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I'm doing a slightly involved Monte Carlo approximation of a quantity $E$ where I end up with the following formula:

$E=\frac{\sum_{i=1}^np_ie_iG_i}{\sum_{i=1}^np_iG_i}\ .\ \ \ \ \ \ \ \ \ $ (1)

The quantities $e_i$ and $p_i$ can be calculated directly from the MC configurations but the quantity $G_i$ is computed with another, secondary, Monte Carlo - and so comes with an error $\delta_i$. Taking the denominator and numerator of (1) separately I tend to think of it like this: We have a bunch of measurements of the quantities $G_i$ which we use together with the parameters $e_i$ and $p_i$ to find the averages as estimates of the true values (although this shouldn't be taken too far, there is the statistical error associated with approximating the MC integral with a sum averaged over MC samples). How to estimate the error on the value $E$?

Normally the error estimate for $E$ in a MC is just the standard deviation of the mean $\sigma_M=\sigma/\sqrt{n-1}$ (possibly computed using bootstrap for a robust estimate). However in this case this does not give the right value as we'll see below.

A few possibilities for the error estimate come to mind, but so far all of them have some problem. I'll describe them in turn:

I) The standard deviation of the mean, as suggested above. The problem with this can be described with the following example: Say I compute (1) using 1000 MC samples, i.e. $n=1000$. With a bootstrap I calculate $\sigma_M=\sqrt{\overline{E^2}-\overline{E}^2}/\sqrt{n-1}$ as the error of the mean. I then do this whole calculation 100 times and so get 100 values $\{E_k\}$, and calculate the actual standard deviation from this - and get a different (much higher) value than $\sigma_M$. I think the reason could be related to systematic error on $E$ depending on the number of samples in the secondary MC computing $G_i$ (in other words $\delta_i$ inducing a systematic error in $E$).

II) Error propagation of the $\delta_i$'s. I could use the formula $\delta_E=\sqrt{\sum_{i=1}^n(\frac{\partial E}{\partial G_i}\delta_i)^2}$ (either with the numerator and denominator separately or the whole of $E$ as above). However this would give zero error $\delta_E$ if $\delta_i=0$, when this situation should correspond to a normal "single" MC where the error is given by the standard deviation, and so cannot be right.

III) Thinking of $G_i$ as measurements. This I alluded to earlier, where as an analogy I think of each $G_i$ as measured with errors $\delta_i$ and then want to find the total error on $E$. It can't just be the standard deviation of E, because imagine having $n$ a very low number, e.g. $n=3$ - it could then happen that the three values are so close as to give a better error than the individual $\delta_i$'s should permit. I assume there is some textbook answer to this problem but I cannot find it.

Does anyone have a suggestion for the error?

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