All the problems in this assignment should be solved and handed in individually. You should be prepared to answer questions about your solutions yourself. The full set of solutions should be submitted as a single PDF document in Canvas. If you don't have Canvas access, submit your solution as a pdf in an email to the examiner. Feel free to use any software of your choosing for preparing illustrations and drawings.
Problem 1 [Probability]:
You are suspecting that the grades awarded to assignments in Calculus 101 depend on how long before the deadline they were handed in. To test this hypothesis, you collect samples of times (in minutes before deadline) and grades (0-100). You fit a linear regression model of onto and find that the slope is 0.
Q 1: Can you conclude that and are independent? Why/why not? What assumptions have you made? Are there additional assumptions that could change your conclusion? Which?
Problem 2 [Probabilistic graphical models]:
Q 2: Draw the set of DAGs with d-connections corresponding exactly to (only) the independencies:
a) , for variables A, B, C
b) , for variables A, B, C, D
c) , , , for variables A, B, C, D
Problem 3 [Structural causal models]:
Q 3: Consider the following set of structural equations with noise variables , all distributed according to a standard Normal distribution
a) Draw the corresponding DAG
b) Draw the mutilated graphs following the interventions
Problem 4 [Backdoor criterion]:
Q 4: Consider the DAG below. Give two valid adjustment sets for identifying the effect of intervening on on , that is, . Hint: Use the backdoor criterion. Are there smaller adjustment sets?