Rare-Event SimulationSplitting for Variance Reduction

IE 680, Spring 2007

Bryan Pearce

What is a Rare Event?

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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 l0, l1, …, lm

h(x) = l0 = l1 = l2 = l3 = lm = l

More formally:

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How to choose h?

• Defining the importance function can be difficult.

• Ideally our h should reflect:– The most likely path to the rare event

– pk(x) = pk (indep. of state)

– pk = p (indep. of level)

• Presumes apriori knowledge of the system.

First sub-interval

time

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MC Sim N0 independent chains. R0 reach l1.

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Second sub-interval: Splitting

time

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MC Sim N1 chains, splitting from the previously achieved threshold states.

R1 reach l2.

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…and so on for each sub-interval

Notation

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Splitting policy – fixed splitting

• Each chain that reaches level k is cloned ck times.

• Nk will be random for each level k > 0

• Stratified sampling from the entrance distribution of level k

Splitting policy – fixed effort

• Fix Nk in advance. Choose the states represented in the entrance distribution by:

Random assignment– Choose these Nk states randomly from the entrance

distribution

Fixed assignment– Choose an equal quantity of each state– Better stratification

Pros & cons of splitting method

• Fixed splitting – – Asymptotically more efficient under optimal

conditions

– Efficiency very sensitive to splitting factor ck

• Fixed effort– Higher memory requirement– More robust

Efficiency

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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.

Truncation - Motivation

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l2

l3

l4

Simulation time spent reaching l1

Simple (biased) Truncation

Choose β:

• If a chain falls below the level lk-β then terminate.

• Estimator becomes biased, moreso with small β.

• Large β does not reduce workload very much.

• RESTART

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Terminate

} β = 2

Unbiased Truncation

Use the ‘Russian Roulette’ principle:

The first time a chain ‘down-crosses’ a level threshold it dies with probability (1 – 1/rk,j). If it survives then its weight is increased by a factor of rk,j.

(these rk,j are user-defined and determine the ‘strength’ of the truncation)

How to choose the rk,js

• The selection of the rk,js at each level of the process will control the aggressiveness of the truncation policy.

• A tried-and-true value:

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,

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l2

l3

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Dies with prob. (1 – 1/r3,2)

Weight increases by a factor of r3,2 if the chain survives.

Russian Roulette, cont.

• There are various methods by which to use the chain weights can compensate for this truncation bias.

– Probabilistic

– Tag-based

– Periodic

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.

Conclusions and notes

• Potential performance– With γ = 10-20,

Var[MC] = 10-23 while Var[split] = 10-41

• Poorly-behaved systems– Inefficient to apply

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.