Dependence between epsilons

[josejhlee]

Hi authors,
I fail to understand why the randomly sampled epsilons are dependent in respect to the timestep axis. And likewise, I fail to understand why does epsilon_hat is independent. Thank you for the great work and eager for your reply!

[inbarhub]

Hi,

Eq1 establishes a connection between x_t and x_{t-1}, incorporating independent noise (eta). Upon rephrasing it as Eq2, we depict a connection from x_0 to x_t leading to a noise (epsilon) that takes into account all preceding etas from Eq 1.
Consequently, the epsilon linking x_0 to x_{t-1} demonstrates a significant correlation with the epsilon linking x_0 to x_t. These epsilons lack independence.

However, in our scenario, the sampling process generating x_t was devised as a separate auxiliary process between all timesteps. There exists no connection between the creation of x_t and the creation of x_{t-1}. This suggests that the epsilons creating x_t are independent across all timesteps.