Excerpt

Judea Pearl’s classic paper is motivated by a foundational puzzle about reasoning in humans. The puzzle begins with a data point: it seems that we engage in something like probabilistic updating on the basis of evidence. If I look outside and see that the sky is cloudy, I will increase the degree to which I believe that it will rain. This updating is done easily, without significant cognitive load or attention.

Probability theory provides prescriptive guidance for how one ought to engage in this kind of belief updating. If the current state of my beliefs about the world is represented by a joint probability distribution p over a set of random variables X1, X2, . . . , Xn, and we learn that one of these variables, Xm, takes the value xm (e.g., a variable representing the possible states of the sky could take the value “cloudy”), then probability theory says that we ought to update our beliefs via conditionalization. One consequence of this is that if all of the random variables X1, X2, . . . , Xn are discretely-valued (i.e., they are not continuous) then the equation for this updated joint distribution is given by the so-called “ratio formula”:

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