(a) (5pts) For each pair of nonadjacent nodes in this graph, fifind a set of variables that d-separates

that pair. What does this list tell us about independences in the data?

(b) (4pts) For each pair of nonadjacent nodes in the graph, determine whether they are independent

conditional on all other variables.

(c) (6pts) For every variable V in the graph, fifind a minimal set of nodes that renders V independent

of all other variables in the graph.

(d) (2pts) Suppose we wish to estimate the value of Y from measurements taken on all other vari

ables in the model. Find the smallest set of variables that would yield as good an estimate of Y

as when we measured all variables.

(e) (2pts) Repeat the last question assuming that we wish to estimate the value of Z2.

(f) (3pts) Suppose we wish to predict the value of Z2 from measurements of Z3. Would the quality

of our prediction improve if we add measurement of W? Explain.

(g) (3pts) List all of the sets of variables that satisfy the backdoor criterion to determine the causal

effect of X on Y .

(h) (2pts) List all of the minimal sets of variables that satisfy the backdoor criterion to determine

the causal effect of X on Y (i.e., any set of variables such that, if you removed any one of the

variables from the set, it would no longer meet the criterion).

(i) Verify your answers with R code

4. (27pts) For the faithful DAG in Figure 1 Figure 1: Another causal graph. 4. (27pts) For the faithful DAG in Figure 1 Figure 1: Another causal graph