Monte Carlo Integration: Error Estimate Reliability for Non-Square Integrables

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Discussion Overview

The discussion revolves around the reliability of error estimates in Monte Carlo integration for functions that are not square integrable, specifically focusing on the function 1/sqrt(x) over the interval from 0 to 1. Participants explore the implications of non-square integrability on error estimates and the behavior of other functions like exp(-x^2).

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant defines a non-square integrable function as one for which the integral of its absolute value squared is not finite, citing 1/sqrt(x) as an example.
  • Another participant notes that the integral of 1/x is infinite, suggesting that this leads to an infinite variance in the Monte Carlo estimator, which affects its reliability.
  • A participant questions whether the same issues apply to the function exp(-x^2), reporting a NaN error estimate in this case.
  • Further inquiries are made about the domain of x, the probability distribution function, and the meaning of NaN.
  • A participant clarifies that NaN stands for "not a number," indicating an undefined value in computations.
  • Another participant mentions that if exp(-x^2) is used as a density function, it requires a constant to ensure it integrates to 1, and provides information about the variance of the standard normal distribution.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the behavior of different functions under Monte Carlo integration, particularly concerning the implications of non-square integrability and the specific case of exp(-x^2). There is no consensus on the reliability of error estimates for these functions.

Contextual Notes

Limitations include the lack of clarity on the specific conditions under which the Monte Carlo method fails for certain functions and the assumptions regarding the probability distribution functions being used.

trelek2
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Hi!

I need help with the monte carlo integration: reliability of the error estimate for functions that are not square integrable.

I'm supposed to investigate this topic.*Hence my first question is what is a function that is not square integrable? I found that such a function is 1/sqrt(x) on the interval 0 to 1. Apparently a function is not square integrable if the integral of its absolute value squared is not finite on that integral... I thought the for f(x)= 1/sqrt(x) that will be -1?
Anyway I evaluated the integrals for 1/sqrt(x) from 0 to 1 (which is 2 analytically) for dofferent number of sample points. Indeed the estimated errors are nowhere close the actual errors...

Can anyone explain why does this happen? And why is 1/sqrt(x) not square integrable?
 
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trelek2 said:
Hi!

I need help with the monte carlo integration: reliability of the error estimate for functions that are not square integrable.

I'm supposed to investigate this topic.*Hence my first question is what is a function that is not square integrable? I found that such a function is 1/sqrt(x) on the interval 0 to 1. Apparently a function is not square integrable if the integral of its absolute value squared is not finite on that integral... I thought the for f(x)= 1/sqrt(x) that will be -1?
Anyway I evaluated the integrals for 1/sqrt(x) from 0 to 1 (which is 2 analytically) for dofferent number of sample points. Indeed the estimated errors are nowhere close the actual errors...

Can anyone explain why does this happen? And why is 1/sqrt(x) not square integrable?

The integral of 1/x is infinite. Therefore the variance of your estimator is infinite. That is why Monte Carlo doesn't work.
 


Will that be also true for exp(-x^2)? I get a NaN as the error estimate oddly only in this case.
 


trelek2 said:
Will that be also true for exp(-x^2)? I get a NaN as the error estimate oddly only in this case.
Several questions:

1. What is the domain of x?
2. Exactly what is the prob. distribution function (or prob. density function)?
3. What is NaN (Sodium Nitride?)?
 


In computing, NaN stands for "not a number", which usually means an infinity or undefined value has popped up.
 


If exp(-x^2) is supposed to be your density function (although there will be a constant attached to it to make it integrate to 1), then the variance is known.
exp(-.5x^2)/ √(2π) is the density for the standard normal with mean 0 and variance 1.
 
Last edited:

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