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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?

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  • $\begingroup$ In some cases, like the one you describe, the only honest (if of course disappointing) answer to a question will be "we can't say": if you have little or no prior knowledge and little data, the posterior should indeed simply reflect the sad truth that we currently cannot say much about the problem under investigation. A somehwat unrelated aside: acknowledging that all we can come up with is a vague prior strikes me as superior to the type of person that - e.g. during the current pandemic - you more often than you would like come across that really has no clue and yet has a strong prior... $\endgroup$ – Christoph Hanck Dec 11 '20 at 13:52
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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:

  1. No experts information and big/small data set, in that case, a vague prior would do the job

  2. 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.

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  • $\begingroup$ So the prior setting is not influenced by the data size? what I mean is in a small data set, the vague prior may not working. Is that your suggestions is that always set the informative prior when you have some information $\endgroup$ – hard worker Dec 10 '20 at 17:08
  • $\begingroup$ @hardworker As you said you only set an informative prior, whenever you have some a priori information. $\endgroup$ – Fiodor1234 Dec 10 '20 at 17:13

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