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According to Gelman, a weakly informative prior is defined in the following way:

We characterize a prior distribution as weakly informative if it is proper but is set up so that the information it does provide is intentionally weaker than whatever actual prior knowledge is available

I humbly consider this definition a bit misleading, Wouldn't more correct to say that a weakly informative prior is a proper prior which doesn't carry a lot of information before observing the data?
Thank you.

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The name "weakly informative prior" is motivated by fact that "uninformative prior" is misleading term since it suggests that such prior does not carry information, while every prior carries some information. Even if you use "flat" prior it carries information on that you assume that all the values are possible with similar probability (what in most cases is actually not true). So you can use the name "weakly informative prior" to describe what your prior is: that it does not carry much information, however it carries some since it is not "uninformative".

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