameter \(\theta\). Suppose that each \(X_{i} \, | \, \theta \sim U(0, \theta)\).

  1. Show that \(M = \max_{i}(X_{i})\) is a sufficient statistic for \(X = (X_{1}, \ldots, X_{n})\).
  2. Show that the Pareto distribution, \(\theta \sim Pareto(a, b)\), \[\begin{eqnarray*} f(\theta) & = & \frac{a b^{a}}{\theta^{a+1}}, \ \ \theta \geq b \end{eqnarray*}\] is a conjugate prior distribution.