Question

Consider the Bayes model \(X_{i} \mid \theta, i=1,2, \ldots, n \sim\) iid with distribution \(\Gamma(1, \theta), \theta>0\) $$ \Theta \sim h(\theta) \propto \frac{1}{\theta} $$ (a) Show that \(h(\theta)\) is in the class of Jeffreys' priors. (b) Show that the posterior pdf is $$ h(\theta \mid y) \propto\left(\frac{1}{\theta}\right)^{n+2-1} e^{-y / \theta}, $$ where \(y=\sum_{i=1}^{n} x_{i}\) (c) Show that if \(\tau=\theta^{-1}\), then the posterior \(k(\tau \mid y)\) is the pdf of a \(\Gamma(n, 1 / y)\) distribution. (d) Determine the posterior pdf of \(2 y \tau\). Use it to obtain a \((1-\alpha) 100 \%\) credible interval for \(\theta\). (e) Use the posterior pdf in part (d) to determine a Bayesian test for the hypotheses \(H_{0}: \theta \geq \theta_{0}\) versus \(H_{1}: \theta<\theta_{0}\), where \(\theta_{0}\) is specified.

Short answer

The exercise involves showing the properties of PDFs in Bayes model. Critical steps include showing that the given \(h(\theta)\) is a Jeffreys' prior, deducing the posterior pdf, applying a transformation for an equivalent Gamma distribution, constructing credible interval for \(\theta\), and performing a Bayesian hypothesis test.

Step-by-step solution

Show that \(h(\theta)\) is in the class of Jeffreys' priors

Jeffrey’s prior is a type of noninformative prior which has the form \(h(\theta) \propto \sqrt{I(\theta)}\), where \(I(\theta)\) is the Fisher information. To show that \(h(\theta)\) falls into this category, we need to calculate the Fisher information. Fisher information \(I(\theta)\) for Gamma distribution \(\Gamma(1, \theta)\) is \(I(\theta) = \frac{1}{\theta^2}\). Hence sqrt(\(I(\theta)\)) = \(\frac{1}{\theta}\) which is exactly equal to \(h(\theta)\), indicating that \(h(\theta)\) is a Jeffrey’s prior.

Determine the posterior pdf \(h(\theta | y)\)

The posterior distribution is determined by multiplying the likelihood and the prior. In this case, the likelihood is the product of n iid Gamma distributions, and the prior is \(h(\theta) = 1/ \theta\). Integrating over this product will result as \(h(\theta | y) \propto(\frac{1}{\theta})^{n+2-1} e^{-y / \theta}\)

Convert the variable \(\theta\) to \(\tau = \theta^{-1}\) and find the posterior distribution

\(\tau=\) is the substitition the question has asked to make and find an equivalent Gamma distribution. Converting yields, the posterior \(k(\tau | y)\) which follows \(\Gamma(n, 1 / y)\) distribution.

Determine the posterior pdf for \(2y\tau\) and Construct a credible interval for \(\theta\)

In order to determine the posterior pdf of \(2y\tau\), we need to conduct a transformation. If we define \(Z = 2y\tau\), it follows an equivalent Gamma distribution. Then, we use these results to compute a credible interval for \(\theta\). In particular, a credible interval of \(1 - \alpha\) can be obtained as the range of values where the cumulative posterior density is between \(\alpha / 2\) and \(1 - \alpha / 2\).

Test for the hypotheses

To conduct a Bayesian test for the given hypotheses, we need to compute the Bayesian factor (BF). If \(H_0: \theta \geq \theta_0\) is true, the value of the posterior probability will be high and the BF will be greater than 1. We reject \(H_0\) if the BF is less than 1, meaning that \(H_1\) has higher posterior probability than \(H_0\).