This actually works if you condition every probability, including the probability of the parent nodes, on the observed information. For example, say that option one is you could start with all options possible in the marble game and then observe that the result was not Heads and White. And option two is you could determine the marble color causally, in a way that never even has the possibility of White when Heads. And these two options result in different probabilities.
This really reinforces how the information about how a node’s value is causally generated is different from observed information about that node.
This actually works if you condition every probability, including the probability of the parent nodes, on the observed information. For example, say that option one is you could start with all options possible in the marble game and then observe that the result was not Heads and White. And option two is you could determine the marble color causally, in a way that never even has the possibility of White when Heads. And these two options result in different probabilities.
This really reinforces how the information about how a node’s value is causally generated is different from observed information about that node.