"But Judea Pearl tells me just how to compute a counterfactual, given only my beliefs about the actual world."

This is actually a subtle issue. The procedure given in the book assumes (a) full knowledge of the precise causal mechanisms (which you never know in practice) and (b) the distribution over all unobserved variables (which you don't know by definition). Surprisingly, it is possible to compute certain counterfactuals using ONLY the distribution over observable variables (which is typically what you get). You can check my thesis for details if you wish.