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We wish to compute the expected value of a random variable X = h(Y ), E[h(Y )], where Y is assumed to be a random variable with density f (with respect to the Lebesgue measure) in the d-dimensional real space Rd (In our examples, we will have d = 1) Then the crude Monte Carlo method estimates E[h(Y )] = h(y)f (y)dy by 1 n

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where Y1 , , Yn are independently and identically distributed copies of Y , and the integral is over Rd IS, on the other hand, samples Y from another density f rather than f Of 1 course, the same estimator n n h(Yi ) then becomes biased in general, but we i=1 can recover an unbiased estimator by weighting the simulation output as follows Assuming that f (y) > 0 whenever h(y)f (y) = 0, E[h(X)] = = h(y)f (y)dy = h(y) f (y) f (y)dy f (y)

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h(y)L(y)f (y)dy = E[h(Y )L(Y )],

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where L(y) = f (y)/f (y) is the likelihood ratio of the density f ( ) with respect to the density f ( ), and E[ ] is the expectation under density f An unbiased estimator of E[h(Y )] is then 1 n

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(21)

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where Y1 , , Yn are independently and identically distributed random variables sampled from f The case where Y has a discrete distribution can be handled analogously; it suf ces to replace the densities by probability functions and the integrals by sums That is, if P[Y = yk ] = pk for k N, then IS would sample n copies of Y , say Y1 , , Yn , using probabilities pk instead of pk , for k N, where pk > 0 whenever pk h(yk ) = 0 An unbiased IS estimator of E[h(Y )] is again (21), but with L(yk ) = pk /pk Indeed, E[h(Y )L(Y )] =

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In full generality, if Y obeys some probability law (or measure) P, and IS replaces P by another probability measure P, we must multiply the original estimator by the likelihood ratio (or Radon--Nikod m derivative) L = dP/dP y

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Clearly, the above procedure leaves us a huge amount of freedom: any alternative P yields an unbiased estimator (as long as the above-mentioned regularity conditions are ful lled) Therefore, the next question is: based on what principle should we choose the IS measure P The aim is to nd a change of measure for which the IS estimator has small variance, preferably much smaller than for the original estimator, and is also easy (and not much more costly) to compute (in that it should be easy to generate variates from the new probability law) We denote these two variances by 2 (h(Y )L(Y )) = E[(h(Y )L(Y ))2 ] (E[h(Y )])2 and 2 (h(Y )) = E[(h(Y ))2 ] (E[h(Y )])2 , respectively Under the assumptions that the IS estimator has a normal distribution (which is often a good approximation but not always), a con dence interval at level 1 for E[h(Y )] is given by 1 n

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h(Yi )L(Yi ) z /2

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where z /2 = 1 (1 /2) and is the standard normal distribution function For xed and n, the width of the con dence interval is proportional to the standard deviation (the square root of the variance) So reducing the variance by a factor K improves the accuracy by reducing the width of the con dence interval by a factor K The same effect is achieved if we multiply n by a factor K, but this requires (roughly) K times more work In the rare event context, one usually simulates until the relative accuracy of the estimator, de ned as the ratio of the con dence-interval half-width and the quantity to be estimated, is below a certain threshold For this, we need 2 (h(Y )L(Y ))/n approximately proportional to 2 Thus, the number of samples needed is proportional to the variance of the estimator In the case where is a small probability and h(Y ) is an indicator function, without IS, 2 (h(Y )L(Y )) = 2 (h(Y )) = (1 ) , so the required n is roughly inversely proportional to and often becomes excessively large when is very small The optimal change of measure is to select the new probability law P so that L(Y ) = dP E[|h(Y )|] , = |h(Y )| dP

which means f (y) = f (y)|h(y)|/E[|h(Y )|] in the continuous case, and pk = pk |h(yk )|/E[|h(Y )|] in the discrete case Indeed, for any alternative IS measure