- 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. 

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

Notational Definition of Confounding 

- 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. 

University of California, Santa Cruz

Below are some of the potential issues when adding variables to a regression model without a clear idea of the causal model - 

- Multicollinearity
- Post-Treatment Bias
- Collider Bias
- How  to recognize and prevent these confounding issues

Perils of Multiple Regression

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)

Multicollinearity = Very  strong correlation between two or more predictor variables. 

- As a result the posterior distribution will appear to suggest that none of the variables are reliably  associated with the outcome - when in fact they might be strongly associated with the outcome.
- The model will still have accurate predictions, but interpreting it will lead to confusion.  You wont be able to determine the actual effect of the variables on the outcome. 

- See next node for an example 

Solutions:
- Identifying highly correlated variables, eliminating all but one
- Combining them into one variable
- However, it is best to take a causal approach -  to determine conditional associations, and not just correlations. 

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