6. Let X ? {0, 1}, Y ? {0, 1}, Z ? {0, 1, 2}. Suppose the distribution of (X, Y, Z) is Markov to: X.

6. Let X ? {0, 1}, Y ? {0, 1}, Z ? {0, 1, 2}. Suppose the distribution of (X, Y, Z) is Markov to: X -? Y -? Z Create a joint distribution f(x, y, z) that is Markov to this DAG. Generate 1000 random vectors from this distribution. Estimate the distribution from the data using maximum likelihood. Compare the estimated distribution to the true distribution. Let ? = (?000, ?001,…,?112) where ?rst = P(X = r, Y = s, Z = t). Use the bootstrap to get standard errors and 95 percent confidence intervals for these 12 parameters. 7. Consider the DAG in Figure 17.12. (a) Write down the factorization of the joint density. (b) Prove that X Zj .