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Let be $f(x)$ a function which can be expressed as a linear combination of other functions: $$f(x)=f_1(x)+f_2(x)+...+f_n(x)$$ would it be possible to apply Monte Carlo integration to each one of the factors and mix them somehow in order to get the right result of $f(x)$? If it's so, how?

Example x 1-dimension: Let's say you want to integrate $f(x)$ between on the interval $[x0,x1]$ using a uniform Monte Carlo estimator: $$f(x)=f_1(x)+f_2(x)$$ $$f_1(x)=ax^2\\f_2(x)=bx+c$$ $$a,b,c\ \ \text{constants}$$

How would you proceed?

Bernard
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BPL
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1 Answers1

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A key thing underlying many MC methods is that expected values of sums are the sum of the expected values:

E(X + Y) = E(X) + E(Y)

WHETHER OR NOT X and Y are INDEPENDENT!

I personally find that kind of amazing. What this means is you can do pretty much anything (including re-using samples between sum components) are you have a unbiased estimate.

So for your example estimate f1 and f2 separately and then just add the estimates.

Peter Shirley
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