# Transitivity of Bayesian inference

To recall, I'm trying to estimate the ratio of execution times from two different versions of the same software (a baseline and an improved one).

However, there is a complication that I didn't mention in that question. I actually have three versions of the software: a serial version, a baseline parallel version, and an improved parallel version. Let's say the execution time for each version is $$t_0$$, $$t_1$$ and $$t_2$$, respectively. I'd like to compute the following quantities (both as point estimates and credible intervals):

1. $$t_0/t_1$$, the parallel speedup of the baseline version
2. $$t_1/t_2$$, the speedup from the baseline to the improved version
3. $$t_0/t_2$$, the parallel speedup of the improved version

Currently I'm using the BEST model, which is a two sample test, and currently I'm running the model for cases (1) and (2) above. I'm also interested in case (3), but would like to avoid paying the price of computing the model again with $$t_2$$ and $$t_0$$ as inputs.

My question is: is there some sort of transitive property, such that I could use the results for $$t_0$$ (obtained from the model for case (1)) and $$t_2$$ (obtained from the model for case (2)) to compute case (3) without running the model for a third time?