I'm trying to understand intuitively how Carlo Estimator was derived/invented. Is it basically derived from taking the expectation of an expression and letting it be the definite integral of f(x)(in this case the expression happens to be the Monte Carlo Estimator formula)?

jierui-cell

I think Monte Carlo Estimator is derived from the idea that we uniformly take points from a function (curve) to estimate its integral. It is just the simplest way we could think of to estimate an integral using randomness. From my understanding it just happens to be an unbiased estimator, which is great from a statistical standpoint. I believe the original reasoning should come from the opposite around instead of intentionally design it to be unbiased. It is not always preferable to take an unbiased estimator compared to a biased estimator. I found this post to explain well on the tradeoffs: https://stats.stackexchange.com/questions/207760/when-is-a-biased-estimator-preferable-to-unbiased-one.

I'm trying to understand intuitively how Carlo Estimator was derived/invented. Is it basically derived from taking the expectation of an expression and letting it be the definite integral of f(x)(in this case the expression happens to be the Monte Carlo Estimator formula)?

I think Monte Carlo Estimator is derived from the idea that we uniformly take points from a function (curve) to estimate its integral. It is just the simplest way we could think of to estimate an integral using randomness. From my understanding it just happens to be an unbiased estimator, which is great from a statistical standpoint. I believe the original reasoning should come from the opposite around instead of intentionally design it to be unbiased. It is not always preferable to take an unbiased estimator compared to a biased estimator. I found this post to explain well on the tradeoffs: https://stats.stackexchange.com/questions/207760/when-is-a-biased-estimator-preferable-to-unbiased-one.