I aim to achieve variance reduction in Random Walk Metropolis Hastings algorithm by introducing stratification to the random walk jumps. What I have tried is to make use of Latin Hypercube Sampling in each iteration.
lhs_1 <- randomLHS(1, 5) ym_1 <- qnorm(lhs_1, sd = sqrt(5))
In above code I predefine the Random Walk jumps by using Latin Hypercube sampler. I have five groups in stratification. Values in ym_1 can serve as different proposals. I can try each of these proposals and pick the one with the highest density. This naturally increases the acceptance ratio but chain explores a limited region.
I am not sure about how to make use of these proposals. More generally, I am not sure about how to correctly achieve variance reduction here by using stratification in Random Walk Jumps. Any idea would be greatly appreciated.