I got stock somewhere in my work for about two weeks more or less and I could not find a reasonable answer for my problem. I would gratefully appreciate it if you could help me out in this regard.
I am using HMC method for sampling from a certain domain in parameter space (let’s assume 2D parameter space for simplicity). I want to have more sample from that domain and then using importance sampling to obtain the interested statistical inference. To this end, I am considering the bayesian approach for sampling. I am working in U space (bivariate standard normal prior) and have my likelihood model.
I found that by reducing the sigma of my likelihood model gradually, I can have more sample points on my interested region in parameter space and it helps me a lot!. But at the end after doing importance sampling, my interested inference does not converge to the correct value. If I use the constant sigma for the likelihood, it eventually converge to the correct inference.
I am wondering if there is any remedy for the first case in terms of correction factor or so on.
Any help would be greatly appreciated.