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Variance Reduction Techniques in Monte Carlo Integration Methods

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This review focuses on variance reduction methods used in Monte Carlo integration, an essential technique in numerical analysis. Key approaches include stratified sampling, importance sampling, control variates, and antithetic variates. Each method aims to enhance the efficiency of integration by providing unbiased estimators with reduced variance compared to crude Monte Carlo. Through comparative analysis and theoretical insights, the course illustrates how proper stratification and sampling functions can significantly improve integration results. This summary serves as a guide for students tackling assignments related to these concepts.

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Variance Reduction Techniques in Monte Carlo Integration Methods

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  1. Variance ReductionFall 2012 By Yaohang Li, Ph.D.

  2. Review • Last Class • Numerical Integration • Monte Carlo Integration • Crude Monte Carlo • Hit-or-Miss Monte Carlo • Comparison • General Principle of Monte Carlo • This Class • Variance Reduction Methods • Assignment #2 • Next Class • Random Numbers

  3. Variance Reduction Methods • Variance Reduction Techniques • Employs an alternative estimator • Unbiased • More deterministic • Yields a smaller variance • Methods • Stratified Sampling • Importance Sampling • Control Variates • Antithetic Variates

  4. Stratified Sampling • Idea • Break the range of integration into several pieces • Apply crude Monte Carlo sampling to each piece separately • Analysis of Stratified Sampling • Estimator • Variance • Conclusion • If the stratification is well carried out, the variance of stratified sampling will be smaller than crude Monte Carlo

  5. Importance Sampling • Idea • Concentrate the distribution of the sample points in the parts of the interval that are of most importance instead of spreading them out evenly • Importance Sampling • where g and G satisfying • G(x) is a distribution function

  6. Importance Sampling • Variance • How to select a good sampling function? • How about g=cf? • g must be simple enough for us to know its integral theoretically.

  7. Control Variates • Control Variates • (x) is the control variate with known integral • Estimator • t-t’+’ is the unbiased estimator • ’ is the first integral • Variance • var(t-t’+’)=var(t)+var(t’)-2cov(t,t’) • if 2cov(t,t’)<var(t’), then the variance is smaller than crude Monte Carlo • t and t’ should have strong positive correlation

  8. Antithetic Variates • Main idea • Select a second estimate that has a strong negative correlation with the original estimator • t’’ has the same expectation of t • Estimator • [t+t’’]/2 is an unbiased estimator of  • var([t+t’’]/2)=var(t)/4+var(t’’)/4+cov(t,t’’)/2 • Commonly used antithetic variate • (t+t’’)/2=f()/2+ f(1-)/2 • If f is a monotone function, f() and f(1-) are negatively correlated

  9. Summary • Analysis of Monte Carlo Integration • Curse of Dimensionality • Error Analysis of Monte Carlo Integration • Variance Reduction Methods • Stratified Sampling • Importance Sampling • Control Variates • Antithetic Variates

  10. What I want you to do? • Review Slides • Work on your Assignment 1 if you haven’t finished • Work on your Assignment 2

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