The exact example you’ve given (binomial versus negative binomial sampling distribution) is actually a counterexample to the above assertion. Those two distributions have the same likelihood function, so the evaluation of the evidence is the same under both scenarios. It’s true that the prior predictive distributions are different, but that doesn’t affect the posterior distribution of the parameter.
So it doesn’t matter whether the data were sampled according to Pr1 or Pr2. You can check that the binomial and negative binomial distributions satisfy the proportionality condition by looking them up in Wikipedia.
The exact example you’ve given (binomial versus negative binomial sampling distribution) is actually a counterexample to the above assertion. Those two distributions have the same likelihood function, so the evaluation of the evidence is the same under both scenarios. It’s true that the prior predictive distributions are different, but that doesn’t affect the posterior distribution of the parameter.
Really? I find that counterintuitive; could you show me the calculation?
Suppose that there are two sampling distributions that satisfy (sorry about the lousy math notation) the proportionality relationship,
Pr1(data | parameter) = k * Pr2(data | parameter)
where k may depend on the data but not on the parameter. Then the same proportionality relationship holds for the prior predictive distributions,
Pr1(data) = Integral { Pr1(data | parameter) Pr(parameter) d(parameter) }
Pr1(data) = Integral { k Pr2(data | parameter) Pr(parameter) d(parameter) }
Pr1(data) = k Integral { Pr2(data | parameter) Pr(parameter) d(parameter) }
Pr1(data) = k Pr2(data)
Now write out Bayes’ theorem:
Pr(parameter | data) = Pr(parameter) Pr1(data | parameter) / Pr1(data)
Pr(parameter | data) = Pr(parameter) k Pr2(data | parameter) / (k Pr2(data) )
Pr(parameter | data) = Pr(parameter) * Pr2(data | parameter) / Pr2(data))
So it doesn’t matter whether the data were sampled according to Pr1 or Pr2. You can check that the binomial and negative binomial distributions satisfy the proportionality condition by looking them up in Wikipedia.
Your argument is convincing; I sit corrected.