Is there really a difference between saying, “I don’t know anything, a priori, about the parameters of this distribution”, and using a uniform prior?
What about, “I don’t know anything about that value” versus “As far as I’m concerned, every possibility for that value is equally likely”?
I think so.
Sometimes I really have no clue what true value is, but a uniform prior is too assertive.
And what about uniform priors over a real-valued parameter? (You can do that if you’re happy with an improper prior; it often still works after the bayes update even.) Do I really think theta=1million is just as likely as theta=25? There are lots of situations where I would say “no”, but I would be unwilling to say the exact ratio between their prior probabilities.
On the other hand, if all a prior is is a placeholder before you update it with a data likelihood, then it doesn’t matter. I think it’s hard to call this “bayesian” though.
I think the answer is no.
You’re talking about a case where you DO have an idea about what the prior probabilities are, you just don’t know the particular shape. In that case, a uniform prior is “too assertive”, but only because it forces a particular distribution.
I’m talking about the case where you say something like “I don’t know anything about this, and I refuse to commit to any prior belief at all”. That’s what I don’t see as any different from a uniform prior.
And maybe this is nit-picking but you only have to resort to improper priors if your real-value parameter is unbounded. :)