I have been digging causality recently, and I cannot understand what the "do-operator" does. Like I understand how it relates to the notion of intervention, like "what happens to our system when this happens", but I don't understand the difference with a simple "conditioned on/knowing" operator.
I feel like the semantic difference is that the "knowing" operator always relies on the probability that X happens, whereas do(X) is more deterministic.
Is my intuition right or somehow right? Or did I get something wrong?
AFAIK the distinction is that:
* When you condition on a particular outcome for X, it affects your probabilities for every other variable that's causally related to X, in either direction.
* You gain information about variables that are causally downstream from X (its "effects"). Like, if you imagine setting X=x and then "playing the tape forward," you'll see the sorts of events that tend to follow from X=x and not those that tend to follow from some other outcome X=x′.
* And, you gain information about variables that are causally upstream from X (its "causes"). If you know that X=x, then the causes of X must have "added up to" that outcome for X. You can rule out any configuration of the causes that doesn't "add up to" causing X=x, and that affects your probability distributions for all of these causative variables.
* When you use the do-operator to set X to a particular outcome for X, it only affects your probabilities for the "effects" of X, not the "causes." (The first sub-bullet above, not the second.)
For example, suppose hypothetically that I cook dinner every evening. And this process consists of these steps in order:
* "W": considering what ingredients I have in the house
* "X": deciding on a particular meal to make, and cooking it
* "Y": eating the food
* "Z": taking a moment after the meal to take stock of the ingredients left in the kitchen
Some days I have lots of ingredients, and I prepare elaborate dinners. Other days I don't, and I make simple and easy dinners.
Now, suppose that on one particular evening, I am making instant ramen (X=making instant ramen). We're given no other info about this evening, but we know this.
What can we conclude from this? A lot, it turns out:
* In Y, I'll be eating instant ramen, not something else.
* In W, I probably didn't have many ingredients in the house. Otherwise I would have made something more elaborate.
* In Z, I probably don't see many ingredients on the shelves (a result of what we kno
I have been digging causality recently, and I cannot understand what the "do-operator" does. Like I understand how it relates to the notion of intervention, like "what happens to our system when this happens", but I don't understand the difference with a simple "conditioned on/knowing" operator.
I feel like the semantic difference is that the "knowing" operator always relies on the probability that X happens, whereas do(X) is more deterministic.
Is my intuition right or somehow right? Or did I get something wrong?