Exercise 6

6.1 Linear Gaussian SCM with  an open backdoor

Consider the DAG  and SCM from slide 9 on Lecture 6.

 $\begin{align*} X_1 &:= N_1 \\ X_2 &:= 7X_1 + N_2 \\ X_3 &:= 4X_2 + N_3 \\ X_4 &:= 2X_1 + 3X_3 + N_4 \\ \end{align*}$

where each $N_i$ is Gaussian with mean 0 and variance 1.  Note that the causal effect of $X_2$ on $X_4$ is 12. Which of the following regressions give the correct value (i.e. the estimated constant before $X_2$ becomes 12 )?

a) $X_4 \sim X_2$

b)  $X_4 \sim X_1 + X_2$

c)  $X_4 \sim X_2 + X_3$

d) $X_4 \sim X_1 + X_2 + X_3$

Hint: Check your answer with the code ex06_01_check.ipynb Links to an external site..

6.2 Linear Gaussian SCM - another example

Consider the SCM and DAG from Lecture 6 below. Write code, using the ols function performing ordinary least squares, similar to the previous exercise, to check the statements made on the slide (from Lecture 6) below.

6.3 Parental Adjustment

Explain why the theorem for getting a valid adjustment set using parental adjustment (see slide from Lecture 6 below) follows from the backdoor criterion.

6.4 The backdoor criterion in another Linear Gaussian SCM

Solve the exercise below using the backdoor criterion, and check your results using the code ex06_04_check.ipynb Links to an external site.

6.5 The frontdoor criterion and smoking

a) Use the frontdoor criterion and the information below to analytically determine

$P(\textrm{cancer=1} | \mathbf{do}(\textrm{Smoke} = 0)) \\ P(\textrm{cancer=1} | \mathbf{do}(\textrm{Smoke}= 1))$

b) Suggest  changes in the DAG that would mean that conditions (i), (ii) or (iii) in the frontdoor condition would be violated. Also describe what these changes could correspond to in real life.

6.6 Finding the causal structure - Myopia example

On Lecture 6 we described how statistical independence  tests could be used to determine that the causal structure explaining the myopia data is the following (figure to the left), ie that the use of night light (NL) does not cause child myopia (CM) even if these two variables are correlated.

But at the end of the lecture we claimed that the 3 structures in the figure to the right could not be separated purely from statistical independence tests. Why could we in this example rule out the two other possible structures ? (Hint: What would these alternative directions of the arrows imply in real life?)

Solution: solu6.pdf Download solu6.pdf

Code solving 6.2 : ex06_02_solution.ipynb Links to an external site.