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Bounds on the probability of the union of events

Bounds on the probability of the union of events . International Colloquium on Stochastic Modeling and Optimization dedicated to the 80th Birthday of Professor András Prékopa. József Bukszár. Medical College of Virginia, Virginia Commonwealth University

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Bounds on the probability of the union of events

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  1. Bounds on the probability of the union of events International Colloquium on Stochastic Modeling and Optimization dedicated to the 80th Birthday of Professor András Prékopa József Bukszár Medical College of Virginia, Virginia Commonwealth University email: jbukszar@vcu.edu

  2. Let be arbitrary events. Our purpose is to give lower and upper bounds for based on intersection probabilities an integer given in advance. where Example: second and first order Bonferroni-bounds S2 S1 The underlying problem

  3. An A1 We can give lower and upper bounds for based on the marginal distribution function values. For that we need tight bounds that are based on only few intersection probabilities. Motivation Estimate values of multivariate distribution functions:

  4. U V A1 is the event that we can call Y from X through the red line. Each event corresponds to a path connecting X and Y. Motivation Network reliability Vertices represent stations. Edges represent phone lines, each of them is busy with a certain probability. What is the probability that we can call Y from X? The probability that we can call Y from X is Pr(A1U…U An)

  5. A1 A2 A3 A7 A4 A6 Pr(A4A5 ) A5 Hunter-Worsley bound: edges of the maximum weight spanning tree Hunter-Worsley bound Given a complete graph whose vertices are identified by the events. The weight of the edge connecting Ai and Aj is Pr(AiAj ). Let T = (V,E) be the maximum weight spanning tree on the complete graph.

  6. Advantages: • quickly computable: the maximum weight spanning tree can be obtained by a greedy algorithm (e.g. by Prim’s algorithm) • requires only n-1 intersection probabilities Disadvantages: • no improvement is available (problematic if the bound is greater than 1) • provides upper bound only

  7. v1 v2 vm+1 (middle vertex) vm-1 vm v1 Example: 2-multicherry (cherry) v3 E2= { {v1,v3}, {v2,v3} } E3= { {v1,v2,v3} } v2 m-multicherry m > 0 integer DEF: An m-multicherry is a hypergraph of the form (V,E2,…,Em+1), where V=(v1,…,vm+1) is the set of vertices, Ei= { H | vm+1 H  {v1,…,vm}, |H|=i } are the set of hyperedges.

  8. ”old multitree” v m-multitree (recursive def.) DEF: An m-multitree is a hypergraph of the form (V,E2,…, Ei,…,Em+1), set of vertices set of hyperedges with i vertices i.)The smallest m-multitree has m vertices. Ei is the set of all subsets of V with i vertices (Em+1 = ). ii.) From an m-multitree we can obtain another m-multitree by adding a new vertex v and an m-multicherry with middle vertex v.

  9. (V,E2,E3): Buildingup a cherry tree 1 1 1 3 3 2 4 2 2 5 5 6 3 1 3 1 2 4 2 4 7 Example of a 2-multitree (cherry tree) E2= { {1,2}, {1,3}, {2,3} } E3= { {1,2,3} } E2= { {1,2} } E3=  E2= { {1,2}, {1,3}, {2,3}, {2,4}, {3,4} } E3= { {1,2,3}, {2,3,4} } Recursion is not unique. We can add the vertices for example in the order 2,3,7,4,1,5,6 to obtain the same cherry tree.

  10. DEF: The weight of an m-multitree =(V,E2, …,Em+1) with V={1,…,n} is defined by THEOREM: For any m-multitree =(V,E2, …,Em+1) with V={1,…,n} we have Upper bounds by m-multitrees Special case m=1 provides us the Hunter-Worsley bound.

  11. An m-multitree is completely determined by its set of vertices and set of edges. In other words, although an m-multitree is a hypergraph, it can be identified by its graph. A5 A3 A4 A7 A6 A1 A2 Some properties of m-multitrees m-multitrees provide us m+1 order upper bounds. There are only O(n) intersection probabilities involved in an m-multitree bound. The number of intersection probabilities with at most m+1 events is O(nm+1), an m-multitree bound uses only O(n) out of them. Useful when the intersection probabilities have to be evaluated, e.g. estimate of multivariate distr. function.

  12. ALGORITHM TO FIND HEAVY M-MULTITREE The theorem enables us to find a heavy m-multitree by starting from the heaviest (1-multi)tree and increasing m step by step to improve on the bound. Only the intersection probabilities involved in the multitree bound are needed, i.e. O(n). Unfortunately, the greedy algorithm generally does not provide the maximum weight m-multitree if m > 1. THEOREM: An m-multitree =(V,E2, …,Em+1) with V={1,…,n} can be extended to an m+1-multitree ’=(V,E’2, …,E’m+2) with w(’) w(). (Extension means that EiE’i for every i.)

  13. 0.8 if i  j Covariance rij = 1.0 if i = j Lower bounds for a 30-variate normal distribution function value sec. bound name sec. bound name Marginal function values were computed by Genz’s Fortran code SADMVN and IMSL subroutines MDNOR and MDBNOR. x1=1.55, x2=1.6, …,x29=2.95, x30=3.0

  14. Covariance for all 1 i  j  30. Lower bounds for a 30-variate normal distribution function value sec. bound name sec. bound name Normal random variables with this covariance matrix are used for estimatingAmerican Option Price. x1 = x2 = … = x29 = x30= 2.5

  15. h 0, m  1 integers (h,m)-hypermultitree (recursive def.) DEF: An (h,m)-hypermultitree is a hypergraph of the form (V, hE2,…, hEi,…,hEm+1), set of vertices set of hyperedges with h+i vertices i.) (0,m)-hypermultitrees are the same as m-multitrees, 0Ei Ei . ii.) The smallest (h,m)-hypermultitree has h+m vertices, and hEi is the set of all subsets of V with h+i vertices (hEm+1 = ).

  16. and Example: Take a (0,1)-hypermultitree (i.e. tree) =( {a,b,c,d}, { {a,b},{a,c},{a,d} } ) on . (1,1)-hypermultitree =( {a,b,c,d}, 1E2) ”old multitree” b Hyperedges added at this step: {a,b,v}, {a,c,v} and {a,d,v}. a c v d iii.) From an (h,m)-hypermultitree =(V, hE2,…,hEm+1), we can obtain another (h,m)-hypermultitree ’=(V’, hE’2,…,hE’m+1), by adding a new vertex v and some hyperedges in the following way. Let =(V, h-1E*2,…,h-1E*m+1) be an arbitrary (h-1,m)-hypermultitree (on ). We add the hyperedges of  extended by v to obtain ’, i.e.

  17. DEF: The weight of an (h,m)-hypermultitree =(V,hE2, …,hEm+1) with V = {1,…,n} is defined by THEOREM: For any (h,m)-hypermultitree =(V, hE2, …, hEm+1) with V = {1,…,n} the following inequalities hold i.) if h is even ii.) if h is odd where Bounds by (h,m)-hypermultitrees Special case m = 1 Tomescu bounds; h = 0 the multitree bounds.

  18. Some properties of (h,m)-hypermultitrees • h+m+1 order bounds, • lower bounds if h is even, upper bounds if h is odd, • the heavier is the hypermultitree, the better is the bound, • based on O(nh+1) intersection probabilities. Remark: Consequently, for upper bounds h = 0, for lower bounds h = 1 is a cost-effective choice, especially in applications where the intersection probabilities have to be evaluated.

  19. ALGORITHM TO FIND HEAVY (1,m)-HYPERMULTITREE We find a heavy (1,1)-hypermultitree by a greedy algorithm. Based on the theorem we extend this (1,1)-hypermultitree to a (1,2)-hypermultitree that we extend to a (1,3)-hypermultitree etc. At the end of the algorithm we obtain a (1,m)-hypermultitree. This stepwise extension can be done in a single step, i.e. the initial (1,1)-hypermultitree can be extended in a single step to a (1,m)-hypermultitree that has higher weight. THEOREM: An (h,m)-hypermultitree =(V, hE2 , …, hEm+1) with V={1,…,n} can be extended to an (h,m+1)-hypermultitree ’=(V, hE’2, …,hE’m+1) with w(’) w(). (Extension means that EiE’i for every i.)

  20. There is a short formula to compute m-multitree ( (1,m)-hypermultitree ) bounds containing n-m ( n-m 2 ) intersection probabilities of the type altogether m+1 (m+2) events Short formulae for the bounds In other words, there are some complement events included in the above intersection probabilities. They can be evaluated in applications where bounds for values of multivariate distribution function values are sought.

  21. Lower bounds Upper bounds seconds seconds Computation were made by a CELERON II 850MHz computer. x1=1.84, x2=1.88, …,x29=2.96, x30=3.0 Covariance Marginal function values were computed by Genz’s Fortran code SADMVN and IMSL subroutines MDNOR and MDBNOR. for all 1 i  j  30.

  22. real value  = simulation based on lower bounds  = simulation based on upper bounds Simulating multivariate normal distribution function values Tamás Szántai developed and implemented a method to simulate multivariate normal distribution function values based on multitrees and hypermultitrees. The code simulates the difference between a lower (upper) bound and the real function value and calculates  () / see the figure /. Szántai showed that  and  are negatively correlated unbiased estimators, thus a + b is an unbiased estimator of the function value with lower variance, where a+b =1, a>0,b>0. Values of a and b are chosen optimally (variance is minimized).

  23. Simulating multivariate normal distribution function values (cont’d) Another version: Let  be simulated function value obtained by the crude Monte-Carlo method. Then a + b + c is an unbiased estimator of the function value, where a+b+c =1, a > 0, b > 0 and c > 0. Szántai’s code turned out to be several thousand times more effective than the crude Monte-Carlo simulation when the function value is high and the dimension is 20-50. The gain in effectiveness is somewhat less but still significant for medium (low) function values 20-50 (20-30) dimension. Some care must be taken to select m for the m-multitree ( (1,m)-hypermultitree ) bounds.

  24. t-cherry trees t-cherry tree not t-cherry tree vertex 1 and 4 are not adjacent DEF: A cherry tree (2-multitree) is called a t-cherry tree if the two non-middle vertices of every cherry are adjacent.

  25. t-cherry trees (cont’d) THEOREM: A t-cherry tree bound can always be identified as the objective function value of the dual feasible basis in the Boolean probability bounding problem. REM: The Boolean probability bounding problem is a linear programming problem with 2n- 1 number of variables (n is the number of events). REM: The same is not true for an arbitrary cherry tree. CONJECTURE: The above theorem can be generalized to m-multitrees.

  26. Open Questions Are there tight lower bounds for Pr(A1… An) of arbitrary order that are based on O(n) number of intersection probabilities? Is there a polynomial time algorithm that finds the maximum weight m-multitree if m > 1? If not, then can the family of all m-multitrees on n vertices be extended to a matroid? Same question for (h,m)-hypermultitrees. What are the best lower or upper bounds of a certain order? We have seen that t-cherry trees provide us the best third order upper bounds on certain examples, but not on all of them.

  27. . This is the probability of the intersection of events Applying lower (upper) bound instead of the intersection probability shrinks (extends) the set of feasible solutions. The underlying Stoch. Optim. Problem Subject to Where Y is a random variable with known distribution, and p is a constant, typically between 0.9 and 1.

  28. Strategy 2 (based on upper bounds): Solve the problem with an upper bound in the place of the intersection probability Iterate Step 1. using a better bound until feasibility holds or using the original probabilistic constraint As another application, Tamás Szántai restricted the search interval with bounds in his line search method to find the boundary points of feasible solutions. Strategy 1 (based on lower bounds): • Solve the problem with a lower bound in the place of the intersection probability • Iterate Step 1. using a better bound until optimality holds or using the original probabilistic constraint

  29. If are concave functions andY has a Corr.: the set of x satisfying the probabilistic constraints is convex. Prékopa’s theorem continuous probability distribution with logarithmically concave probability density function, then the function is also logarithmically concave. Exa: Let the probabilistic constraints in the underlying problem be whereY has multivariate joint normal distribution.

  30. Is the set of feasible solutions convex when bounds are used? Def: An m-multitree is called an m-multistar if the non-middle vertex set of its multicherries are identical. Rem: An m-multistar can be extended to an (m+1)-multistar that provides us a better bound. Th: Bounds based on multistars yield logarithmically concave function in the probabilistic constraints, i.e. and are arbitrary positive numbers. is logarithmically concave if the correlations of Yj are Exa: Covariance for all 1 i < j.

  31. REFERENCES Bukszár, J. Upper Bounds for the Probability of a Union by Multitrees, Advances in Applied Probability 33 (2), 437-452, 2001. Bukszár, J. Prékopa, A. Probability Bounds with Cherry Trees, Mathematics of Operations Research, 26 (1), 174-192, 2001. Szántai, T. Bukszár, J. Probability Bounds given by Hypercherry Trees, Optimization Methods and Software, 17 (3), 409-422, 2002. Bukszár, J. Hypermultitrees and Bonferroni Inequalities, Mathematical Inequalities and Applications, 6 (4), 727-743,2003. Galambos, J. Simonelli, I. Bonferroni-type Inequalities with Applications, Springer-Verlag, NY, 1996. Genz, A Numerical Computation of the Multivariate Normal Probabilities, J. Comput. Graph. Stat. 1,141-150, 1992. Grable, DA. Sharpened Bonferroni Inequalities, J. Combin. Theory Ser. B 57, 131-137, 1993. Hoppe, FM., Seneta, E. A Bonferroni-type Identity and Permutation Bounds, International Statistical Review 58, 3, 253-261, 1990. Hunter, D. An Upper Bound for the Probability of a Union, J. Appl. Prob. 13, 597-603, 1976. Prékopa, A. Stochastic Programming, Kluwer Academic Publishers, Dordrecht, 1995. Prékopa, A. Boole-Bonferroni Inequalities and Linear Programming, Oper. Res. 36, 145-162,1988. Prékopa, A. Sharp Bounds on Probabilities Using Linear Programming, Oper. Res. 38, 227-239, 1990. Prékopa, A. The Discrete Moment Problem and Linear Programming, Discrete Applied Mathematics, 27, 235-254, 1990. Sobel, M. Uppuluri, VRR On Bonferroni-type Inequalities of the Same Degree for the Probability of Unions and Intersections, Ann. Math. Statist. 43, 1549-1558, 1972. Tomescu, I. Hypertrees and Bonferroni Inequalities, J. Combin. Theory Ser. B 41, 209-217, 1986. Worsley, KJ. An Improved Bonferroni Inequality and Applications, Biometrika 69, 297-302, 1982.

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