There are countless examples of diffuse priors being used to 'allow the data to speak.' However, what if one's past experience leads you to be skeptical of new data, without necessarily having a prior as to where the bulk of the density is located. Is there some way to have a diffuse prior that nevertheless strongly influences the posterior?
Edit for clarification:
I'm thinking of a situation in which you want to capture some of the "wisdom" of an experienced researcher who knows nothing about a particular domain but has priors about the amount of evidence required before he/she accepts a finding. Something like the Ioannidis story. So it's diffuse because you don't have any reason to believe one value over another, but it's informative in that you want to down-weight the data.
Best Answer
Diffuse priors and noninformative priors are not necessarily the same: a very diffuse prior (like the (improper) infinite uniform distribution) could be quite "informative" on e.g. a variance.
That being stated: your intent of using a diffuse prior that strongly influences the prior seems to be contradictive. I'm assuming you want to use a diffuse prior exactly for the purpose of empowering the data, no? The only "other" reason I see to use a vague prior is because you have little prior information – but then indeed you put all your bayesian hope in the data.
In short: either you trust your prior information, and then you use whatever prior expresses this information, or you don't, in which case you use as noninformative a prior as possible.
In the case where you have something like an "outdated" prior (e.g. a prior that comes from research from a long time ago, or in different circumstances), you can, however, make the prior more diffuse to express this. In some cases this can be done in a controlled manner (I believe this is the case for a conjugate prior) so you can add diffusion matching a set of "virtual observations", but most analytically expressed distributions have straightforward ways of increasing the variance. In any case, it is a matter of feeling and judgment how much diffusion to add, and it will definitely be worth your while to do a sensitivity analysis for this effect afterwards.