Not sure I understand. My question was, what kind of probability theory can support things like “P(X|Y) is defined but P(Y) isn’t”. The snippet you give doesn’t seem relevant to that, as it assumes both values are defined.
The kind of probability theory that defines P(X|Y) axiomatically as a primitive entity and only then defines P(X&Y) as a multiplication of P(X|Y) and P(Y), instead of defining conditional probability as a ratio between P(X&Y) and P(Y).
While it’s mathematically equivalent, the former method is more resembling the way people deal with probabilities in practice—usually conditional probability is known and probability of an intersection isn’t—and formally allows us to talk about conditional probabilities, even when the probability of an event we condition on is not defined.
Not sure I understand. My question was, what kind of probability theory can support things like “P(X|Y) is defined but P(Y) isn’t”. The snippet you give doesn’t seem relevant to that, as it assumes both values are defined.
The kind of probability theory that defines P(X|Y) axiomatically as a primitive entity and only then defines P(X&Y) as a multiplication of P(X|Y) and P(Y), instead of defining conditional probability as a ratio between P(X&Y) and P(Y).
While it’s mathematically equivalent, the former method is more resembling the way people deal with probabilities in practice—usually conditional probability is known and probability of an intersection isn’t—and formally allows us to talk about conditional probabilities, even when the probability of an event we condition on is not defined.