Question about discrete Monte Carlo Summation

[mgamito]

Hello all,

I'm aware of the Monte Carlo Summation method in discrete spaces, where you can approximate a very long summation over the entire space by a shorter one with only a few randomly selected terms from the original summation (weighted by the inverse probability density of them being chosen).

My question is: should the random selection of summation terms include replacement or not? That is, once one term is selected can it go back into the pool to be selected again? Or, stated yet another way: can one term from the original summation be selected more than once?

If both techniques are possible (with or without replacement), are there any known advantages or disadvantages to each one?

Thank you all,
manuel

 

[mighty2000]

Hello manuel,

Great question! The decision to use replacement or not in the Monte Carlo Summation method depends on the specific problem you are trying to solve. Let me explain the advantages and disadvantages of each technique:

1. With replacement: This means that a term can be selected more than once from the original summation. The advantage of this technique is that it allows for a larger number of terms to be sampled, which can lead to a more accurate approximation of the original summation. However, the disadvantage is that this can lead to higher computational costs, as the same terms may be selected multiple times.

2. Without replacement: This means that each term can only be selected once from the original summation. The advantage of this technique is that it can reduce computational costs, as each term is only selected once. However, the disadvantage is that it may not accurately represent the original summation, especially if the number of terms sampled is small.

In general, the decision to use replacement or not depends on the trade-off between accuracy and computational costs. If accuracy is the top priority, then using replacement may be the better option. However, if reducing computational costs is more important, then using without replacement may be a better choice.

I hope this helps! Let me know if you have any further questions.