Understanding parameter sharing within Bayesian Networks

Question

I am learning about probabilistic graphical models and am a bit confused by the idea of parameter sharing. In the image below, I have been told that the parameters of time slice 0 are copied to time slice 1 and 2 etc. Now, I understand on a high level that, the way weather at t-1 effects velocity at time t will be the same regardless of our position in the sequence. But when we say that each of these time slices have the same "parameters" I get confused because, won't the probabilities associated with the CPD at time slice 100 be different than at time slice 1? How should I understand the concept of parameter sharing in this context?

Answer 1

Suppose we model $$ \text{Velocity}^{t+1} \sim \text{Normal}(\text{Velocity}^{t}, \sigma^2), $$ i.e., your current velocity is your velocity in the last time slice plus some Gaussian noise. Certainly your velocity will change over time, but the noise parameter $\sigma^2$ is shared over all time slices.

The main advantage of parameter sharing is that all the velocity data you collect at every time point can be pooled to learn good estimates of $\sigma^2$.