P(Dog barks | do(kick the dog) )
The `do` modifies your model to treat "kick the dog" as observed, breaking its dependence on any other variables, e.g. the doorbell ringing or the cat hissing. With those links broken, the model becomes simpler, and often the causal question can be answered using observational rather than experimental quantities.
At least, that's the idea. Not sure how broadly accepted it is, but Pearl is a towering figure and his work seemed robust at least to my not-so-expert eyes.