- The first of the four relations, its one of the most common ones. 
- Conditioning on Z, updating X doesn't change Y. So X and Y are independent of each other, only conditional on Z. 

Fork

- The second type of relation, it is similar to the Fork in the sense conditioning of the middle variable, Z, blocks the path. 
- Conditioning on Z, the path from X to Y becomes blocked. 

Pipe 

- Third type of variable relation
- In a collider there is no association between X and Y, unless we condition on Z. Z opens this path up. 
- Once its open, information can move between X and Y without having direct causal influence on each other. 

Collider 

- Fourth and final variable relation
- Conditioning on the descendent leads to partially conditioning on the parent as the descendent contains some information about the parent. 
- The consequences of this conditioning on the descendent is dependent on the parent
- A common variable relation as we cannot often directly measure a variable, and instead have to rely on its proxy. 

Descendent 

- List the paths connecting the potential cause of interest and the outcome.
- Classify each path as open or closed; paths are open unless a collider is present. 
- Classify each path by whether its a backdoor path or not; backdoor paths have an arrow entering its potential cause of interest.
- Then, open any backdoor paths, and decide which variables to condition on in order to close it. 

The method to figure out what variables to include and which ones to not

- When blocking confounding paths between a predictor and an outcome, we are shutting what's known as the backdoor 
- Causal DAGs allow us to determine which variables to control for (to close the backdoors) and which ones to not control for (to not create new confounds). 
- There are four types of variable relations that merge to create all possible confounding paths; DAGs regardless of their causal complexity are generated from these relations. 

University of California, Santa Cruz

- Confounding is defined as any context where the association between an outcome and a predictor of interest is different from the association we would get if we had experimentally determined the values of the predictor. 
- Multiple regression and confounding develop a relationship where multiple regression can be used to deal with confounding, however, if you controlling for the wrong variables ruins the inference of a multiple regression; multiple regression causes the confounding. 

Confounding

McElreath, R. (2020). Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd ed.). Chapman and Hall/CRC.

Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd edition)

Backdoor 

Considering how confounding by definition impacts the association between the outcome and predictor of interest, its important to control for the correct variables to avoid spurious associations. Even though most ML models are geared more towards prediction rather than the way variables impact one another, confounding becomes an issue when feedback loops are generated and prediction happens using confounded data. 

Confounding effects and Machine Learning 

- Defining confounding with precise notation requires something called with the Do-Operator. Confounding happens when Pr(Y | X) \= Pr(Y | do(X))
- do(X) cuts of all backdoor paths into X, changing the graph and closing the backdoors
- Pr(Y | do(X)) is the expected result of updating X on Y 
- This Do-Operator gives the total causal effect on all forward paths, which can be used to derive causal inference strategies when we can't close all the backdoors

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