Solved – Monte Carlo integration error

monte carlosimulation

I have a question regarding the Monte Carlo integration method. This is probably a "noob" question, but I have searched the internet and haven't been able to find an answer…

Let's say I want to estimate an integral of some function over an area $D$, that I'll denote as $I$. I can choose $N$ points to randomly sample from $D$ in order to obtain the integral, the error should "decay" as $err\sim \frac{\sigma}{\sqrt{N}}$, so that $I\sim \overline{I}\pm\frac{\sigma}{\sqrt{N}}$. But what if I just ran the simulation $N$ times and would get different values like: $I_1,I_2,….I_N$ those should be distributed normally as well, right? So I can estimate the error from those values again as $I\sim\overline{I} \pm \frac{\sigma}{\sqrt{N}}$, would this give the same results or is my thinking wrong?

Best Answer

As rightly stressed in the comments of J Bowman, there is a fair amount of confusion in this question. Considering

  1. $N$ simulations from a uniform distribution on the set $\mathfrak{D}$, $x_1,\ldots,x_N$, the estimator$$I_N=\frac{1}{N}\sum_{i=1}^N h(x_i)$$is an unbiased estimator of$$\mathfrak{I}=\int_\mathfrak{D} h(x)\text{d}x$$ that a.s. converges at the speed $\sqrt{N}$ in the sense of the Central Limit Theorem:$$\frac{1}{\sqrt{N}}\sum_{i=1}^N \{h(x_i)-\mathfrak{I}\}\stackrel{\mathcal{L}}{\longrightarrow}\mathcal{N}(0,\sigma^2)$$ which variance $\sigma^2$ can be estimated by the empirical variance of the $h(x_i)$'s. This means that $I_N$ is approximately Gaussian for large $N$'s, with variance $\sigma^2/N$
  2. When considering an iid sequence $I_N^1,\ldots,I^N_N$ of such estimators, they all are unbiased estimators of $\mathfrak{I}$ with the same variance $\sigma^2/N$ that can be estimated by the empirical variance of the $I_N^i$'s, and again satisfy a CLT: $$\frac{\sqrt{N}}{\sqrt{N}}\sum_{i=1}^N \{I^i_N-\mathfrak{I}\}\stackrel{\mathcal{L}}{\longrightarrow}\mathcal{N}(0,\sigma^2)$$ This means that $I_N^1,\ldots,I^N_N$ is approximately a Gaussian sample for large $N$'s, again with variance $\sigma^2/N$
  3. Letting $N$ grow to infinity in $I_N^1,\ldots,I^N_N$ does not serve a clear purpose, as opposed to computing a single $I_{N^2}$ based on the same number of simulations, $N^2$, which has a variance of $\sigma^2/N$
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