Graphical model question

Problem 1Let X
~
N(μ, Σ) where X = (X1,, Xp). We saw in class that marginal independence in the multivariate normal
model corresponds to zeros in the covariance matrix, i.e., X¿ ■ X; if and only if Σij = 0. We also saw that conditional
independence where the conditioning set is “everything else” corresponds to zeros in the precision matrix: X₁ 1
X;|X \ {Xi, X; } if and only if Kij = 0. Let Xs © X \ {Xi, X; } be some arbitrary subset of the variables. Explain
why the following is true: X₁ X;|Xs if and only if (Σ{ijS},{ijS})₁₁ = 0 (Hint: that conditional independence
statement amounts to a marginal independence X; 1 X; in the conditional distribution where Xs
conditional distribution?)
Problem 2
Show that the binary Ising model (pairwise MRF) implies that
1
=xs. What is that
P(X¿ = 1|X_i = x_i) =
1+ exp(-010-Σink Oikxk)
where X denotes all the vertices except Xi. That is, the Ising model implies a logistic model for each vertex
conditional on all the rest. (This is excersize 17.11 in Hastie et al.)Problem 6
Construct the DAG in Figure 1 as a daggity object. Simulate 10000 observations from this graph using simulateSEM()
as you did on the first homework. Estimate the effect of E on F and the effect of B on A using backdoor adjustment
and linear regression. If there is more than one sufficient adjustment set, try each of the ones identified by dagitty
and compare them. Are the point estimates similar? Do the estimates have similar variance (or confidence interval
length)? Compare also these estimates against an approach which simply adjusts for all other variables in the graph.
How are the results different (if they are) and what is the explanation?D E
G
B
F
Figure 1
H