Introduction to Rare Event Simulation
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1 Introduction to Rare Event Simulation Brown University: Summer School on Rare Event Simulation Jose Blanchet Columbia University. Department of Statistics, Department of IEOR. Blanchet (Columbia) 1 / 31
2 Goals / questions to address in this lecture The need for rare event simulation by introducing several problems. Blanchet (Columbia) 2 / 31
3 Goals / questions to address in this lecture The need for rare event simulation by introducing several problems. Why do we care about rare event simulation? Blanchet (Columbia) 2 / 31
4 Goals / questions to address in this lecture The need for rare event simulation by introducing several problems. Why do we care about rare event simulation? How does one measure the effi ciency of rare event simulation algorithms? Blanchet (Columbia) 2 / 31
5 Goals / questions to address in this lecture The need for rare event simulation by introducing several problems. Why do we care about rare event simulation? How does one measure the effi ciency of rare event simulation algorithms? Describe basic rare event simulation techniques. Blanchet (Columbia) 2 / 31
6 Why are Rare Events Diffi cult to Assess? Typically no closed forms But crudely implemented simulation might not be good Blanchet (Columbia) 3 / 31
7 Why are Rare Events Diffi cult to Assess? Blanchet (Columbia) 4 / 31
8 Why are Rare Events Diffi cult to Assess? Relative mean squared error (MSE) PER TRIAL = stdev / mean P (RED) (1 P (RED)) P (RED) 1 P (RED). Blanchet (Columbia) 5 / 31
9 Why are Rare Events Diffi cult to Assess? Relative mean squared error (MSE) PER TRIAL = stdev / mean P (RED) (1 P (RED)) P (RED) 1 P (RED). Moral of Story: Naive Monte Carlo needs ( ) 1 N = O P (Rare Event) replications to obtain a given relative accuracy. Blanchet (Columbia) 5 / 31
10 Introduction to Rare Event Simulation Rare event simulation = techniques for effi cient estimation of rare-event probabilities AND conditional expectations given such rare events. Blanchet (Columbia) 6 / 31
11 Introduction to Rare Event Simulation Rare event simulation = techniques for effi cient estimation of rare-event probabilities AND conditional expectations given such rare events. Arises in settings such as: Blanchet (Columbia) 6 / 31
12 Introduction to Rare Event Simulation Rare event simulation = techniques for effi cient estimation of rare-event probabilities AND conditional expectations given such rare events. Arises in settings such as: 1 Finance and Insurance, Blanchet (Columbia) 6 / 31
13 Introduction to Rare Event Simulation Rare event simulation = techniques for effi cient estimation of rare-event probabilities AND conditional expectations given such rare events. Arises in settings such as: 1 Finance and Insurance, 2 Queueing networks (operations and communications), Blanchet (Columbia) 6 / 31
14 Introduction to Rare Event Simulation Rare event simulation = techniques for effi cient estimation of rare-event probabilities AND conditional expectations given such rare events. Arises in settings such as: 1 Finance and Insurance, 2 Queueing networks (operations and communications), 3 Environmental and physical sciences, Blanchet (Columbia) 6 / 31
15 Introduction to Rare Event Simulation Rare event simulation = techniques for effi cient estimation of rare-event probabilities AND conditional expectations given such rare events. Arises in settings such as: 1 Finance and Insurance, 2 Queueing networks (operations and communications), 3 Environmental and physical sciences, 4 Statistical inference and combinatorics... Blanchet (Columbia) 6 / 31
16 Introduction to Rare Event Simulation Rare event simulation = techniques for effi cient estimation of rare-event probabilities AND conditional expectations given such rare events. Arises in settings such as: 1 Finance and Insurance, 2 Queueing networks (operations and communications), 3 Environmental and physical sciences, 4 Statistical inference and combinatorics... 5 Stochastic optimization, Blanchet (Columbia) 6 / 31
17 Insurance Reserve Setup V i s i.i.d. (independent and identially distributed) claim sizes Blanchet (Columbia) 7 / 31
18 Insurance Reserve Setup V i s i.i.d. (independent and identially distributed) claim sizes τ i s i.i.d. inter-arrival times Blanchet (Columbia) 7 / 31
19 Insurance Reserve Setup V i s i.i.d. (independent and identially distributed) claim sizes τ i s i.i.d. inter-arrival times A n = time of the n-th arrival Blanchet (Columbia) 7 / 31
20 Insurance Reserve Setup V i s i.i.d. (independent and identially distributed) claim sizes τ i s i.i.d. inter-arrival times A n = time of the n-th arrival Constant premium p Blanchet (Columbia) 7 / 31
21 Insurance Reserve Setup V i s i.i.d. (independent and identially distributed) claim sizes τ i s i.i.d. inter-arrival times A n = time of the n-th arrival Constant premium p Reserve process N (t) R (t) = b + pt j=1 V j Blanchet (Columbia) 7 / 31
22 Insurance Reserve Setup V i s i.i.d. (independent and identially distributed) claim sizes τ i s i.i.d. inter-arrival times A n = time of the n-th arrival Constant premium p Reserve process N (t) = # arrivals up to time t N (t) R (t) = b + pt j=1 V j Blanchet (Columbia) 7 / 31
23 Insurance Reserve Plot Plot of risk reserve R(t) b A 1 A 2 A 3 t Blanchet (Columbia) 8 / 31
24 Insurance Reserve and Random Walks Evaluating reserve at arrival times we get random walk Blanchet (Columbia) 9 / 31
25 Insurance Reserve and Random Walks Evaluating reserve at arrival times we get random walk Suppose Y 1, Y 2,... are i.i.d. S (n) = b + Y Y n Blanchet (Columbia) 9 / 31
26 Insurance Reserve and Random Walks Evaluating reserve at arrival times we get random walk Suppose Y 1, Y 2,... are i.i.d. S (n) = b + Y Y n R (A n ) = S (n) reserve at arrival times with Y n = pτ n V n. R(t) b A 1 A 2 A 3 t Blanchet (Columbia) 9 / 31
27 Insurance Reserve and Random Walks Evaluating reserve at arrival times we get random walk Suppose Y 1, Y 2,... are i.i.d. S (n) = b + Y Y n R (A n ) = S (n) reserve at arrival times with Y n = pτ n V n. R(t) b A 1 A 2 A 3 t Question: What is the chance that inf{r (t) : t 0} < 0? And how does this happen? Blanchet (Columbia) 9 / 31
28 How Does Ruin Occur with Light-Tailed Increments? Y i s are N(0, 1), EY i = 1 Blanchet (Columbia) 10 / 31
29 How Does Ruin Occur with Light-Tailed Increments? Y i s are N(0, 1), EY i = 1 Random walk conditioned on ruin Blanchet (Columbia) 10 / 31
30 How Does Ruin Occur with Light-Tailed Increments? Y i s are N(0, 1), EY i = 1 Random walk conditioned on ruin Light tails: Exponential, Gamma, Gaussian, mixtures of these, etc. Blanchet (Columbia) 10 / 31
31 How Does Ruin Occur with Light-Tailed Increments? Y i s are N(0, 1), EY i = 1 Random walk conditioned on ruin Light tails: Exponential, Gamma, Gaussian, mixtures of these, etc. Picture generated with Siegmund s 76 algorithm Blanchet (Columbia) 10 / 31
32 Back to Heavy-tailed Insurance Ruin Ruin with with Power law Pareto Type Claims Increments 400 Random Reserve Walk Value B Time Y i s are t-distributed, EY i = 1, Var (Y i ) = 1, 4 degrees of freedom. Blanchet (Columbia) 11 / 31
33 Back to Heavy-tailed Insurance Ruin Ruin with with Power law Pareto Type Claims Increments 400 Random Reserve Walk Value B Time Y i s are t-distributed, EY i = 1, Var (Y i ) = 1, 4 degrees of freedom. Heavy-tailed random walk conditioned on ruin. Blanchet (Columbia) 11 / 31
34 Back to Heavy-tailed Insurance Ruin Ruin with with Power law Pareto Type Claims Increments 400 Random Reserve Walk Value B Time Y i s are t-distributed, EY i = 1, Var (Y i ) = 1, 4 degrees of freedom. Heavy-tailed random walk conditioned on ruin. Picture generated with Blanchet and Glynn s 09 algorithm. Blanchet (Columbia) 11 / 31
35 A Two-node Tandem Network Poisson arrivals, exponential service times, and Markovian routing... Blanchet (Columbia) 12 / 31
36 Example: A Simple Stochastic Network Questions of interest: How would the system perform IF TOTAL POPULATION reaches n inside a busy period? How likely is this event under different parameters / designs? Blanchet (Columbia) 13 / 31
37 Naive Benchmark: Crude Monte Carlo Say λ =.1, µ 1 =.5, µ 2 =.4, p =.1 Blanchet (Columbia) 14 / 31
38 Naive Benchmark: Crude Monte Carlo Say λ =.1, µ 1 =.5, µ 2 =.4, p =.1 Overflow probability 10 7 Blanchet (Columbia) 14 / 31
39 Naive Benchmark: Crude Monte Carlo Say λ =.1, µ 1 =.5, µ 2 =.4, p =.1 Overflow probability 10 7 Each replication in crude Monte Carlo 10 3 secs Blanchet (Columbia) 14 / 31
40 Naive Benchmark: Crude Monte Carlo Say λ =.1, µ 1 =.5, µ 2 =.4, p =.1 Overflow probability 10 7 Each replication in crude Monte Carlo 10 3 secs Time to estimate with 10% precision 10 7 /1000 = 10 4 secs 2.7 hours Blanchet (Columbia) 14 / 31
41 Naive Benchmark: Crude Monte Carlo Say λ =.1, µ 1 =.5, µ 2 =.4, p =.1 Overflow probability 10 7 Each replication in crude Monte Carlo 10 3 secs Time to estimate with 10% precision 10 7 /1000 = 10 4 secs 2.7 hours What if you want to do sensitivity analysis for a wide range of parameters? Blanchet (Columbia) 14 / 31
42 Naive Benchmark: Crude Monte Carlo Say λ =.1, µ 1 =.5, µ 2 =.4, p =.1 Overflow probability 10 7 Each replication in crude Monte Carlo 10 3 secs Time to estimate with 10% precision 10 7 /1000 = 10 4 secs 2.7 hours What if you want to do sensitivity analysis for a wide range of parameters? What about different network designs? Blanchet (Columbia) 14 / 31
43 Rare Events in Networks Question: How does the TOTAL CONTENT OF THE SYSTEM is most likely to reach a give level in an operation cycle? Blanchet (Columbia) 15 / 31
44 Rare Events in Networks Question: How does the TOTAL CONTENT OF THE SYSTEM is most likely to reach a give level in an operation cycle? Say λ =.2, µ 1 =.3, µ 2 =.5... How can total content reach 50? Blanchet (Columbia) 15 / 31
45 Rare Events in Networks Question: How does the TOTAL CONTENT OF THE SYSTEM is most likely to reach a give level in an operation cycle? Say λ =.2, µ 1 =.3, µ 2 =.5... How can total content reach 50? Naive Monte Carlo takes 115 days Blanchet (Columbia) 15 / 31
46 Rare Events in Networks Question: How does the TOTAL CONTENT OF THE SYSTEM is most likely to reach a give level in an operation cycle? Say λ =.2, µ 1 =.3, µ 2 =.5... How can total content reach 50? Naive Monte Carlo takes 115 days Each picture below took.01 seconds Blanchet (Columbia) 15 / 31
47 Rare Events in Networks λ =.1, µ 1 =.3, µ 2 =.5 Plot of Conditional Path X 2 OF X 1 Blanchet (Columbia) 16 / 31
48 Rare Events in Networks λ =.1, µ 1 =.5, µ 2 =.2 Plot of Conditional Path OF X 2 X 1 Blanchet (Columbia) 17 / 31
49 Rare Events in Networks λ =.1, µ 1 =.4, µ 2 =.4 Plot of Conditional Path X 2 OF X 1 Blanchet (Columbia) 18 / 31
50 Several Notions of Effi ciency Assume the estimator is unbiased. Blanchet (Columbia) 19 / 31
51 Several Notions of Effi ciency Assume the estimator is unbiased. E (Z ) = P (A) and we assume P (A) 0. Blanchet (Columbia) 19 / 31
52 Several Notions of Effi ciency Assume the estimator is unbiased. E (Z ) = P (A) and we assume P (A) 0. Logarithmic Effi ciency (weakly effi ciency or asymptotic optimality): log E ( Z 2) lim P (A) 0 2 log P (A) = 1. Blanchet (Columbia) 19 / 31
53 Several Notions of Effi ciency Assume the estimator is unbiased. E (Z ) = P (A) and we assume P (A) 0. Logarithmic Effi ciency (weakly effi ciency or asymptotic optimality): log E ( Z 2) lim P (A) 0 2 log P (A) = 1. Bounded Relative Error E ( Z 2) = O 2 (1) as P (A) 0. P (A) Blanchet (Columbia) 19 / 31
54 Effi ciency of Naive Monte Carlo If Z = I (A) then E ( Z 2) = E (Z ) = P (A) Blanchet (Columbia) 20 / 31
55 Effi ciency of Naive Monte Carlo If Z = I (A) then E ( Z 2) = E (Z ) = P (A) Therefore lim P (A) 0 log P (A) 2 log P (A) = 1 2. Blanchet (Columbia) 20 / 31
56 Effi ciency of Naive Monte Carlo If Z = I (A) then E ( Z 2) = E (Z ) = P (A) Therefore lim P (A) 0 log P (A) 2 log P (A) = 1 2. How to design effi cient rare event simulation estimators? Importance Sampling and other variance reduction techniques... Blanchet (Columbia) 20 / 31
57 Graphical Interpretation of Importance Sampling Importance sampling (I.S.): sample from the important region and correct via likelihood ratio RED AREA PROPORTION DARTS IN RED AREA Blanchet (Columbia) 21 / 31
58 Importance Sampling Goal: Estimate P (A) > 0 Blanchet (Columbia) 22 / 31
59 Importance Sampling Goal: Estimate P (A) > 0 Choose P and simulate ω from it Blanchet (Columbia) 22 / 31
60 Importance Sampling Goal: Estimate P (A) > 0 Choose P and simulate ω from it Importance Sampling (I.S.) estimator per trial is I.S.Estimator = L (ω) I (ω A), where L (ω) is the likelihood ratio (i.e. L (ω) = dp (ω) /d P (ω)). Blanchet (Columbia) 22 / 31
61 Importance Sampling Goal: Estimate P (A) > 0 Choose P and simulate ω from it Importance Sampling (I.S.) estimator per trial is I.S.Estimator = L (ω) I (ω A), where L (ω) is the likelihood ratio (i.e. L (ω) = dp (ω) /d P (ω)). NOTE: P ( ) is called a change-of-measure Blanchet (Columbia) 22 / 31
62 Importance Sampling Suppose we choose P ( ) = P ( A) L (ω) = dp (ω) I (ω A) = P (A) I (ω A) dp (ω) /P (A) Blanchet (Columbia) 23 / 31
63 Importance Sampling Suppose we choose P ( ) = P ( A) L (ω) = dp (ω) I (ω A) = P (A) I (ω A) dp (ω) /P (A) Estimator has zero variance, but requires knoweledge of P (A) Blanchet (Columbia) 23 / 31
64 Importance Sampling Suppose we choose P ( ) = P ( A) L (ω) = dp (ω) I (ω A) = P (A) I (ω A) dp (ω) /P (A) Estimator has zero variance, but requires knoweledge of P (A) Lesson: Try choosing P ( ) close to P ( A)! Blanchet (Columbia) 23 / 31
65 Large Deviations of the Empirical Mean Suppose X 1, X 2,... are i.i.d. and H (θ) = log E exp (θx ) < for θ in a neighborhood of the origin. Blanchet (Columbia) 24 / 31
66 Large Deviations of the Empirical Mean Suppose X 1, X 2,... are i.i.d. and H (θ) = log E exp (θx ) < for θ in a neighborhood of the origin. Assume that E (X i ) = 0. Blanchet (Columbia) 24 / 31
67 Large Deviations of the Empirical Mean Suppose X 1, X 2,... are i.i.d. and H (θ) = log E exp (θx ) < for θ in a neighborhood of the origin. Assume that E (X i ) = 0. Suppose that for each a > 0, there is θ a such that H (θ a ) = a. Blanchet (Columbia) 24 / 31
68 Large Deviations of the Empirical Mean Suppose X 1, X 2,... are i.i.d. and H (θ) = log E exp (θx ) < for θ in a neighborhood of the origin. Assume that E (X i ) = 0. Suppose that for each a > 0, there is θ a such that H (θ a ) = a. Consider the problem of estimating via simulation where S (n) = X X n. P (S (n) > na), Blanchet (Columbia) 24 / 31
69 Exponential Tilting Suppose that X i has density f ( ) and define f θ (x) = exp (θx H (θ)) f (x). Blanchet (Columbia) 25 / 31
70 Exponential Tilting Suppose that X i has density f ( ) and define f θ (x) = exp (θx H (θ)) f (x). Note that log E θ (exp (ηx )) = log f θ (x) exp (ηx) dx = log f (x) exp ((η + θ) x H (θ)) dx = H (η + θ) H (θ). Blanchet (Columbia) 25 / 31
71 Exponential Tilting Suppose that X i has density f ( ) and define f θ (x) = exp (θx H (θ)) f (x). Note that log E θ (exp (ηx )) = log f θ (x) exp (ηx) dx = log f (x) exp ((η + θ) x H (θ)) dx = H (η + θ) H (θ). Therefore E θ (X ) = H (θ) and Var θ (X ) = H (θ). Blanchet (Columbia) 25 / 31
72 Exponential Tilting in Action Cramer s theorem (strong version): (Sketch) = = P (S n > na) f (x 1 )...f (x n ) I (s n > na) dx 1:n f θa (x 1 )...f θa (x n ) exp ( θ a s n + nh (θ a )) dx 1:n {s n >na} = exp ( n[aθ a H (θ a )]) E θa [exp ( θ a (S n na)) I (S n > na)] = exp ( ni (a)) E θa [exp ( θ a (S n na)) I (S n > na)]. Blanchet (Columbia) 26 / 31
73 Exponential Tilting in Action Cramer s theorem (strong version): (Sketch) = = P (S n > na) f (x 1 )...f (x n ) I (s n > na) dx 1:n f θa (x 1 )...f θa (x n ) exp ( θ a s n + nh (θ a )) dx 1:n {s n >na} = exp ( n[aθ a H (θ a )]) E θa [exp ( θ a (S n na)) I (S n > na)] = exp ( ni (a)) E θa [exp ( θ a (S n na)) I (S n > na)]. Use CLT and approximate (S n na) by N (0, nh (θ a )) to obtain E θa [exp ( θ a (S n na)) I (S n > na)] 1 n 1/2 (2πH (θ a )) 1/2 θ a. Blanchet (Columbia) 26 / 31
74 Why is Exponential tilting Useful? Theorem Under the assumptions imposed Proof. P (X 1 dx 1 S n > na) P θa (X 1 dx 1 ). In simple words, exponential tilting approximates the zero-variance (optimal!) change-of-measure! P (X 1 dx 1 S n > na) = P (S n 1 > na x 1 ) f (x 1 ) dx. P (S n > na) Now apply Cramer s theorem with a (na x 1 ) /(n 1) and n n 1 in the numerator and simplify, observing that ( ) ( na x1 θ = θ a + a x ) 1 1 a x 1 θ (a) + n 1 n 1 H (θ a ) n 1. Blanchet (Columbia) 27 / 31
75 First Asymptotically Optimal Rare Event Simulation Estimator Recall the identity used in Cramer s theorem P (S n > na) = f (x 1 )...f (x n ) I (s n > na) dx 1:n = f θa (x 1 )...f θa (x n ) exp ( θ a s n + nh (θ a )) dx 1:n {s n >na} = E θa [exp ( θ a S n + nh (θ a )) I (S n > na)]. Blanchet (Columbia) 28 / 31
76 First Asymptotically Optimal Rare Event Simulation Estimator Recall the identity used in Cramer s theorem P (S n > na) = f (x 1 )...f (x n ) I (s n > na) dx 1:n = f θa (x 1 )...f θa (x n ) exp ( θ a s n + nh (θ a )) dx 1:n {s n >na} = E θa [exp ( θ a S n + nh (θ a )) I (S n > na)]. IMPORTANCE SAMPLING ESTIMATOR: Z = exp ( θ a S n + nh (θ a )) I (S n > na), with X 1,..., X n simulated using the density f θa ( ). Blanchet (Columbia) 28 / 31
77 First Asymptotically Optimal Rare Event Simulation Estimator Recall the identity used in Cramer s theorem P (S n > na) = f (x 1 )...f (x n ) I (s n > na) dx 1:n = f θa (x 1 )...f θa (x n ) exp ( θ a s n + nh (θ a )) dx 1:n {s n >na} = E θa [exp ( θ a S n + nh (θ a )) I (S n > na)]. IMPORTANCE SAMPLING ESTIMATOR: Z = exp ( θ a S n + nh (θ a )) I (S n > na), with X 1,..., X n simulated using the density f θa ( ). Let us verify now that Z is asymptotically effi cient! Blanchet (Columbia) 28 / 31
78 First Asymptotically Optimal Rare Event Simulation Estimator IMPORTANCE SAMPLING ESTIMATOR: Z = exp ( θ a S n + nh (θ a )) I (S n > na), with X 1,..., X n simulated using the density f θa ( ). Blanchet (Columbia) 29 / 31
79 First Asymptotically Optimal Rare Event Simulation Estimator IMPORTANCE SAMPLING ESTIMATOR: Z = exp ( θ a S n + nh (θ a )) I (S n > na), with X 1,..., X n simulated using the density f θa ( ). Therefore, ( E θa Z 2 ) = E θa exp ( 2θ a S n + 2nH (θ a )) I (S n > na) = exp ( 2nI (a)) E θa [exp ( 2θ a (S n na)) I (S n > na)]. Blanchet (Columbia) 29 / 31
80 First Asymptotically Optimal Rare Event Simulation Estimator IMPORTANCE SAMPLING ESTIMATOR: Z = exp ( θ a S n + nh (θ a )) I (S n > na), with X 1,..., X n simulated using the density f θa ( ). Therefore, ( E θa Z 2 ) = E θa exp ( 2θ a S n + 2nH (θ a )) I (S n > na) = exp ( 2nI (a)) E θa [exp ( 2θ a (S n na)) I (S n > na)]. Thus ( log E θa Z 2 ) 2nI (a) = + o (1) 1. 2 log P (S n > na) 2nI (a) Blanchet (Columbia) 29 / 31
81 How to Simulate under Exponential Tilting Key is to identify: H θ (η) = H (η + θ) H (θ). Blanchet (Columbia) 30 / 31
82 How to Simulate under Exponential Tilting Key is to identify: H θ (η) = H (η + θ) H (θ). Example 1: Gaussian if X N (µ, 1) H (θ) = θ 2 /2 + µθ, H θ (η) = η 2 /2 + (θ + µ) η. Thus, under f θ, X N (µ, 1). Blanchet (Columbia) 30 / 31
83 How to Simulate under Exponential Tilting Key is to identify: H θ (η) = H (η + θ) H (θ). Example 1: Gaussian if X N (µ, 1) H (θ) = θ 2 /2 + µθ, H θ (η) = η 2 /2 + (θ + µ) η. Thus, under f θ, X N (µ, 1). Example 2: If X Poisson (λ), then H (θ) = λ (exp (θ) 1), H θ (η) = λ exp (θ) (exp (η) 1). Thus, under f θ, X Poisson (λ exp (θ)). Blanchet (Columbia) 30 / 31
84 Conclusions 1 Rare event simulation is a power numerical tool applicable in a wide range of examples. Blanchet (Columbia) 31 / 31
85 Conclusions 1 Rare event simulation is a power numerical tool applicable in a wide range of examples. 2 Importance sampling can be used to construct effi cient estimators. Blanchet (Columbia) 31 / 31
86 Conclusions 1 Rare event simulation is a power numerical tool applicable in a wide range of examples. 2 Importance sampling can be used to construct effi cient estimators. 3 Best importance sampler is the conditional distribution given the event of interest. Blanchet (Columbia) 31 / 31
87 Conclusions 1 Rare event simulation is a power numerical tool applicable in a wide range of examples. 2 Importance sampling can be used to construct effi cient estimators. 3 Best importance sampler is the conditional distribution given the event of interest. 4 Effi cient algorithms come basically in two flavors: logarithmically and strongly effi cient. Blanchet (Columbia) 31 / 31
88 Conclusions 1 Rare event simulation is a power numerical tool applicable in a wide range of examples. 2 Importance sampling can be used to construct effi cient estimators. 3 Best importance sampler is the conditional distribution given the event of interest. 4 Effi cient algorithms come basically in two flavors: logarithmically and strongly effi cient. 5 Exponential tilting can be used to approximate the conditional distribution given the event of interest. Blanchet (Columbia) 31 / 31
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