Suppose at t1 there are countably infinitely many people with red hats and countably infinitely many people with black hats. You’re one of them and you can’t see anybody hat (including your own). What probability you should attach to the proposition that your hat is red depends on the causal history rather than on what the world is like at t1.
For consider two causal histories, each of which results in the same time slice at t1. In the first history, we start with infinitely many hatless people at t0, and for each one we flip a fair coin to see if they get a red or a black hat. Then we arrange the people in a (bidirectionally infinite) line of alternating hat colors. In the second history, we start the same way, but now our coin is unfair, so any given person has only a 1/4 chance of getting a red hat and a 3/4 chance of getting a black hat. But again after the fact the people are arranged in an infinite line of alternating hat colors.
In the two scenarios, the outputs at t1 are relevantly alike—an infinite line of people of alternating hat colors—but what probability you should assign to the proposition that your hat is red depends on which causal history actually took place. So probabilities don’t just depend on how things are now, but also on how things were. At least when we’re dealing with infinities.