# Choosing non-informative priors

I am working on a model relying on an ugly parametrized function acting as a calibration function on a part of the model. Using a Bayesian setting, I need to get non-informative priors for the parameters describing my function. I know that ideally, I should derivate reference or at least Jeffreys priors but the function is very ugly, have many parameters and I am pessimistic on the possibility to obtain actually a result. So I decided to drop this possibility and to empirically choose my priors prying for them to be quite non-informative. Here are my two questions.

1. Can I make more than prying and give insights of their non-informativness from inference results ? Edit : I guess that plotting posterior Vs prior would be a first point. Maybe comparing MAP and ML estimations could be a second argument ?

2. Moreover, is that make sense to justify some aspect of the choice from a "dimensional analysis"? As an example, if I consider a likelihood structure of the form (in a simple regression setting): $$Y | a,b,x = a.x+b.e^{-x} + \epsilon$$ Do you think that I can guess any "structure" for the prior on $a$ and $b$ based on the fact that one weighs $x$ while the other weighs $e^x$ ?

1. Using the actual data to produce or select a "prior" is a contradiction in terms! However you can use the sampling distribution to simulate pseudo-data sets and check the impact of various priors on those datasets. For instance, looking at distances between prior and posteriors. For instance, you can use simulated data associated with a parameter $\theta$ to derive an approximate asymptotic variance $\hat{I}(\theta)$ for the associated MLE or MAP $\hat(\theta)$ and then use $|\hat{I}(\theta)|^{-1/2}$ as your substitute Jeffreys prior.
2. In a regression setting like the one presented here, Zellner's G-prior would handle the difference of scale for $X$ and $\exp(-X)$ rather naturally.