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exponentially with the number of levels, which will result in computational problems A compromise has to be found In the next subsection, we investigate this issue in a simpli ed setting In xed-effort splitting, no explosion is possible, as a xed total number Nk of offspring are allocated at level k to the collection of successful trajectories that have managed to reach Bk Nonetheless, deciding how many offspring to create, as well as the number of successful trajectories in the case of a xed-performance implementation, are important issues Finally, given the importance function h, how many intermediate regions should be introduced and how should the increasing sequence of thresholds be de ned The next subsection investigates this point However, the precise optimal strategy depends on the implementation considered There is also the option to learn the levels, as is done in the xed-probability-of-success method of [8]
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Analysis in a simpli ed setting: a coin- ipping model
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Suppose we have already selected an importance function and one of the splitting implementations discussed in the previous section For a given total computation budget, we would like to nd the number and the locations of the thresholds, or equivalently the numbers n, p0 , , pn , that minimize the variance of the estimator We are also interested in convergence results for the variance and the work-normalized variance, under various asymptotic regimes, such as when N while n and p0 , , pn are xed, or when 0 and n Here we study these questions and provide partial answers under a very simpli ed (but tractable) model, for the xed-effort and xed-splitting strategies The main focus is on the asymptotic behavior when N Our simpli ed setting is a coin- ipping model uniquely characterized by the initial probability p0 = P(A0 ) (ie, the occurrence of the event A0 depends only on the outcome of a {0, 1} Bernoulli trial with parameter p0 ), and by the transition probabilities pk = P(Ak | Ak 1 ) (ie, the occurrence of Ak , conditional on Ak 1 , depends only on the outcome of a {0, 1} Bernoulli trial with parameter pk ), for k = 1, , n This model is equivalent to assuming that there is only a single entrance state at each level For the work-normalized analysis, we need to make some assumptions on how much work it takes, on average, to run a trajectory from a given level k 1 until it reaches either the next level or the set A = \ B (ie, the stopping time T without reaching B) If there is a natural drift toward A, it appears reasonable to assume that the chains will reach A in O(1) expected time, independently of n, if A and B (and therefore ) are xed If we use truncation and/or Russian roulette, we still have O(1) expected time Then the total expected work for all stages is proportional to n E[Nk ] This is the assumption we will make k=0 everywhere in this section, unless stated otherwise If E[Nk ] = N for all k, then this sum is N (n + 1) For simplicity, we will further assume that the constant of proportionality (in the O(1) expected time mentioned above) is 1
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In a different asymptotic regime, where 0 and n jointly, and if truncation and/or Russian roulette are not applied, the average time to reach A should increase when 0, typically as O( ln ), in which case the total work will be proportional to ( ln )(n + 1) n Nk If we further assume that k=0 p0 , , pn are all equal to a xed constant p, then = p (n+1) , so ln = (n + 1) ln p and the total work is proportional to ( ln p)(n + 1)2 n Nk As k=0 it turns out, the extra linear factor (n + 1) has a negligible role in the asymptotic behavior [18, 20, 26]
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