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Is there a precise definition of weakly informative prior?

How is it different from a subjective prior with broad support?

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    $\begingroup$ My understanding is that a weakly-informative prior expresses more about the researcher's attitude towards the prior, rather than any mathematical properties of the prior itself. The canonical example would be Gelman's recommendation of a Cauchy prior with location 0 and scale 5/2 for logistic regression. $\endgroup$ – Sycorax Oct 22 '13 at 16:24
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The above comment is accurate. For a quantitive discussion, there are a number of "uninformative" priors in the literature. See for example Jeffreys' prior; see earlier post What is an "uninformative prior"? Can we ever have one with truly no information?

They are defined in different ways, but the key is that they do not place too much probability in any particular interval (and hence favor those values) with the uniform distribution being a canonical example. The idea is to let the data determine where the mode is.

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  • $\begingroup$ So weakly informative prior is just a better name for slightly informative "uninformative prior"? $\endgroup$ – Memming Oct 22 '13 at 17:09
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    $\begingroup$ I usually use "uninformative" as that is more common and indicates intent. Weakly informative is probably more accurate though, as all distributions carry some information (unless they are improper priors..but that is another discussion) $\endgroup$ – user31668 Oct 22 '13 at 17:13
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    $\begingroup$ I had the impression that weakly informative priors aim to avoid having to commit to uninformative priors formally defined according to some theory or other - they're proper priors that work for inference, while not taking into account all prior knowledge as a fully subjective prior would. $\endgroup$ – Scortchi Oct 23 '13 at 10:49
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    $\begingroup$ @Scortchi: I think your comment highlights the ambiguity inherent in "weakly informative prior." Your interpretation makes sense and is in a similar vein to user777. The relationship between "informativeness" and probability is a tricky thing, with only partially satifactory solutions (e.g., Shannon entropy). I get your point though...they are not necessarily synonymous, as weakly informative priors may use only some of the information, while uninformative priors explicity ignore all available information. $\endgroup$ – user31668 Oct 23 '13 at 12:07
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Further to Eupraxis1981's discussion of informative priors, you can think of the "information" in a prior as inversely proportional to its variance. Consider a prior with near zero variance: you're basically saying "before looking at the data, I'm almost positive I already know the location of the true value of the statistic." Conversely, if you set a really wide variance, you're saying "without looking at the data, I have really no assumptions about the true value of the parameter. It could be pretty much anywhere, and I won't be that surprised. I've got hunch it's probably near the mode of my prior, but if it turns out to be far from the mode I won't actually be surprised."

Uninformative priors are attempts to bring no prior assumptions into your analysis (how successful they are is open to debate). But it's entirely possible and sometimes useful for a prior to be only "weakly" informative.

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