Literature on design of importance sampling distribution using MLE or point-estimates of highest modes

Suppose I have many distributions $$p_i(\theta)$$ I wish to take expectations over $$\mathbb{E}_{p_i}[\mathbf{f}_i(\theta)]$$ where the $$\mathbf{f}_i$$ are vector-valued. In my problem the $$p_i$$ share some structure and this lets me find $$\theta^*_i = \textrm{argmax}_\theta\ p_i(\theta)$$ for all $$i$$ very efficiently. My $$p_i$$ are multimodal and sparse in that $$\log p_i(\theta_1) \gg \log p_i(\theta_0)$$ if $$\theta_1$$ is near a mode and $$\theta_0$$ isn't. Also I can find the locations of the top $$M$$ modes of the $$p_i$$ efficiently. I should also say $$\theta$$ lives in a discrete space with a spatial structure which in my case are indexes of pixels in an image.

As the $$\mathbf{f}_i$$ are vector-valued I can't try to choose importance distributions close to the optimal $$q_i(\theta) \propto |\mathbf{f}_i(\theta) - \mu| p_i(\theta)$$ I'm trying to choose $$q_i(\theta) \propto p_i(\theta)$$ with fatter tails.

My question is whether this is a well-studied problem and if so can you point me to any relevant literature? Are there any examples of the use of MLE estimates to design importance distributions I can cite?

I'm also wondering once I have identified a pixel in a mode for a $$p_i$$ should I try to find the edges of the mode through thresholding $$p_i$$ or is finding the curvature of $$p_i$$ at the mode peak and dropping a t distribution on it a more popular/better technique? Perhaps this depends very much on $$p_i$$.

• Yes, this is rather common. – Xi'an May 22 at 19:42
• Thanks for the comment! I will try to make it a little more concrete. – John May 23 at 8:37