# Enhancing Monte-Carlo convergence (crude method)

I am currently doing a project involving Monte-Carlo method. I wonder if there is papers dealing with a "learning" refinement method to enhance the MC-convergence, example :

Objective : estimate of $E(X)\thickapprox \sum _{i=1}^{10 000}X_i$

-> step 1 (500 simulations): $approx_1=\sum _{i=1}^{500}X_i$

(i) Defining and 'Acceptance interval'

$A_1 = [approx_1-\epsilon_1,approx_1+\epsilon_1]$

where $\epsilon_1$ could be a function of the empirical variance and other statistics

-> step 2 (500 other simulations): "throwing" all simulation out of the interval $A_1$ , $approx_2=\sum _{i=1}^{500}X_i^{(2)}$

New 'acceptance interval'

$A_2 = [approx_2-\epsilon_2,approx_2+\epsilon_2]$

where $\epsilon_2 < \epsilon_1$

...

$\epsilon \xrightarrow {} 0$

• I am not sure that you want to discard observation $X_i$ because it produces an estimate that falls outside your target interval. You raise the number of iterations $n$ to lower the standard error of your estimate. I don't think there's a way around this. (Although if you wanted a better idea of what the distribution of estimates looks like in a certain neighborhood, then you could provide $X_i$ from a specific portion of the distribution to provide estimates in the correct neighborhood.) Mar 8, 2011 at 1:17

Having shifted your samples, you need to then track their likelihood ratio (or Radon-Nikodym derivative) versus the original distribution, because your samples now need to be weighted by that ratio in your Monte Carlo sum. In the case of a shift in normal distributions, this is fairly easy to compute for each sample $\vec{x}$, as
$$\frac{1}{\sqrt{2\pi\ \text{det}A^{-1}}}\exp{\left[-\frac12 (x-b)A(x-b)^t\right]}$$
where $A$ and $b$ are the covariance matrix and mean of your change to the original multivariate normal.