Estimating%20Rare%20Event%20Probabilities%20Using%20Truncated%20Saddlepoint%20Approximations

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Estimating Rare Event Probabilities Using
Truncated Saddlepoint Approximations

• Timothy I. Matis, Ph.D
• Ivan G. Guardiola
• Department of Industrial Engineering
• Texas Tech University

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Overview of Presentation(ordering is not strict)

• How I got here
• What are saddlepoint approximations
• Are truncated approximations any good
• Why we should care
• Numerical demonstrations
• What is next

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Texas Tech Trivia

• TTU is a comprehensive university
• TTU has Bobby Knight, and Mike Leach
• Cotton, cotton, and oil everywhere

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Evolution of Research Topic

• Preliminary investigations into stochastic
  shortest path problems
• The path that is shortest in expectation is not
  necessarily the shortest in probability
• Convolution of random path lengths
• Preliminary investigations into Cross-Entropy
  Methods R.Y. Rubinstein
• Used to find the change of measure of the
  importance sampling density

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Analytical Approximation

• CE methods are efficient, yet considerable
  computational effort is still required
• Can I quickly approximate the (rare)
  probabilities to at least the correct order?
• Truncated saddlepoint approximations are
  relatively simple and robustly accurate

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Motivating Example

• Example in Simulation (2002) by S.M. Ross
• Consider the sum of of independent random
  variables
• where
• For Num large, find the rare event probability ?
  

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Motivating Example

• In this example, the change of measure for the IS
  density may be calculated exactly, yielding a
  Monte Carlo IS point estimate of ?3.17x10-4 when
  Num16
• By contrast, a 3rd order truncated saddlepoint
  approximation yields the estimate ?2.378x10-4
  when Num16

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Mathematica Code

• Note that K is the CGF of the convolution, h is a
  list of solutions, and f is the truncated
  saddlepoint approximated density, which is
  subsequently integrated numerically

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Saddlepoint Approximations

• Daniels (seminal), Wang (bivariate), Renshaw
  (truncated)
• Saddlepoint approximations are accurate in the
  tails of the distribution (as opposed to
  Edgeworth or Guassian approximations)

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Mathematical Development

• Let Ki(?) be the cumulant generating function
  (CGF) of Xi
• The CGF of SX1Xn is
• K(?) K1(?) Kn(?)
• It follows that a saddlepoint approximation of
  the density function of S is given by
• where ?o is the positive real solution of

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Why Truncate ?

• It is likely that the solution to
  for ?o will be messy, if even attainable
• Second order truncation reduces to Gaussian
  approximation, third brings in skewness, fourth
  brings in kurtosis, etc.
• Truncated saddlepoint approximations may be
  complex over some of the support, yet are often
  not in the tails

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Truncation Development

• Let be the jth order truncated CGF
• The individual cumulants of the truncated CGF are
  found through differentiation of the CGF of the
  convolution

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Truncation Development

• It follows that the truncated CGF is given by
• where ?o is the positive real solution of the
  polynomial

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Optimal Truncation Level?

• Truncated saddlepoint approximations converge to
  full saddlepoint approximations in the limit
• In a finite sense, however, increasing the
  truncation level does not monotonically decrease
  the error
• In practice, evaluate the truncated saddlepoint
  at multiple levels! (if possible)

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Truncation Level Example

• For X1Normal(2,.5) and X2Exponential(1),
  estimate the rare event probability
• using truncated saddlepoint approximations.
• The solution to of the full
  saddlepoint is not reportable, thereby motivating
  the truncated approach

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Truncation Level Example

• The table on the right gives jth order truncated
  saddlepoint approximations of ?
• An IS based estimate of this probability is
  ?5.5x10-4

j ?
3 5.18x10-5
4 2.17x10-4
5 3.82x10-4
6 5.04x10-4
7 5.71x10-4
8 6.71x10-4
9 6.41x10-4
10 6.55x10-4
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What is Next?

• Truncated saddlepoint approximations of bivariate
  distributions
• Accuracy of truncated saddlepoint approximations
  when only moment closure estimates of the
  cumulants of a distribution are known

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Questions

• Contact Information
• Timothy I. Matis
• timothy.matis_at_ttu.edu
• [phone]