A clinical trial was conducted to determine if hormone treatment benefits women who were previously treated for breast cancer. Each subject entered the clinical trial when they had a recurrence. They were then treated with irradiation and assigned to a hormone treatment group or a control group. The observation of interest is the time until a second recurrence, which can be assumed to follow an exponential distribution with parameter $($hormone therapy group$)$ or $($control group$).$In the table, a censoring time $\backslash ($ Q $\backslash )$ means that a woman was observed for $\backslash ($ Q $\backslash )$ months and did not have a recurrence during that time, so the recurrence time exceeds $\backslash ($ Q $\backslash )$ months. For example, $15$ women who received hormone treatment had recurrences, and the total recurrence time was $280$ months.Let $\backslash ($ y$\_$i$^$H $=($x$\_$i$^$H$,\backslash$delta$\_$i$^$H$)\backslash )$ be the data for the $\backslash ($ i $\backslash )-$th person in the hormone group, where $\backslash ($ x$\_$i$^$H $\backslash )$ is the time $\backslash ($ y$\_$i$^$H $\backslash )$ if it is a recurrence time and $0$ if it is a censoring time. The data for the control group can be written similarly.The likelihood is given by:$\backslash [$ p$($y$|\backslash$theta$,\backslash$tau$)\backslash$propto $(\backslash$theta $\backslash$tau$)^\{\backslash$sum $\backslash$delta$\_$i$^$H $+\backslash$sum $\backslash$delta$\_$i$^$C$\}\backslash$exp $\backslash$left$\backslash \{-\backslash$theta $\backslash$left$(\backslash$sum x$\_$i$^$C $+\backslash$tau $\backslash$sum x$\_$i$^$H $\backslash$right$)\backslash$right$\backslash \}\backslash ]$You have been hired by the pharmaceutical company to analyze their data. They want to know if hormone treatment works. Your task is to find the marginal posterior distribution of $\backslash (\backslash$tau $\backslash )$ using Gibbs sampling.The conjugate prior is given by:$\backslash [$ p$(\backslash$theta$,\backslash$tau$)\backslash$propto $\backslash$theta$^\{$a$-1\}\backslash$tau$^\{$b$-1\}\backslash$exp $\backslash \{-$c$\backslash$theta $-$ d$\backslash$tau $\backslash$theta $\backslash \}\backslash ]$Doctors who have worked extensively with this treatment have indicated that reasonable values for the hyperparameters are $\backslash (($a$,$ b$,$ c$,$ d$)=(3,1,60,120)\backslash ).$

$1.$ Derive the necessary conditional distributions to implement Gibbs sampling.

$2.$ Program and run your Gibbs sampler. Use a set of convergence diagnostics to evaluate the convergence and mixing of your algorithm. Interpret the diagnostics.

$3.$ Calculate summary statistics of the estimated joint posterior distribution, including marginal means, standard deviations, and a $95\%$ credible interval for each parameter. Create a table with these results.