Consider this Bayesian hierarchical model: will; ~ indep. N(1;,?) j =1, ..., 12 ; I wo, oo ~ iid N(10, 07) j = 1,..., 12 o ~ N(0, 10002) Co ~ U(0, 1000) with vo and oo independent, and the values oj, j = 1, ..., 12, regarded as fixed and known. (a) [2 pts] Specify improper densities that the proper hyperpriors given above appear to be approximationich parameters are the hyperparameters?) (b) [5 pts] Draw a directed acyclic graph (DAG) appropriate for this model. (Use the notation introduced in lecture, including "plates.") You may draw it neatly by hand or use software. (c) [5 pts] Using the template asgn2template. bug provided on the course website, form a JAGS model statement (corresponding to your graph). Also, set up any R (rjags) statements appropriate for creating a JAGS model. [Remember: JAGS "dnorm" uses precisions, not variances!] (d) [5 pts] Run at least 10,000 iterations of burn-in, then 100,000 iterations to use for inference. For both vo and of (not oo), produce a posterior numerical summary and also graphical estimates of the posterior densities. Explicitly give the approximations of the posterior expected values, posterior standard deviations, and 95% central posterior intervals. (Just showing R output is not enough!)