In statistics, I think 'weakly informative priors' are becoming more popular. Weakly informative priors are distributions like a t distribution (or normal) with a really wide standard deviation and low degrees of freedom. This allows us to avoid spending all out data on merely narrowing down the correct order of order of magnitude, which can be a problem in many problems using non-informative priors. It's almost never the case that we literally know nothing prior to the data.