where x(t) is a value is sampled from X(t) distribution for all t.
In plain English, given the last value you get a probability distribution for the next value.
In the AI example: Given x(2025), estimate probability distribution X(2030) where x is the AI capability level.
Possibilities
a) x(t+1) value is determined by x(t) value. There is no randomness. No new information is learned from x(t).
b) X(t+1) distribution is conditional on the value of x(t). Learning which value x(t) was sampled from distribution X(t) distribution gives you new information. However you sampled one of those values such that P(x(t+1) | x(t-1), x(t-2), ...) = P(x(t+1) | x(t), x(t-2) ). You got lucky, and the value sampled ensures distribution remains the same.
c) You learned new information and the probability distribution also changed.
a is possible but seems to imply overconfidence to me.
b is possible but seems to imply extraordianry luck to me, especially if it’s happening multiple times.
Update based on the replies:
I basically see this as a Markov process.
X(t+1) = P(x(t+1) | x(t), x(t-1), x(t-2), ...) = F(x(t))where x(t) is a value is sampled from X(t) distribution for all t.
In plain English, given the last value you get a probability distribution for the next value.
In the AI example: Given x(2025), estimate probability distribution X(2030) where x is the AI capability level.
Possibilities
a) x(t+1) value is determined by x(t) value. There is no randomness. No new information is learned from x(t).
b) X(t+1) distribution is conditional on the value of x(t). Learning which value x(t) was sampled from distribution X(t) distribution gives you new information. However you sampled one of those values such that
P(x(t+1) | x(t-1), x(t-2), ...) = P(x(t+1) | x(t), x(t-2) ). You got lucky, and the value sampled ensures distribution remains the same.c) You learned new information and the probability distribution also changed.
a is possible but seems to imply overconfidence to me.
b is possible but seems to imply extraordianry luck to me, especially if it’s happening multiple times.
c seems like the most likely situation to me.