lConsider this Bayesian hierarchical model: 133;: | is N indcp. New?) 3': 1,...,12 is Warm ~ is Maria) j:1,...,12 an m N(D,l(}2) an ~ 11(1), 1009) with if)\" and on independent, and the valnas 03-, j = 1, . . . , 12, regarded as xed and known. {a} [2 pts] Specify improper densities that the proper hyperpriors given above appear to be approximating. (Which parameters are the hyperparameters?) (b) [5 pts] Draw a directed aeyciic 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. [Remembers JAGS \"dnorm\" uses precisions, not Wiriimeesl] (d) [5 pts] Run at least l iterations of burnin, then 100,000 iterations to use for inference. For both 1,03 and :73 (not or\"), produce a posterior numerical summary and also graphical estimates of the posterior densities. Eapticitty give the approximations of the posterior expected values, posterior standard deviations, and 95% central posterior intervals. l[Just showing R. output is not enough!) {e} Suppose a new casecontrol study is to be performed, and assume that its logodds standard error (new a} wiil be 0.25. Assume the if; for the new study is exchangeable with those Jfor the previous studies. (i) [2 pts] Redraw your DAG, adding new nodes to represent the new it and new It}. (ii) [2 pts] Correspondingly modify your JAGS model to answer the following parts. Show the modied JAGS and R. code and output that you used. (iii) [3 pts] Estimate the posterior mean and posterior standard deviation, and form a 95% central posterior predictive interval for the estimated logodds ratio that the new study will obtain. (Remember, this new estimated logodds ratio will be the new it}, not the new iii.) (iv) [1 pt] Estimate the posterior predictive probability that the new estimated logodds ratio will be at least twice its stimdard error, i.e., at least two standard errors (20] greater than zero. [This is roughly the posterior probability that the new study will nd a statistically signicant result, and in the positive direction.) Suggestion: Add an indicator variable to your JAGS model one that equals 1 when the event occurs, and 0 otherwise. {What is its mean?) Use at least lil iterations of burnin, and 100,000 for inference as before