# Intuitive explanation of rules of do-calculus

Do-calculus has 3 rules: https://plato.stanford.edu/entries/causal-models/do-calculus.html

I understand them on a mathematical level, but they seem so arbitrary. I can not wrap my head around what the intuitive explanation behind them is, and why they are useful. I guess there is some intuitive explanation, as they are called 'Rule 1 (Insertion/deletion of observations)', 'Rule 2 (Action/observation exchange)', and 'Rule 3 (Insertion/deletion of actions)', but why?

• Well, they're not totally arbitrary. They have been proved to be complete, as the Stanford article mentions. I don't know if they've been shown to be independent, though. They're complete in the sense that any expression with the $\operatorname{do}$ operator in it that can be reduced to an expression without the operator, can be manipulated with those three rules to achieve that. As for intuition, I'm trying to learn that myself. Can't help you there. Oct 8, 2020 at 2:49
• Mhh, what do you mean with reduced to an expression without the do operator? In all three rules the left hand side and right hand side still have the do operator in it? Oct 8, 2020 at 4:03
• True, but if you notice, there are never the same number of terms in the RHS as in the LHS. For Rules 2 and 3, there are not the same number of do operators. Here's a big idea in all of this: you cannot directly observe probability distributions that have the do operator in them. Therefore, you would like to eliminate them, if possible, from any expression you may have. Given a particular causal graph, and a particular expression, the do calculus can let you eliminate instances of the do operator, thus allowing you to measure the value of an expression using data. Oct 8, 2020 at 12:23