What's the purpose of do-calculus? I understand backdoor adjustment blocks backdoor paths and front door adjustment combines the causal effect of different nodes. The purpose of both is to eventually identify the causal effect of a treatment on the outcome variable.
How do the 3 rules of do-calculus help in doing the same?


What's the purpose of do-calculus?

The do-calculus is an axiomatic system for replacing probability formulas containing the do operator with ordinary conditional probabilities.
Criterions like backdoor and frontdoor can be considered as consequence of do-calculus. Them are verifiable under some conditions. However, as suggested by Adrian Keister, those conditions sometimes do not holds.
Read here for more details: https://plato.stanford.edu/entries/causal-models/do-calculus.html
