The do-notation extends the language of conditional probability to include intervention on some variables and observing other variables.

A causal network is a belief network where $P(X\mid parents(X))=P(X\mid do(parents(X)))$ for each variable $X$ – intervening on the parents of a variable has the same effect as observing them.

D-separation characterizes which conditional independencies follow from the independencies of a directed graphical model (belief network). The do-calculus extends d-separation to include interventions.

The do-calculus can be used to show cases where the effect of interventions can be computed from observational data, including the backdoor and front-door criteria.

There are cases, such as in Simpson’s paradox, where the probabilistic inferences depend on the causal model and not just the data.

Counterfactual reasoning can be used to answer “what-if” queries.

Causal assumptions can be used to go beyond randomized clinical trials, if the assumptions are accepted.