I had asked some similar questions in the past, but I never got either the answers or the discussion I was hopping for. So I will rephrase the problem to see if I can understand it myself.
I'm trying to fit a complex model to some data that take a large amount of time to run. I'm also unable to write down a Likelihood function to this problem and so I turned to approximate Bayesian computation (ABC).Now, given the slowness of my simulations, I used Sequential ABC (a strategy where the prior are updated at each iteration), as implemented in R EasyABC.
I had realised that the result posterior distribution is highly variable (Sequential ABC get stuck in "local minimum"). This lead me to tun ABC_sequential more than once, and from different prior. In the end I gather all simulation that were done and now I want to analysed this.
I also need to state two other facts:
I have very little knowledge to build the priors. In fact, contrary to the concept of Bayesian statistic (new knowledge updating old knowledge) I would like to remove all the influence of the priors from my estimates.
The resulting distributions had several peaks (given from the different runs of ABC_sequential).
I designed also my custom distance function and computed a simple rejection algorithm of the best 5% simulations. However, as is probably obvious by now, this "posterior distribution" is highly influenced by the non-uniform original distributions. How can I remove this influence?
A common suggestion is importance sampling. As far as I understand, this implies computing the ratio between the density in my posterior and my prior and use this as a weight for my simulations. I did this with R package densratio and the results are unsatisfactory... Do you have any suggestions, corrections, etc?