Here's How Vague Priors Can Be Informative
Data analysis has to start from some set of assumptions. Bayesian prior distributions drive some people crazy because they make assumptions explicit that people prefer to leave implicit. But there’s no escaping the need to make some sort of prior assumptions, whether you’re doing Bayesian statistics or not.
One attempt to avoid specifying a prior distribution is to start with a “non-informative” prior. David Hogg gives a good explanation of why this doesn’t accomplish what some think it does.
In practice, investigators often want to “assume nothing” and put a very or infinitely broad prior on the parameters; of course putting a broad prior is not equivalent to assuming nothing, it is just as severe an assumption as any other prior. For example, even if you go with a very broad prior on the parameter a, that is a different assumption than the same form of very broad prior on a2 or on arctan(a). The prior doesn’t just set the ranges of parameters, it places a measure on parameter space. That’s why it is so important.