Time Series Forecasting concept check

Dear all,

I have some doubt about the concept for time series forecasting using probabilistic way.

Given a sample data with time in x-axis and speed error in y-axis.

In normal practice, we generate the predictions using sample posterior predictive, where
p(y_new |x_new,y,x)

In this case, x_new refers to time and y_new refer to speed_error.

But my objective is to predict y_new based on y_current.
Another example like stock prediction case, our objective is to predict next day stock price based on price today.
So based on my understanding, y should refers to future value while x refers to current value.
Am I correct? please correct me if I am wrong.

Thank you very much

I think you are right. When you already construct a parametric model for p(y|x) (i.e. p(future|current) and perform inference for parameters of the model (i.e. get to know that p), you can use that model to get prediction p(y_new|x_new).

Is there examples in the documentation for out of sample prediction?

Thank you very much