1. Rare-Event Simulation Splitting for Variance Reduction IE 680, Spring 2007 Bryan Pearce

2. What is a Rare Event? Ω A B

3. Formal Problem Definition

4. Splitting: the beginning Importance function h –Measures “how close” a state is to the rare event Divide the intermediary state space into m ‘levels’ according to the thresholds l 0, l 1, …, l m

5. h(x) = l 0 = l 1 = l 2 = l 3 = l m = l

6. More formally:

7. How to choose h? Defining the importance function can be difficult. Ideally our h should reflect: –The most likely path to the rare event –p k (x) = p k (indep. of state) –p k = p (indep. of level) Presumes apriori knowledge of the system.

8. First sub-interval time h 0 l1l1 MC Sim N 0 independent chains. R 0 reach l 1.

9. Second sub-interval: Splitting time h 0 l1l1 MC Sim N 1 chains, splitting from the previously achieved threshold states. R 1 reach l 2. l2l2 …and so on for each sub-interval

10. Notation

11. Splitting policy – fixed splitting Each chain that reaches level k is cloned c k times. N k will be random for each level k > 0 Stratified sampling from the entrance distribution of level k

12. Splitting policy – fixed effort Fix N k in advance. Choose the states represented in the entrance distribution by: Random assignment –Choose these N k states randomly from the entrance distribution Fixed assignment –Choose an equal quantity of each state –Better stratification

13. Pros & cons of splitting method Fixed splitting – –Asymptotically more efficient under optimal conditions –Efficiency very sensitive to splitting factor c k Fixed effort –Higher memory requirement –More robust

14. Efficiency Our hope is that splitting will allow our variance to shrink faster than our computational time grows. This has indeed been shown to be true in many cases.

15. Truncation - Motivation h 0 l1l1 l2l2 l3l3 l4l4 Simulation time spent reaching l 1

16. Simple (biased) Truncation Choose β: If a chain falls below the level l k-β then terminate. Estimator becomes biased, moreso with small β. Large β does not reduce workload very much. RESTART

17. h 0 l1l1 l2l2 l3l3 l4l4 Terminate } β = 2

18. Unbiased Truncation Use the ‘Russian Roulette’ principle: The first time a chain ‘down-crosses’ a level threshold it dies with probability (1 – 1/r k,j ). If it survives then its weight is increased by a factor of r k,j. (these r k,j are user-defined and determine the ‘strength’ of the truncation)

19. How to choose the r k,j s The selection of the r k,j s at each level of the process will control the aggressiveness of the truncation policy. A tried-and-true value:

20. h 0 l1l1 l2l2 l3l3 l4l4 Dies with prob. (1 – 1/r 3,2 ) Weight increases by a factor of r 3,2 if the chain survives.

21. Russian Roulette, cont. There are various methods by which to use the chain weights can compensate for this truncation bias. –Probabilistic –Tag-based –Periodic

22. Truncation w/o weights Chain weighting truncation methods can inflate the variance of our gamma estimator. We can avoid this problem by allowing our chains to probabilistically re-split upon re- achieving previously achieved goals.

23. Conclusions and notes Potential performance –With γ = 10 -20, Var[MC] = 10 -23 while Var[split] = 10 -41 Poorly-behaved systems –Inefficient to apply

24. References L’Ecuyer, P., V. Demers, B. Tuffin. 2006. Splitting for rare- event simulation. Glasserman, P., P. Heidelberger,and T. Zajic. 1998. A large deviations perspective on the efficiency of multilevel splitting. L’Ecuyer, P., V. Demers, B. Tuffin. 2006. Rare-events, splitting, and quasi-Monte Carlo. Garvels, M. J. J. 2000. The splitting method in rare event simulation.