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[15] P Heidelberger Fast simulation of rare events in queueing and reliability models ACM Transactions on Modeling and Computer Simulation, 5(1): 43 85, 1995 [16] S Juneja and P Shahabuddin Rare event simulation techniques: An introduction and recent advances In S G Henderson and B L Nelson, eds, Simulation, Handbooks in Operations Research and Management Science, pp 291 350 Elsevier, Amsterdam, 2006 [17] C Kollman, K Baggerly, D Cox, and R Picard Adaptive importance sampling on discrete Markov chains Annals of Applied Probability, 9(2): 391 412, 1999 [18] I Kuruganti and S Strickland Importance sampling for Markov chains: computing variance and determining optimal measures In Proceedings of the 1996 Winter Simulation Conference, pp 273 280 IEEE Press, 1996 [19] P L Ecuyer and Y Champoux Estimating small cell-loss ratios in ATM switches via importance sampling ACM Transactions on Modeling and Computer Simulation, 11(1): 76 105, 2001 [20] P L Ecuyer and B Tuf n Effective approximation of zero-variance simulation in a reliability setting In Proceedings of the 2007 European Simulation and Modeling Conference, pp 48 54, Ghent, Belgium, 2007 EUROSIS [21] A Ridder Importance sampling simulations of Markovian reliability systems using cross-entropy Annals of Operations Research, 134: 119 136, 2005 [22] R Rubinstein and D P Kroese A Uni ed Approach to Combinatorial Optimization, Monte Carlo Simulation, and Machine Learning Springer, Berlin, 2004 [23] R Y Rubinstein Optimization of computer simulation models with rare events European Journal of Operations Research, 99: 89 112, 1997 [24] J S Sadowsky On the optimality and stability of exponential twisting in Monte Carlo estimation IEEE Transactions on Information Theory, IT-39: 119 128, 1993 [25] D Siegmund Importance sampling in the Monte Carlo study of sequential tests Annals of Statistics, 4: 673 684, 1976 [26] R Srinivasan Importance Sampling Applications in Communications and Detection Springer, Berlin, 2002
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As already explained in previous chapters, rare event simulation requires acceleration techniques to speed up the occurrence of the rare events under consideration, otherwise it may take unacceptably large sample sizes to get enough positive realizations, or even a single one, on average On the other hand, accelerating too much can be counterproductive and even lead to a variance explosion and/or an increase in the computation time Therefore, an appropriate balance must be achieved, and this is not always easy This dif culty was highlighted in the previous chapter when discussing the importance sampling (IS) technique, the idea of which is to change the probability laws driving the model in order to make the events of interest more likely, and to correct the bias by multiplying the estimator by the appropriate likelihood ratio In this chapter, we review an alternative technique called splitting, which accelerates the rate of occurrence of the rare events of interest Here, we do not change the probability laws driving the model Instead, we use a selection mechanism to favor the trajectories deemed likely to lead to those rare events The main idea is to decompose the paths to the rare events of interest into shorter subpaths whose probability is not so small, encourage the realizations that take these subpaths (leading to the events of interest) by giving them a chance to reproduce (a bit like in selective evolution), and discourage the realizations that
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Rare Event Simulation using Monte Carlo Methods Edited by Gerardo Rubino and Bruno Tuffin 2009 John Wiley & Sons, Ltd ISBN: 978-0-470-77269-0
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go in the wrong direction by killing them with some positive probability The subpaths are usually delimited by levels, much like the level curves on a map Starting from a given level, the realizations of the process (which we also call trajectories or chains or particles) that do not reach the next level will not reach the rare event, but those that do are split (cloned) into multiple copies when they reach the next level, and each copy pursues its evolution from then on This creates an arti cial drift toward the rare event by favoring the trajectories that go in the right direction In the end, an unbiased estimator can be recovered by multiplying the contribution of each trajectory by the appropriate weight The procedure just described is known as multilevel splitting If we assume, for instance, that we are simulating a stochastic process (usually a Markov chain) and that the rare event of interest occurs when we reach a given subset of states before coming back to the initial state, then the levels can be de ned by a decreasing (embedded) sequence of state sets that all contain the rare set of interest In general, these levels are de ned via an importance function whose aim is to represent how close a state is from this rare set Several strategies have been designed to determine the levels, to decide the number of splits at each level, and to handle the trajectories that tend to go in the wrong direction (away from the rare event of interest) The amount of splitting when reaching a new level is an important issue; with too much splitting, the population of chains will explode, while with too little splitting, too few trajectories are likely to reach the rare event There is also the possibility of doing away with the levels, by following a strategy that can either split the trajectory or kill it at any given step One applies splitting (sometimes with some probability) if the weighted importance function is signi cantly larger at the current (new) state than at the previous state, and we apply Russian roulette (we kill the chain with some probability), when the weighted importance function becomes smaller Russian roulette can also be viewed as splitting the chain into zero copies The expected number of clones after the split (which is less than 1 in the case of Russian roulette) is usually taken as the ratio of the importance function value at the new state to that at the old state [13, 22] The most important dif culty in general is to nd an appropriate importance function This function de nes the levels (or the amount of splitting if we get rid of levels), and a poor choice can easily lead to bad results In this sense, its role is analogous to the importance measure whose choice is critical in IS (see the previous chapter) One important advantage of splitting compared with IS is that there is no need to modify the probability laws that drive the system This means (among other things) that the computer program that implements the simulation model can just be a black box, as long as it is possible to make copies (clones) of the model, and to maintain weights and obtain the current value of the importance function for each of those copies It is also interesting to observe that for splitting implementations where all chains always have the same weight at any given level, the empirical distribution of the states of the chains when they hit a given
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