Ej6lyu-lecture11 - Rare Event Sampling Importance Sampling...

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Rare Event Sampling Importance Sampling Statement of Problem Suppose that X is an ( E , d )-valued random variable with distribution π and that we need to calculate the expectation E [ f ( X )] = E f ( x ) π ( dx ) . Unfortunately, analytical and numerical evaluations of this integral may be unfeasible for any of the following reasons: E is high-dimensional; π is known explicitly, but concentrated on a set with complicated geometry; π is only known implicitly. Jay Taylor (ASU) APM 530 - Lecture 11 Fall 2010 1 / 42
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Rare Event Sampling Importance Sampling Monte Carlo Estimation In these cases it may still be possible to estimate the integral using a Monte Carlo method. The most basic estimate is given by: E [ f ( X )] 1 N N i =1 f ( X i ) I N [ f ] , where X 1 , · · · , X N are independent, identically-distributed random variables with the same distribution as X . By the SLLN, we know that I N [ f ] E [ f ( X )] a.s. whenever the expectation exists. For this method to work, we need to be able to generate IID samples of X . Jay Taylor (ASU) APM 530 - Lecture 11 Fall 2010 2 / 42
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Rare Event Sampling Importance Sampling Although the size of the approximation error is random, we can often use the Central Limit Theorem to estimate an upper bound. If σ 2 = Var( f ( X )) < , then the CLT implies that the di ff erence I N [ f ] E [ f ( X )] N 0; σ 2 N is asymptotically normal when N is large. If || f || = sup x | f ( x ) | < , then σ 2 || f || 2 and we can conservatively estimate the distribution of the error as N 0; || f || 2 N . Jay Taylor (ASU) APM 530 - Lecture 11 Fall 2010 3 / 42
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Rare Event Sampling Importance Sampling Importance Sampling Importance sampling is an important alternative to direct MC estimation. Suppose that ν is a probability distribution on E and that π is absolutely continuous w.r. to ν , i.e., π ( A ) = 0 whenever ν ( A ) = 0 . Under this condition, the Radon-Nikodym theorem asserts the existence of an integrable function h 0 such that for every measurable set F E π ( F ) = F h ( x ) ν ( dx ) = F d π d ν ( x ) ν ( dx ) . h = d π / d ν is the Radon-Nikodym derivative of π with respect to ν . Jay Taylor (ASU) APM 530 - Lecture 11 Fall 2010 4 / 42
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Rare Event Sampling Importance Sampling By writing the expectation as an integral w.r.t. ν , we obtain a new estimator for the expectation: E [ f ( X )] = E f ( x ) π ( dx ) = E f ( x ) h ( x ) ν ( dx ) 1 N N i =1 f ( Y i ) h ( Y i ) I N [ f ; ν ] where Y 1 , · · · , Y N are IID RV’s with distribution ν ; The quantities h ( Y i ) are called the importance sampling weights . I N [ f ; ν ] E [ f ( X )] by the SLLN. Jay Taylor (ASU) APM 530 - Lecture 11 Fall 2010 5 / 42
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Rare Event Sampling Importance Sampling We can still use the CLT to estimate the error of the IS estimate: I N [ f ; ν ] E [ f ( X )] N 0 , τ 2 N , where τ 2 = Var( f ( Y ) h ( Y )) In particular, if f > 0 and the function h ( y ) = E [ f ( x )] f ( y ) is in L 1 ( ν ), then τ 2 = Var f ( Y ) E [ f ( X )] f ( Y ) = Var E [ f ( X )] = 0 .
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