For Causal Nodes X,Y:
X and Y will be deconfounded if we block every backdoor, because such paths allow for spurious correlations. 

When eliminating backdoors, we additionally need to ensure that anything closed off from X is not a descendent of X.

Back-Door Criterion

Pearl introduces an axiomatic system containing three rules to remove do operators in Chapter 7. 

Rule 1. Irrelevance
Rule 2. Conditional equivalence of *do(x)* and *see(x)*
Rule 3. Lack of causal paths

Generalizing Do-Calculus

A Randomized Controlled Trial (RCT) or a Controlled Experiment is a method to eliminate unmeasurable confounders effect on causal inference by implementing an intervention on a randomly selected group of individuals (treatment/intervened group) while keeping other individuals unchanged (controlled group, placebo, not-intervened group) to observe the changes in outcomes due to the intervention effect. The RCTs are intended to "disarm" confounders (confounding variables).

Randomized Controlled Trial (RCT) = Controlled Experiment

An observational study, as supposed to an experimental study is a type of study that investigators observe the effect of an intervention without assigning it to the subjects at random.  That is, the subjects choose their own treatment/ intervention.  

Observational studies may prove to be less costly that RCT's, given that the investigator need not actively change the behavior of the subjects


Observational Studies

When the causal effect of a variable $X$ on $Y$ is mediated by a variable $M$, which is shielded from a confounding set $C$, if $C$ is *unmeasurable*, we cannot block the back-door $X \leftarrow C \rightarrow Y$. So, we use the front-door $X \rightarrow M \rightarrow Y$. 

Front-Door Criterion (Unmeasurable confounder of a mediated path)

$\ \ \ \ \ \ \ \ \ \ \ C$

$\ \ \ \ \swarrow \ \ \ \ \ \ \ \ \searrow$

$X \rightarrow M \rightarrow Y$

Pearl notes two ways of eliminating unmeasurable confounders, in the context of:

- RCT
- Observational studies

In each of these methods, we can use the front-door criterion, back-door criterion, or if necessary, do-calculus.

University of Michigan - Ann Arbor

Deconfounding is defined as mitigating the adverse effect of a confounder in causal inference.  There are multiple methods of deconfounding depending on whether the confounder is:
- Measurable
- Unmeasurable

Deconfounding in Causal Inference

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

Given two variables X, Y and a confounder Z (i.e. X<-Z->Y). To obtain the relationship between X and Y not affected by Z, we can separately compare X and Y in every groups of Z. Then we can take an average of the effects, weighing each group according to its percentage. This method of compensation is called "adjusting for Z" or "controlling for Z" by statisticians.

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