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.