Flat and weak prior distributions are two possible types of prior in Bayesian statistics. In layman's terms, what are the key differences between the two, and why do we use one or the other?

Specifically, I use Bayesian methods to estimate variance and covariance components and I am looking for a clear simple way to explain the differences to non-statisticians/Bayesians.

Laymans description of Inverse-Wishart and Parameter-Expanded priors would also be of use! :)

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    $\begingroup$ Flat has a precise definition, it means the prior density is flat, i.e., constant, while weak does not have a clear-cut definition, see for instance in Gelman et al.'s 2013 Bayesian Data Analysis. $\endgroup$ – Xi'an Sep 15 '15 at 13:31
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    $\begingroup$ Purely from the language: 'flat' concerns shape, but 'weak' concerns impact (on inference), so it would be reasonable to identify 'weak' priors with ones that are more or less uninformative. That is, those that do not generate conclusions differing much from those a sampling theory or likelihood based analysis would come up with. ('Uninformative' is, of course, still a somewhat ambiguous term, but at least we have several ways to formalise that one.) The various beta priors for a binomial problem can be used to intuitively demonstrate the difference between weakness and flatness in this sense. $\endgroup$ – conjugateprior Sep 17 '15 at 13:01
  • $\begingroup$ @conjugateprior that's a nice way of explaining it - would you then say that a flat prior can also be weak or strong, i.e. when the prior is flat and strong it has a very large influence on the posterior? $\endgroup$ – rg255 Sep 17 '15 at 14:07
  • $\begingroup$ Exactly. In a low N Binomial problem with a beta(k,k) prior, k=1 is flat but strong, but gets weaker and less flat as $k\rightarrow0$. Obvious caveat: Flatness is relative to the parameterisation, e.g. k=1 is flat in $\theta$ but k=0 is flat in logit($\theta$). And non-informativeness usually too, if I remember correctly. Unless you don't mind being improper, which I do, though I'd defer to @xian on these things. $\endgroup$ – conjugateprior Sep 17 '15 at 15:16
  • $\begingroup$ I should note that defining weak as 'has little effect on inference in this experiment' is a preçis of the statement in Box and Tiao p.44. $\endgroup$ – conjugateprior Sep 17 '15 at 15:21

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