Choosing non-informative priors

Question

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 $e^x$ ?

Answer 1

Jeffreys priors are indeed unmanageable outside standard families and not even necessarily recommended in high dimensions. If the model is complex enough, priors should take advantage of the hierarchical structures underlying this model...

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.