As I understand it, selecting a prior provides something of a starting point for your analysis. From there, the distribution is shaped by the observed data. Obviously, the more data you observe, the more discrepancy is possible between the prior and posterior distributions (especially if the selected prior is inappropriate). As a result, it would seem to make sense that, for some large n, the selection of a prior is essentially irrelevant because the observed data will overwhelm the prior. Is this, in fact, the case? If so, does this actually occur in practice (or does that value of n need to be so ridiculously large that the point is purely theoretical)?
The underlying problem that I’m facing is that if I have m data points and I’m concerned about the appropriateness of my prior, what are some tools at my disposal to determine if my concern is legitimate?
Note: I realize that this question is very theoretical and that a concrete answer isn’t really possible (I’m sure a lot of this depends on the types of distributions, how inappropriate the prior is, etc.), so I’m worried that this might violate the condition that questions must be “practical, answerable questions based on actual problems you face.” If this is the case, please let me know. I’m new to the site and don’t really have a firm grasp on the etiquette yet…