# When should past data be allowed to inform a prior?

The question in the title is usually discussed subjectively, but I'm wondering if there is a way to answer it with some mathematical rigor...

Suppose I have experiment A that generated data $$x_1$$ and later I do experiment B and collect data $$x_2$$. Is it fair to say that an informative prior developed from $$x_1$$ should be used to analyze $$x_2$$ if and only if I consider the combined data $$(x_1, x_2)$$ exchangeable?

• Example: Machine A is old and known to make good widgets that have a $p_0 = 90\%$ probability of not breaking. We have data from a past test that indicates this. Machine B is a brand new update of Machine A that is required to also make widgets that have a $90\%$ chance of not breaking. Machine B is supposed to be better than Machine A, but has been completely redesigned. The makers of Machine B argue that we don't have to test Machine B very much because we can just use the data from Machine A to form a prior on $p$. My inner Bayesian agrees, but it seems risky to subject B to less testing. – JTH Aug 23 at 20:50
• Is there any guidance of how to determine a prior that is "between" uninformative and highly informative? Suppose the test on Machine A produced $10,000$ widgets. The prior is "overwhelmingly informative" in my opinion, and new observations from Machine B might not affect it very much. – JTH Aug 23 at 21:33