Structural Causal Models Flashcards
Directed Acyclic Graphs (DAG)
The parent-child relation is clear with the arrows. Additionally, it is acyclic meaning there is no loop.
Why focusing on acyclic graphs in causality?
We do not want a variable to be the cause of itself. I.e., we do not want an instantaneous feedback loop.
What are recursive models?
Recursive models are same as Acyclic models. Only fully recursive models can have causal or structural interpretation.
Path
Sequence of edges connecting two vertices. Path can go against the direction of the arrows.
Directed path
A path that foes along the direction of the arrows.
Structural Causal Model
A DAG represents an underlying structural causal model
Structural Equation Models
Same as Structural causal models but relationship between nodes are represented with linear equations. Where as with causal models it is fully non-parametric.
epsilon or error term property in structural models
epsilons are jointly independent and not correlated with any of the independent variables (i.e., Markovian)
What is a semi markovian model?
The error terms or background factors of nodes (independent variables can be dependent) can be dependent due to an unobserved mutual parent.
What are the possible configurations in a graph?
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
Provide an example of Collider structure.
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.
What is d-separation
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).
What is d-connected
It is the opposite of d-separated. Meaning there two variables are correlated.
Does conditioning on descendents of a collider also opens up a path?
Yes
D-separation Example
What is a do-operator
It defines interventions. P(Y|do(X=x))
How does the intervention impact the causal graph?
Delete all the natural causes of X or all equations corresponding to variables in X and substituting with x.
Is P(y|do(x)) equal to P(y|x)?
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.
There are 4 open paths from x to y in this graph. Explain the open paths
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.
Describe the cobweb model as illustrated in this graph.
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.
