Asymptotics and Simulation of Heavy-Tailed Processes

Transcription

1 Asymptotics and Simulation of Heavy-Tailed Processes Department of Mathematics Stockholm, Sweden Workshop on Heavy-tailed Distributions and Extreme Value Theory ISI Kolkata January 14-17, 2013

2 Outline 1 Introduction to Rare-Event Simulation The Problem with Monte Carlo Importance Sampling 2 Importance Sampling in a Heavy-Tailed Setting The Conditional Mixture Algorithm for Random Walks Performance of the Conditional Mixture Algorithm 3 Markov Chain Monte Carlo in Rare-Event Simulation Efficient Sampling for a Heavy-Tailed Random Walk Efficient Sampling of Heavy-Tailed Random Sums

3 Simulation of Heavy-Tailed Processes Goal: improve on the computational efficiency of standard Monte Carlo. Design: The large deviations analysis lead to a heuristic way to design efficient algorithms. Efficiency Analysis: The large deviations analysis can be applied to theoretically quantify the computational performance of algorithms.

4 The Problem with Monte Carlo Outline 1 Introduction to Rare-Event Simulation The Problem with Monte Carlo Importance Sampling 2 Importance Sampling in a Heavy-Tailed Setting The Conditional Mixture Algorithm for Random Walks Performance of the Conditional Mixture Algorithm 3 Markov Chain Monte Carlo in Rare-Event Simulation Efficient Sampling for a Heavy-Tailed Random Walk Efficient Sampling of Heavy-Tailed Random Sums

5 The Problem with Monte Carlo The Performance of Monte Carlo Let X 1,...,X N be independent copies of X. The Monte Carlo estimator is Its standard deviation is p = 1 N N I{X k > b}. k=1 std = 1 N p(1 p) To obtain a Relative Error (std/p) of 1% you need to take N such that 1 N p(1 p) p 0.01 N p p

6 Importance Sampling Outline 1 Introduction to Rare-Event Simulation The Problem with Monte Carlo Importance Sampling 2 Importance Sampling in a Heavy-Tailed Setting The Conditional Mixture Algorithm for Random Walks Performance of the Conditional Mixture Algorithm 3 Markov Chain Monte Carlo in Rare-Event Simulation Efficient Sampling for a Heavy-Tailed Random Walk Efficient Sampling of Heavy-Tailed Random Sums

7 Importance Sampling Introduction to Importance Sampling 1 The problem with Monte Carlo is that few samples hit the rare event. This problem can be fixed by sampling from a distribution which puts more mass on the rare event. Must compensate for not sampling from the original distribution. Key issue: How to select an appropriate sampling distribution? 1 See e.g. [1] for an introduction.

8 Importance Sampling Monte Carlo and Importance Sampling The Basics for Computing F(A) = P{X A} Monte Carlo Sample X 1,...,X N from F. Empirical measure F N ( ) = 1 N N δ X k( ). k=1 Plug-in estimator p = F N (A). Importance Sampling Sample X 1,..., X N from F. Weighted empirical measure F w N( ) = 1 N N k=1 Plug-in estimator p = F w N(A) = 1 N N k=1 df d F ( X k )δ X k( ). df d F ( X k )I{ X k A}.

9 Importance Sampling Monte Carlo and Importance Sampling The Basics for Computing F(A) = P{X A} Monte Carlo Sample X 1,...,X N from F. Empirical measure F N ( ) = 1 N N δ X k( ). k=1 Plug-in estimator p = F N (A). Importance Sampling Sample X 1,..., X N from F. Weighted empirical measure F w N( ) = 1 N N k=1 Plug-in estimator p = F w N(A) = 1 N N k=1 df d F ( X k )δ X k( ). df d F ( X k )I{ X k A}.

10 Importance Sampling Importance Sampling The importance sampling estimator is unbiased: [ df ] Ẽ d F ( X)I{ X df A} = A d F d F = df = F(A). A Its variance is ( df ) [( df ) ] Var d F ( X)I{ X A} = 2I{ X Ẽ A} F(A) 2 d F ( df ) = 2d F F(A) 2 A d F df = df F(A)2 A d F [ df ] = E d F I{X A} F(A) 2.

11 Importance Sampling Importance Sampling The importance sampling estimator is unbiased: [ df ] Ẽ d F ( X)I{ X df A} = A d F d F = df = F(A). A Its variance is ( df ) [( df ) ] Var d F ( X)I{ X A} = 2I{ X Ẽ A} F(A) 2 d F ( df ) = 2d F F(A) 2 A d F df = df F(A)2 A d F [ df ] = E d F I{X A} F(A) 2.

12 Importance Sampling Illustration Sampling the tail Monte Carlo Importance Sampling

13 Importance Sampling Quantifying Efficiency Variance Based Efficiency Criteria Basic idea: variance roughly the size of p 2. Embed in a sequence of problems such that p n = P{X n A} 0. Logarithmic Efficiency: for some ǫ > 0 lim sup n Var( p n ) p 2 ǫ n <. Strong Efficiency/Bounded Relative Error: Vanishing relative error: sup n lim sup n Var( p n ) p 2 n Var( p n ) p 2 n <. = 0.

14 Importance Sampling The Zero-Variance Change of Measure There is a best choice of sampling distribution, the zero-variance change of measure, given by For this choice F A ( ) = P{X n X n A}. df df A (x) = P{X n A}I{X n A} = p n I{X n A}, and hence [ ] Var( p n ) = E p n I{X n A} F(A) 2 = 0.

15 Example Problem A Random Walk with Heavy Tails Let {Z k } be iid non-negative and regularly varying (index α) with density f. Put S m = Z 1 + +Z m. Compute P{S m > n}. The zero-variance sampling distribution is P{(Z 1,...,Z m ) S m > n}. By regular variation we know that P{(Z 1,...,Z m ) S m > n} µ( ), where µ is concentrated on the coordinate axis. But µ and F are singular!

16 Example Problem A Random Walk with Heavy Tails Let {Z k } be iid non-negative and regularly varying (index α) with density f. Put S m = Z 1 + +Z m. Compute P{S m > n}. The zero-variance sampling distribution is P{(Z 1,...,Z m ) S m > n}. By regular variation we know that P{(Z 1,...,Z m ) S m > n} µ( ), where µ is concentrated on the coordinate axis. But µ and F are singular!

17 The Conditional Mixture Algorithm for Random Walks Outline 1 Introduction to Rare-Event Simulation The Problem with Monte Carlo Importance Sampling 2 Importance Sampling in a Heavy-Tailed Setting The Conditional Mixture Algorithm for Random Walks Performance of the Conditional Mixture Algorithm 3 Markov Chain Monte Carlo in Rare-Event Simulation Efficient Sampling for a Heavy-Tailed Random Walk Efficient Sampling of Heavy-Tailed Random Sums

18 The Conditional Mixture Algorithm for Random Walks The Conditional Mixture Algorithm 2 Suppose S i 1 = s. Sample Z i as follows 1 If s n, sample Z i from the original density f. 2 If s n, sample Z i from the mixture p i f(z)+(1 p i ) f i (y s), 1 i m 1, fm(y s), i = m. Idea: take f i to produce large values of Z i. 2 This algorithm is due to [4]

19 The Conditional Mixture Algorithm for Random Walks The Conditional Mixture Algorithm We can take f i s to be the conditional distributions of the form f(z)i{z > a(n s)} fi (z s) = P{Z > a(n s)}, i m 1, fm (z s) = where a (0, 1). f(z)i{z > n s} P{Z > n s}, Note: f i is the conditional distribution of Z given that Z is large.

20 Performance of the Conditional Mixture Algorithm Outline 1 Introduction to Rare-Event Simulation The Problem with Monte Carlo Importance Sampling 2 Importance Sampling in a Heavy-Tailed Setting The Conditional Mixture Algorithm for Random Walks Performance of the Conditional Mixture Algorithm 3 Markov Chain Monte Carlo in Rare-Event Simulation Efficient Sampling for a Heavy-Tailed Random Walk Efficient Sampling of Heavy-Tailed Random Sums

21 Performance of the Conditional Mixture Algorithm The Performance of the Conditional Mixture Algorithm The normalized second moment is E[ p n ] pn 2 = 1 df pn 2 d F (z 1,...,z m )I{s m > n}f(dz 1,...,dz m ) 1 df m 2 P{Z 1 > n} 2 d F (z 1,...,z m )I{s m > n}f(dz 1,...,dz m ) 1 1 df m 2 P{Z 1 > n} d F (nz 1,...,nz m )I{s m > 1} F(ndz 1,...,ndz m ). P{Z 1 > n}

22 Performance of the Conditional Mixture Algorithm The Performance of the Conditional Mixture Algorithm Weak convergence: F(n ) m P{Z 1 > n} k=1 I{ze k }µ α (dz) The normalized likelihood ratio is bounded: sup n 1 df P{Z 1 > n} d F (nz 1,...,nz m )I{s m > 1} <.

23 Performance of the Conditional Mixture Algorithm The Performance of the Conditional Mixture Algorithm The above enables us to show that the normalized second moment converges: lim n E[ p n ] p 2 n It is minimized at = 1 ( m 1 m 2 i=1 a α 1 p i i 1 j=1 p i = (m i 1)a α/2 + 1 (m i)a α/2 + 1, m ) +. p j p j j=1 with minimum 1 ( 2. m 2 (m 1)a α/2 + 1)

24 Performance of the Conditional Mixture Algorithm The Performance of the Conditional Mixture Algorithm The conditional mixture algorithm has (almost) vanishing relative error. Heavy-tailed heuristics indicate how to design the algorithm. Heavy-tailed asymptotics needed to prove efficiency of the algorithm.

25 MCMC for Rare Events 3 Let X be random variable with density f and compute p = P{X A}, where A is a rare event. The zero-variance sampling distribution is F A ( ) = P{X X A}, df A dx f(x)i{x A} (x) =. p It is possible to sample from F A by constructing a Markov chain with stationary distribution F A (e.g. Gibbs sampler or Metropolis-Hastings). Idea: sample from F A and extract the normalizing constant p. 3 This part is based on [5]

26 MCMC for Rare Events Let X 0,...,X T 1 be a sample of a Markov chain with stationary distribution F A. Consider a non-negative function v(x) with A v(x)dx = 1. The sample mean q T = 1 T is an estimator of [ v(x)i{x A} ] E FA = f(x) T 1 t=0 A v(x t )I{X t A}, f(x t ) v(x) f(x) Take q T as the estimator of 1/p. f(x) p dx = 1 v(x)dx = 1 p A p.

27 MCMC for Rare Events The rare-event properties of q T are determined by the choice of v. The large sample properties of q T are determined by the ergodic properties of the Markov chain.

28 The Normalized Variance The rare-event efficiency (p small) is determined by the normalized variance: ( v(x) ) p 2 Var FA f(x) I{X A} = p 2( E FA [( v(x) f(x) I{X A} ) 2 ] 1 p 2 ) = p 2( v 2 (x) f 2 (x) = p A v 2 (x) dx 1. f(x) f(x) p dx 1 p 2 )

29 The Optimal Choice of v It is optimal to take v as f(x)/p. Indeed, then ( v(x) ) p 2 Var FA f(x) I{X A} = p A v 2 (x) dx 1 = 0. f(x) Insight: take v as an approximation of the conditional density given the event.

30 Efficient Sampling for a Heavy-Tailed Random Walk Outline 1 Introduction to Rare-Event Simulation The Problem with Monte Carlo Importance Sampling 2 Importance Sampling in a Heavy-Tailed Setting The Conditional Mixture Algorithm for Random Walks Performance of the Conditional Mixture Algorithm 3 Markov Chain Monte Carlo in Rare-Event Simulation Efficient Sampling for a Heavy-Tailed Random Walk Efficient Sampling of Heavy-Tailed Random Sums

31 Efficient Sampling for a Heavy-Tailed Random Walk Efficient Sampling for a Heavy-Tailed Random Walk Let {Z k } be iid with density f. Put S n = Z 1 + +Z n. Compute P{S n > a n }. Heavy-tailed assumption: P{S n > a n } lim n P{M n > a n } = 1. (includes subexponential distributions, see [3]).

32 Efficient Sampling for a Heavy-Tailed Random Walk Gibbs Sampling of a Heavy-Tailed Random Walk 1 Start from a state with Z 1 > a n. 2 Update the steps in a random order according to j 1,...,j n (uniformly without replacement). 3 Update Z jk by sampling from P{Z Z + i j k Z i > n}. The vector (Z t,1,...,z t,n ) forms a uniformly ergodic Markov chain with stationary distribution F A ( ) = P{(Z 1,...,Z n ) Z Z n > a n }.

33 Efficient Sampling for a Heavy-Tailed Random Walk Rare-Event Efficiency The Choice of v Take v to be the density of P{(Z 1,...,Z n ) M n > a n }. Then v(z 1,...,z n ) f(z 1,...,z n ) = 1 P{M n > a n } I{ n i=1z i > a n }. Rare event efficiency: ( pnvar 2 v(z1,...,z n ) ) FA f(z 1,...,Z n ) I{Z 1 + +Z n > a n } = P{S n > a n } 2 P{M n > a n } 2P{M n > a n S n > a n }P{M n a n S n > a n } = P{S n > a n } ( 1 P{M n > a n } ) 0. P{M n > a n } P{S n > a n }

34 Efficient Sampling for a Heavy-Tailed Random Walk Rare-Event Efficiency The Choice of v Take v to be the density of P{(Z 1,...,Z n ) M n > a n }. Then v(z 1,...,z n ) f(z 1,...,z n ) = 1 P{M n > a n } I{ n i=1z i > a n }. Rare event efficiency: ( pnvar 2 v(z1,...,z n ) ) FA f(z 1,...,Z n ) I{Z 1 + +Z n > a n } = P{S n > a n } 2 P{M n > a n } 2P{M n > a n S n > a n }P{M n a n S n > a n } = P{S n > a n } ( 1 P{M n > a n } ) 0. P{M n > a n } P{S n > a n }

35 Efficient Sampling for a Heavy-Tailed Random Walk Numerical illustration n = 5, α = 2, a n = an. b = 25, T = 10 5, α = 2, n = 5, a = 5, p max = 0.737e-2 MCMC IS MC Avg. est e e e-2 Std. dev. 3e-5 9e-5 27e-5 Avg. time per batch(s) b = 25, T = 10 5, α = 2, n = 5, a = 20, p max = 4.901e-4 MCMC IS MC Avg. est e e e-4 Std. dev. 6e-7 13e-7 770e-7 Avg. time per batch(s) b = 20, T = 10 5, α = 2, n = 5, a = 10 3, p max = e-7 MCMC IS Avg. est e e-7 Std. dev. 3e-11 20e-11 Avg. time per batch(s) b = 20, T = 10 5, α = 2, n = 5, a = 10 4, p max = e-9 MCMC IS Avg. est e e-9 Std. dev. 7e e-14 Avg. time per batch(s)

36 Efficient Sampling for a Heavy-Tailed Random Walk Numerical Illustration n = 20, α = 2, a n = an. b = 25, T = 10 5, α = 2, n = 20, a = 20, p max = e-4 MCMC IS MC Avg. est e e e-4 Std. dev. 2e-7 3e-7 492e-7 Avg. time per batch(s) b = 25, T = 10 5, α = 2, n = 20, a = 200, p max = e-6 MCMC IS MC Avg. est e e e-6 Std. dev. 4e-10 12e-10 33,166e-10 Avg. time per batch(s) b = 20, T = 10 5, α = 2, n = 20, a = 10 3, p max = e-8 MCMC IS Avg. est e e-8 Std. dev. 7e-12 66e-12 Avg. time per batch(s) b = 20, T = 10 5, α = 2, n = 20, a = 10 4, p max = e-10 MCMC IS Avg. est e e-10 Std. dev. 2e-14 71e-14 Avg. time per batch(s)

37 Efficient Sampling of Heavy-Tailed Random Sums Outline 1 Introduction to Rare-Event Simulation The Problem with Monte Carlo Importance Sampling 2 Importance Sampling in a Heavy-Tailed Setting The Conditional Mixture Algorithm for Random Walks Performance of the Conditional Mixture Algorithm 3 Markov Chain Monte Carlo in Rare-Event Simulation Efficient Sampling for a Heavy-Tailed Random Walk Efficient Sampling of Heavy-Tailed Random Sums

38 Efficient Sampling of Heavy-Tailed Random Sums Efficient Sampling for a Heavy-Tailed Random Sum Let {Z k } be iid with density f. Put S n = Z 1 + +Z n and let {N n } be independent of {Z k } Compute P{S Nn > a n }. Heavy-tailed assumption: 4 P{S Nn > a n } lim n P{M Nn > a n } = 1. 4 See e.g. [6] for examples.

39 Efficient Sampling of Heavy-Tailed Random Sums Constructing the Gibbs Sampler Suppose we are in state (N t, Z t,1,...,z t,nt ) = (k t, z t,1,...,z t,kt ). Let kt = min{j : z t,1 + +z t,j > a n }. Update the number of steps N t+1 from the distribution p(k t+1 kt ) = P{N = k t+1 N k t } If k t+1 > k t, sample Z t+1,kt+1,...,z t+1,kt+1 independently from F Z. Proceed by updating all steps as before. Permute the steps.

40 Efficient Sampling of Heavy-Tailed Random Sums Properties of the algorithm The Markov chain {(N t, Z t,1,...,z t,nt )} is uniformly ergodic with stationary distribution F A ( ) = P{(Z 1,...,Z n ) Z Z n > a n }. With v as the density of P{(N, Z 1,...,Z N ) M N > a n } we have v(k, z 1,...,z k ) f(k, z 1,...,z k ) = 1 P{M N > a n } I{max z 1,...,z k > a n } = 1 1 g N (F Z (a n )) I{max z 1,...,z k > a n }. The algorithm has vanishing normalized variance.

41 Efficient Sampling of Heavy-Tailed Random Sums Numerical illustration Geometric(ρ) number of steps: ρ = 0.05, α = 1, a n = a/ρ. b = 25, T = 10 5, α = 1, ρ = 0.2, a = 10 2, p max = 0.990e-2 MCMC IS MC Avg. est e e e-2 Std. dev. 4e-5 6e-5 35e-5 Avg. time per batch(s) b = 25, T = 10 5, α = 1, ρ = 0.2, a = 10 3, p max = 0.999e-3 MCMC IS MC Avg. est e e e-3 Std. dev. 1e-6 3e-6 76e-6 Avg. time per batch(s) b = 20, T = 10 6, α = 1, ρ = 0.2, a = , p max = e-8 MCMC IS Avg. est e e-8 Std. dev. 6e e-14 Avg. time per batch(s) b = 20, T = 10 6, α = 1, ρ = 0.2, a = , p max = e-10 MCMC IS Avg. est e e-10 Std. dev. 0 13e-14 Avg. time per batch(s)

42 Efficient Sampling of Heavy-Tailed Random Sums Numerical Illustration Geometric(ρ) number of steps: ρ = 0.05, α = 1, a n = a/ρ. b = 25, T = 10 5, α = 1, ρ = 0.05, a = 10 3, p max = 0.999e-3 MCMC IS MC Avg. est e e e-3 Std. dev. 1e-6 4e-6 105e-6 Avg. time per batch(s) b = 25, T = 10 5, α = 1, ρ = 0.05, a = , p max = e-6 MCMC IS MC Avg. est e e e-6 Std. dev. 1e-10 53e-10 55,678e-10 Avg. time per batch(s)

43 Appendix For Further Reading I S. Asmussen and P.W. Glynn Stochastic Simulation: Algorithms and Anal- ysis. Springer-Verlag, New York, S. Resnick Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York, D. Denisov, A.B. Dieker, and V. Shneer Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Probab. 36(5), , 2008.

44 Appendix For Further Reading II P. Dupuis, K. Leder, and H. Wang Importance sampling for sums of random variables with regularly varying tails. ACM Trans. Model. Comput. Simul. 17(3), T. Gudmundsson and H. Hult Markov chain Monte Carlo for computing rare-event probabilities for a heavy-tailed random walk. C. Klüppelberg and T. Mikosch Large deviations for heavy-tailed random sums with applications to insurance and finance. J. Appl. Probab. 37(2), , 1997.