If the data size is small? should we give the noninformative prior(vague prior) to the data set? I think if the data size is small. It is hard for data to tell the whole story.If you do think we can not set noninfomative prior, is there any other way we can get the information for setting the prior?
The use of vague or informative prior depends on the amount of knowledge that you have for the parameters that you want to assign the prior.
I consider the following cases:
No experts information and big/small data set, in that case, a vague prior would do the job
Experts information and big/small data set, in that case, a more informative prior based on the beliefs of the expert would do the job
So, the choice of the prior depends on the a priori information that you have and not the size of the data set. However, in the case where you have a small data set and experts information is available, then the bayesian narrative can work much better than the frequentists narrative.