1. Improved Cross Entropy Method For Estimation Presented by: Alex & Yanna

2. This presentation is based on the paper “Improved Cross-Entropy Method for Estimation” By Dirk P.Kroese & Joshua C.Chan This presentation is based on the paper “Improved Cross-Entropy Method for Estimation” By Dirk P.Kroese & Joshua C.Chan

3. Rare Events Estimation

4. We wish to estimate - Random vector taking values in some set Function on

5. Rare Events Estimation We can rewrite it as - And estimate with a crude Monte Carlo

6. Rare Events Estimation Lets say, for example, that Direct Calculation Simulated

7. Rare Events Estimation

9. Importance Sampling

10. And the importance sampling estimator will be

11. Importance Sampling What would be a good choice for the importance density

12. Importance Sampling We shall take a look at the Kullback Leibler divergence: The zero variance density = The density from the family of with parameter

13. CE Algorithm In the article, 2 problematic issues were mentioned regarding the multilevel CE: The parametric family within which the optimal importance density g is obtained might not be large enough when the dimension of the problem is large, the likelihood ratio involved in obtaining becomes unstable. In the article, 2 problematic issues were mentioned regarding the multilevel CE: The parametric family within which the optimal importance density g is obtained might not be large enough when the dimension of the problem is large, the likelihood ratio involved in obtaining becomes unstable. Importance Sampling

14. Solution Sample directly from g*

15. Importance Sampling Our goal is to find Stochastic Version Deterministic Version

16. Importance Sampling But how the hell are we supposed to sample from ? ? ?

17. Importance Sampling This observation grants us the opportunity to apply the useful tool of gibbs sampling.

18. Gibbs Sampler In Brief

19. an algorithm to generate a sequence of samples from the joint probability distribution Gibbs sampling is a special case of the Metropolis–Hastings algorithm, and thus an example of a Markov chain Monte Carlo algorithm Gibbs sampling is applicable when the joint distribution is not known explicitly, but the conditional distribution of each variable is known It can be shown that the sequence of samples constitutes a Markov chain, and the stationary distribution of that Markov chain is just the sought-after joint distribution an algorithm to generate a sequence of samples from the joint probability distribution Gibbs sampling is a special case of the Metropolis–Hastings algorithm, and thus an example of a Markov chain Monte Carlo algorithm Gibbs sampling is applicable when the joint distribution is not known explicitly, but the conditional distribution of each variable is known It can be shown that the sequence of samples constitutes a Markov chain, and the stationary distribution of that Markov chain is just the sought-after joint distribution Gibbs Sampler In Brief

20. Gibbs Sampler In Brief The Gibbs sampler algorithm Given Generate Return

21. Improved Cross Entropy

22. Improved Cross Entropy The Improved CE consists of 3 steps: 1. Generate via gibbs sampler, N RVs 2. Solve 3. Estimate

23. Improved Cross Entropy Considerwhere and we would like to estimate under the improved cross entropy scheme.

24. Improved Cross Entropy Lets set and imply the new proposed algorithm

25. Improved Cross Entropy Step 1 – generate RVs from First we need to find

26. Improved Cross Entropy Step 1 – generate RVs from cont. Set Generate Set For

27. Improved Cross Entropy Step 2 – Solve the optimization problem

28. Improved Cross Entropy Step 3 – Estimate via importance sampling

29. Improved Cross Entropy Multilevel CE Vs. Improved CE

30. Improved Cross Entropy

31. CE N=10000 4 iterations Total budget 40000 CE N=10000 4 iterations Total budget 40000 Gibbs Sampler 10 parallel chains Each has 1000 length Total budget 10000 Gibbs Sampler In Brief

33. Obligors Probability of the obligor to default for a given threshold Monetary loss if the obligor defaults

35. t Copula Model

36. Known methods for the rare event estimation Exponential Change of MeasureHazard Rate Twisting Bounded relative errorLogarithmically efficient Needs to generate RVs from non standard distribution 10 times more variance reduction then ECM

37. The Improved CE for Estimating the Prob. of a Rare Loss

38. Step I – Sampling from g*

39. Sampling From g* Now we will show how we find the conditional probabilities of g* to apply the gibbs sampler For generating RVs from g*

40. Sampling From g* Define and arrange them is ascending order Let denote the ordered value and the corresponding loss Then the event occurs iff where Via Inverse Transform

41. Sampling From g* Define and arrange them is ascending order Let denote the ordered value and the corresponding loss Then the event occurs iff where Via Inverse Transform

42. Sampling From g* Multivariate truncated normal distribution Sequentially draw from if then else

43. After we got we are ready to move to the next step…

44. Step II – Solving the Opt. Problem

45. Solving Opt. Problem In our model

46. Solving Opt. Problem Since any member of the group is a product of densities, standard techniques of maximum likelihood estimation can be applied to find the optimal v*.

47. Solving Opt. Problem Once we obtain the optimal importance density we are moving to step 3

48. Step III – Importance Sampling

49. Importence Sampling

50. Some Results

52. Pros and Cons Improved CE Pros Rare events 3 basic steps Appropriate in multi dimension settings Fewer simulation effort then the Multi level CE Pros Rare events 3 basic steps Appropriate in multi dimension settings Fewer simulation effort then the Multi level CE Cons Problematic in general performance function not trivial Gibbs sampler requires warm up time Cons Problematic in general performance function not trivial Gibbs sampler requires warm up time

53. Further research Gibbs sampler for the general performance function Applying Sequential Monte Carlo Methods for sampling from g* Gibbs sampler for the general performance function Applying Sequential Monte Carlo Methods for sampling from g*