An idea that keeps bothering me is if there is some parallel between a Bayesian and Frequentist model in using a non-informative/uniform prior. I have heard that the answer is no, but I am wondering if someone could go into more detail why. The intuition is that using a uniform prior gives equal weight to the parameter space as specified by the likelihood, and so it appeals to intuition to think of it that way. What is the reason why this thinking is wrong? Thanks!
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1$\begingroup$ An uninformative prior is not necessarily uniform. It depends on the model. $\endgroup$– Neil GCommented Feb 9, 2017 at 2:56
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1$\begingroup$ I think that the answers given in stats.stackexchange.com/questions/64259/… and stats.stackexchange.com/questions/180420/… contain everything you need $\endgroup$– beuhbbbCommented Feb 9, 2017 at 10:35
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$\begingroup$ As @Neil_G correctly notes, it depends on the model. For example, in the case of the Beta-Binomial model, the prior which leads to the same estimates a frequentist statistician would make, is the Haldane prior (an improper prior), not the uniform one (see here). Actually, a good frequentist statistician would know better than using the Wald confidence interval for a proportion :) $\endgroup$– DeltaIVCommented Feb 10, 2017 at 7:08
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1 Answer
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The simplest argument against this intuition is that the uniform distribution is not invariant by a change of parametrisation: if $\theta\sim \mathrm{U}(0,1)$, the parameter $\eta=\theta^2$ is not uniformly distributed. However, if one knows nothing about $\theta$, one knows nothing about $\eta$ as well.
