Hello, PyMC community!
First of all, thanks a lot for CausalPy and including so many designs to it. I’ve got a couple of questions about Interrupted Time Series design and the Intervals it’s providing and would appreciate your help understanding a couple of points.
1. Please suggest theory on why we can sum up intervals to get cumulative ones
I can see that to calculate cumulative impact of the intervention, the package basically sums up the differences between the actuals and the predictions.
Same is done to calculate cumulative intervals - upper and lower bounds of each prediction are summed up (or the posteriors standing behind them, potentially).
While I understand that it seems an obvious way to do it and is often done in the industry (Causal Impact does it same way, afaik), can you please suggest some theory on why we actually do can sum up intervals. Is it possible in Bayesian setting or in frequentist setting as well?
Are there other situations where adding the intervals (or posteriors) rises up?
2. Why do Confidence Intervals grow with time?
In Interrupted time series example page
there’s a graph showing that cumulative intervals grow with time. Can you please help me understand why this happens?
It seems that the model is a regular Linear Regression, predictions are made for each day separately and probably have same confidence intervals. In other words, having looked at the code, it seems that the model seems to not know that it’s dealing with time. Why is it that interval is growing with time?
3. Confidence Intervals vs Prediction Intervals
In frequentist contexts there are two different intervals - confidence interval (which tell where our betas should be) and prediction intervals (where we expect our predictions to be) (more here).
Prediction Intervals are by definition always wider.
Can you please say whether such concepts exist in Bayesian framework and which concept is closer to what is being used in Interrupted time series?
Very much appreciate your help!