Structural Causal Models

Question 1

Directed Acyclic Graphs (DAG)

Answer 1

The parent-child relation is clear with the arrows. Additionally, it is acyclic meaning there is no loop.

Question 2

Why focusing on acyclic graphs in causality?

Answer 2

We do not want a variable to be the cause of itself. I.e., we do not want an instantaneous feedback loop.

Question 3

What are recursive models?

Answer 3

Recursive models are same as Acyclic models. Only fully recursive models can have causal or structural interpretation.

Question 4

Path

Answer 4

Sequence of edges connecting two vertices. Path can go against the direction of the arrows.

Question 5

Directed path

Answer 5

A path that foes along the direction of the arrows.

Question 6

Structural Causal Model

Answer 6

A DAG represents an underlying structural causal model

Question 7

Structural Equation Models

Answer 7

Same as Structural causal models but relationship between nodes are represented with linear equations. Where as with causal models it is fully non-parametric.

Question 8

epsilon or error term property in structural models

Answer 8

epsilons are jointly independent and not correlated with any of the independent variables (i.e., Markovian)

Question 9

What is a semi markovian model?

Answer 9

The error terms or background factors of nodes (independent variables can be dependent) can be dependent due to an unobserved mutual parent.

Question 10

What are the possible configurations in a graph?

Answer 10

1- Chain: X -> Z -> Y then: X indep Y|Z
2- Fork: X <- Z -> Y then: X indep Y| Z
3- Collider: X -> Z <- Y then X indep of Y but X dep Y|Z

Question 11

Provide an example of Collider structure.

Answer 11

In this structure the unobservable ability can create a collider for management. So conditioning on management can create a collider structure that opens the path and creates bias.

Question 11

What is d-separation

Answer 11

d-separation happens by conditioning in a chain or fork structure or by having a collider structure. In other words if every path between X & Y are blocked they are d-separated (i.e., conditionally independent).

Question 12

What is d-connected

Answer 12

It is the opposite of d-separated. Meaning there two variables are correlated.

Question 12

Does conditioning on descendents of a collider also opens up a path?

Answer 12

Yes

Question 12

D-separation Example

Answer 12

Question 13

What is a do-operator

Answer 13

It defines interventions. P(Y|do(X=x))

Question 14

How does the intervention impact the causal graph?

Answer 14

Delete all the natural causes of X or all equations corresponding to variables in X and substituting with x.

Question 15

Is P(y|do(x)) equal to P(y|x)?

Answer 15

No. The data generating distribution changes when we do the intervention as we break the links to x from other variables. p(Y|x) is when we have observational data and hence the links to X will still be in place. When you do not have an intervention (say it is costly or infeasible) we need to use observational methods to be able to remove the do operation.

Question 16

There are 4 open paths from x to y in this graph. Explain the open paths

Answer 16

Top left is a directed causal path (usually what we are interested in).

Top right z1 is a common parent of x and y and hence will create a non-causal association between x and y.

Bottom left: z4 is common parent of z4 where its effect is mediated with z1 and z5 (that transmit the effect of the parent to x and y). but it suffers from the same issue as the top right panel.

Botton right: same as the bottom left panel.

Question 17

Describe the cobweb model as illustrated in this graph.

Answer 17

The equilibrium assumption created a cyclic pattern such that the endogenous variables are determined jointly by all equations in the system. So the recursive pattern breaks and once cannot for example manipulate price to change quantity. That is why we focus on recursive and acyclic models for causality.