3. [25 points total] The National Longitudinal Study of Adolescent to Adult Health collected data on adolescent friendship networks at several high schools in the US. At each of the NS schools, all juniors were asked to provide a list of all other juniors at their school that they consider to be a "good friend." Let Ys be an indicator of whether the ith pair of juniors at the sth school are good friends, for i = 1,..., NP, where NP is the number of pairs of students at each school (assume each of the schools has the same number of juniors). We also know whether the pair of students are of the same sex or not: Xis = 0 if ith pair of students consists of either two boys or two girls, while Xis 1 if the ith pair of students is mixed with respect to sex (i.e., one boy and one girl). Note that Xis is a covariate: it is assumed known for each pair i at school s. = Let Y = {Yis i = 1, information, x = {xis ,Ns = 1,...,NS} and consider the following model for the friendship link data Y given the sex pair covariate i=1,...,NP,s ', s = 1,...,NS}. First, we assume that where and For = (1, NS NP p(y|) = P(Yis |s, s), s=1 i=1 Yis | as, s~ ind Bern(s), for i = 1,...,NP,s = 1,...,N logit(is) = log (1-0 = s + sxis, for i = 1, ,NP,s = 1,..., Ns. 1 - Ois aNs) and = (1,..., BNS), we assume NS p(|) = p( |)p(s|), s=1 where for all s = 1,...,NS, and Finally, , ind - Normal(u, o) Bs, ind Normal(, g). ~ P(, , ) = P()p()p()p(), where ~ Normal(0,9), ~ Unif (0,3), ~ Normal(0,9), and of ~ Unif (0,3). Note that the prior distributions for variance parameters are specified on the standard deviation. a. [5 points] Explain why the model above is a hierarchical Bayesian model. b. [10 points] Draw a DAG representation of this model. c. [10 points] Provide interpretations of all unknown parameters.