It’s worth noting that gradient descent towards maximum entropy (with respect to the Wasserstein metric and Lebesgue measure, respectively) is equivalent to the heat equation, which justifies your picture of probability mass diffusing outward. It’s also exactly right that if you put a barrier at the left end of the possibility space (i.e. ruling out the date of AGI’s arrival being earlier than the present moment), then this natural direction of increasing entropy will eventually settle into all the probability masses spreading to the right forever, so the median will also move to the right forever.
This isn’t the only way of increasing entropy, though—just a very natural one. Even if I have to keep the median fixed at 2050, by keeping fixed all the 0.5 probability mass to the left of 2050, I can still increase entropy forever by spreading out only the probability masses to the right of 2050 further towards infinity.
It’s worth noting that gradient descent towards maximum entropy (with respect to the Wasserstein metric and Lebesgue measure, respectively) is equivalent to the heat equation, which justifies your picture of probability mass diffusing outward. It’s also exactly right that if you put a barrier at the left end of the possibility space (i.e. ruling out the date of AGI’s arrival being earlier than the present moment), then this natural direction of increasing entropy will eventually settle into all the probability masses spreading to the right forever, so the median will also move to the right forever.
This isn’t the only way of increasing entropy, though—just a very natural one. Even if I have to keep the median fixed at 2050, by keeping fixed all the 0.5 probability mass to the left of 2050, I can still increase entropy forever by spreading out only the probability masses to the right of 2050 further towards infinity.