I'm running an A/B test. The test is a funnel that, at some step in the funnel, sends half the population to experience A, the other half to experience B. Traditionally everyone saw experience A at this step in the funnel, so I have plenty of historical data for my prior on A. However, I have no data at this step in the funnel for Group B. I do, however, have data on Group B, just at a different step in the funnel.
I expect both groups to convert at similar rates, but the point of the test is to determine which converts higher.
I can argue for using either historical A data as my prior for B, and at the same time argue for using historical B data (which happens at a different step, but behaves similarly as A).
Conceptually, how would I reconcile between these two choices? Other than using my domain knowledge, is there a more scientific way to select the prior?
And yes, I understand that selecting a prior is inherently subjective, but I'd like to minimize that if possible.