Suppose that (tau=left{x_{1}, ldots, x_{n} ight}) are observations of iid continuous and strictly positive random variables, and

Suppose that \(\tau=\left\{x_{1}, \ldots, x_{n}\right\}\) are observations of iid continuous and strictly positive random variables, and that there are two possible models for their pdf. The first model \(p=1\) is

\[ g(x \mid \theta, p=1)=\theta \exp (-\theta x) \]

and the second \(p=2\) is \[ g(x \mid \theta, p=2)=\left(\frac{2 \theta}{\pi}\right)^{1 / 2} \exp \left(\frac{-\theta x^{2}}{2}\right) \]
For both models, assume that the prior for \(\theta\) is a gamma density \[ g(\theta)=\frac{b^{t}}{\Gamma(t)} \theta^{t-1} \exp (-b \theta) \]
with the same hyperparameters \(b\) and \(t\). Find a formula for the Bayes factor, \(g(\tau \mid p=1) / g(\tau \mid\) \(p=2\) ), for comparing these models.