(i) Outline briefly the Bayesian approach to hypothesis testing based on Bayes factors.

(ii) Let $Y_{1}, Y_{2}$ be independent random variables, both uniformly distributed on $\left(\theta-\frac{1}{2}, \theta+\frac{1}{2}\right)$. Find a minimal sufficient statistic for $\theta$. Let $Y_{(1)}=\min \left\{Y_{1}, Y_{2}\right\}$, $Y_{(2)}=\max \left\{Y_{1}, Y_{2}\right\}$. Show that $R=Y_{(2)}-Y_{(1)}$ is ancilliary and explain why the Conditionality Principle would lead to inference about $\theta$ being drawn from the conditional distribution of $\frac{1}{2}\left\{Y_{(1)}+Y_{(2)}\right\}$ given $R$. Find the form of this conditional distribution.