Choosing Control Variates for Monte Carlo Integration

Summary

The document asks how to improve a Monte Carlo estimate of an integral over the unit interval using a control variate. It describes estimating the integral by drawing uniform random values, evaluating the integrand, and averaging the results. The integrand has a singular factor near zero, which makes the choice of a useful comparison variable a relevant question.

The author understands that a control variate must have a known expected value and asks whether it should be a simplified version of the original integrand. The document gives no proposed control variate, derivation, simulation, or measured variance reduction, so it does not establish which choice works for this integral. The general criterion is that the control should have a known expectation and be strongly correlated with the quantity being estimated; its usefulness should be assessed by the resulting variance reduction. The text is best read as a question about applying that principle, rather than as a worked method or evidence-backed result.

Key ideas

• Monte Carlo integration on the unit interval can be estimated by averaging integrand values at uniform draws.
• A control variate uses a related random variable whose expectation is known.
• A simplified expression is not automatically a good control variate; correlation with the target matters.
• The document poses the selection problem but does not demonstrate a solution or variance reduction.

Tags

# Optimizing Monte Carl integral calculation with control variate


# Optimizing Monte Carl integral calculation with control variate












For an exercise I am asked to calculate an integral with a monte carlo simulation, after that I need to optimize the results with a control variate. This was the given integral:

$\int_0^1 \! \frac{\sin(1-x)}{\sqrt[3]{x}} \, \mathrm{d}x.$

As far as I understand, this isn't that hard. Especially since this is an integral with domain [0,1]. I basically only need to generate a U(0,1) variable, insert this in the follwing formula and in the end take the average result as my solution.

$\frac{\sin(1-x)}{\sqrt[3]{x}}$

However, I am now asked to introduce a control variate to optimize my result. In my understanding, a control variate is another stochastic variable of which the expected value is known. I don't know however what the criteria are when you are searching for a control variate.

For example: in the example of control variates of wikipedia they chose $g(x)=1+x$ as a control variate for $f(x)=\frac{1}{1+x}$. Is this a good control variate because it is a simplified version of the original integral?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.