Adv. Appl. Prob. 33, 437–452 (2001) Printed in Northern Ireland Applied Probability Trust 2001
UPPER BOUNDS FOR THE PROBABILITY OF A UNION BY MULTITREES JÓZSEF BUKSZÁR,∗ University of Miskolc
Abstract The problem of finding bounds for P(A1 ∪ · · · ∪ An ) based on P(Ak1 ∩ · · · ∩ Aki ) (1 ≤ k1 < · · · < ki ≤ n, i = 1, . . . , d) goes back to Boole (1854), (1868) and Bonferroni (1937). In this paper upper bounds are presented using methods in graph theory. The main theorem is a common generalization of the earlier results of Hunter, Worsley and recent results of Prékopa and the author. Algorithms are given to compute bounds. Examples for bounding values of multivariate normal distribution functions are presented. Keywords: Bonferroni-type bound; hypergraph; multivariate normal distribution AMS 2000 Subject Classification: Primary 60C05
1. Introduction Let A1 , . . . , An be arbitrary events in a probability space ( , A, P). Our purpose is to present upper bounds for the probability P(A1 ∪ · · · ∪ An ) based on some of the probabilities P(Ak1 ∩ · · · ∩ Aki ) (1 ≤ k1 < · · · < ki ≤ n, i = 1, . . . , d). The bounds of this type are called dth order upper bounds. For m = d − 1, we introduce the notion of a hypergraph with n vertices, called an m-multitree, and assign an upper bound to each m-multitree. For m = 1 we obtain the Hunter–Worsley bound (see [10] and [16]) and for m = 2 the cherry-tree bound (see [5]). It will be shown that for an m-multitree an (m + 1)-multitree can be constructed such that the upper bound assigned to it is at least as good as the one assigned to the original m-multitree. We show that it is possible to create a bound better than the Hunter–Worsley bound or the cherry-tree bound, at the expense of increasing m, if we know (or if we can compute) some additional intersection probabilities of the events. We prove that a multitree is completely determined by its set of vertices and edges. Thus, the bound is computed by the use of graphs rather than hypergraphs. Since the evaluation of the probabilities P(Ak1 ∩ · · · ∩ Aki ) is quite costly, it is crucial that we use only a few of them for creating the bound. We present an algorithm to find an m-multitree bound based on P(Ak1 ) (1 ≤ k1 ≤ n), P(Ak1 ∩ Ak2 ) (1 ≤ k1 < k2 ≤ n) and O(n) P(Ak1 ∩ · · · ∩ Aki ) (1 ≤ k1 < · · · < ki ≤ n, i = 3, . . . , m + 1) out of all the O(nm+1 ) numbers. The algorithm is a recursion with respect to m where the m-multitree bound is improved whenever m is increased. Lower bounds for multivariate normal distribution function values are presented as examples.
Received 31 August 1999; revision received 15 March 2001. ∗ Postal address: Institute of Mathematics, University of Miskolc, Miskolc, H-3515, Hungary. Email address:
[email protected] Partly supported by OTKA T032369.
437
438
J. BUKSZÁR
1
3
2
5
4 6 Figure 1.
2. Bounds by m-multitrees In this section we introduce the concept of an m-multitree and use it to create upper bounds for the probability of the union. We prove that an m-multitree can be extended to an (m + 1)multitree while the bound is improved. Definition 1. Let m be a positive integer. An m-multicherry is a hypergraph of the form (V , E2 , . . . , Em+1 ), where V = {v1 , . . . , vm+1 } is the set of vertices and the family of hyperedges Ei is the set of all subsets of {v1 , . . . , vm } containing i vertices with vm+1 included, i.e. Ei = {H | vm+1 ∈ H ⊆ {v1 , . . . , vm+1 }, |H | = i}. The vertex vm+1 is called the dominating vertex of the m-multicherry. The m-multicherry with dominating vertex vm+1 and nondominating vertices v1 , . . . , vm is denoted by ({v1 , . . . , vm }, vm+1 ). Note that a 1-multicherry is a single edge together with its incident vertices, and a 2multicherry is a cherry (see [5]). Definition 2. Let m be a positive integer. An m-multitree is a hypergraph of the form (V , E2 , . . . , Em+1 ), where V is the set of vertices and the Ei s are sets of hyperedges containing i vertices. An m-multitree is defined recursively by the following two rules. (i) The smallest m-multitree = (V , E2 , . . . , Em+1 ) has m vertices and Ei is the set of all subsets of V containing i vertices (here Em+1 = ∅). (ii) From an m-multitree = (V , E2 , . . . , Em+1 ) we can obtain a new m-multitree = (V , E2 , . . . , Em+1 ) by adjoining an m-multicherry ({v1 , . . . , vm }, vm+1 ), where v1 , . . . , vm ∈ V and vm+1 is a new vertex (i.e. vm+1 ∈ / V ). In other words, V = V ∪ {vm+1 }, Ei = Ei ∪ {H | vm+1 ∈ H ⊆ {v1 , . . . , vm+1 }, |H | = i}. Figure 1 illustrates how a 3-multitree = (V , E2 , E3 , E4 ) can be obtained from the smallest 3-multitree. Given 1, 2, 3, the edges {1, 2}, {1, 3}, {2, 3} and the hyperedge {1, 2, 3}, we adjoin successively the 3-multicherries ({1, 2, 3}, 4), ({1, 3, 4}, 5) and ({2, 3, 5}, 6), as shown in the figure. The edges of a 3-multicherry are drawn with the same style of line. The vertices and hyperedges of are V = {1, 2, 3, 4, 5, 6}, E2 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {5, 6}}, E3 = {{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 3, 5}, {1, 4, 5}, {3, 4, 5}, {2, 3, 6}, {2, 5, 6}, {3, 5, 6}}, E4 = {{1, 2, 3, 4}, {1, 3, 4, 5}, {2, 3, 5, 6}}.
Union by multitrees
439
Note that a 1-multitree is a normal tree and a 2-multitree is a cherry tree (see [5]). Remark 1. It follows from the definition that if = (V , E2 , . . . , Em+1 ) is an m-multitree with |V | = n, then m m |Ei | = + (n − m), i i−1 for all i = 2, . . . , m + 1. Remark 2. An m-multitree with m + 1 vertices has a unique structure: its hyperedges are all subsets of its vertex set containing at least two vertices. Definition 3. Let A1 , . . . , An be arbitrary events. Then the weight of an m-multitree = (V , E2 , . . . , Em+1 ) is defined to be P(Al1 ∩ Al2 ) − P(Al1 ∩ Al2 ∩ Al3 ) + · · · w() = {l1 ,l2 }∈E2
+ (−1)
m+1
{l1 ,l2 ,l3 }∈E3
P(Al1 ∩ · · · ∩ Alm+1 )
{l1 ,...,lm+1 }∈Em+1
Theorem 1. Let A1 , . . . , An be arbitrary events and = (V , E2 , . . . , Em+1 ) an arbitrary m-multitree with V = {1, . . . , n}. Then we have the inequality
where S1 =
n
P(A1 ∪ · · · ∪ An ) ≤ S1 − w(),
(1)
i=1 P(Ai ).
Proof. Without loss of generality we can assume that is obtained by the following recursion. We start from the m-multitree (m)
(m)
(m) = (V (m) , E2 , . . . , Em+1 ), (m)
where V (m) = {1, . . . , m}, Ei (i = 2, . . . , m) is the set of all subsets of V (m) containing i ele(m) ments and Em+1 = ∅. We construct the sequence of m-multitrees (m+1) , (m+2) , . . . , (n) in (j ) (j ) (j −1) such a way that we obtain (j ) = (V (j ) , E2 , . . . , Em+1 ) from (j −1) = (V (j −1) , E2 ,..., (j −1) Em+1 ) by adjoining the m-multicherry ({ij,1 , . . . , ij,m }, j ), where 1 ≤ ij,1 < · · · < ij,m ≤ j − 1. Finally, we obtain = (V , E2 , . . . , Em+1 ) mentioned in the theorem as = (n) . The recursion gives that Em+1 = ({ij,1 , . . . , ij,m , j } | j = m + 1, . . . , n) and Ei =
n
{K | j ∈ K ⊆ {ij,1 , . . . , ij,m , j }, |K| = i} ∪ {K | K ⊆ {1, . . . , m}, |K| = i},
j =m+1
for all i = 2, . . . , m. We need the following relations P(A1 ∪ · · · ∪ An ) = P(A1 ∪ · · · ∪ An−1 ) + P(An ) − P((A1 ∪ · · · ∪ An−1 ) ∩ An ) = P(A1 ∪ · · · ∪ An−2 ) + P(An−1 ) − P((A1 ∪ · · · ∪ An−2 ) ∩ An−1 ) + P(An ) − P((A1 ∪ · · · ∪ An−1 ) ∩ An ) = ···
440
J. BUKSZÁR
· · · = S1 − = S1 −
n
P((A1 ∪ · · · ∪ Aj −1 ) ∩ Aj )
j =2 m j =2
≤ S1 −
m
P((A1 ∪ · · · ∪ Aj −1 ) ∩ Aj ) +
j =2
= S1 −
n
P((A1 ∪ · · · ∪ Aj −1 ) ∩ Aj ) +
j =m+1 n
P((A1 ∪ · · · ∪ Aj −1 ) ∩ Aj ) P((Aij,1 ∪ · · · ∪ Aij,m ) ∩ Aj )
j =m+1
m
P((A1 ∩ Aj ) ∪ · · · ∪ (Aj −1 ∩ Aj ))
j =2
P((Aij,1 ∩ Aj ) ∪ · · · ∪ (Aij,m ∩ Aj )) .
n
+
j =m+1
If we apply the inclusion–exclusion formula for P((A1 ∩ Aj ) ∪ · · · ∪ (Aj −1 ∩ Aj )) and P((Aij,1 ∩ Aj ) ∪ · · · ∪ (Aij,m ∩ Aj )), then we obtain S1 −
m
P((A1 ∩ Aj ) ∪ · · · ∪ (Aj −1 ∩ Aj )) +
j =2
= S1 −
n
P((Aij,1 ∩ Aj ) ∪ · · · ∪ (Aij,m ∩ Aj ))
j =m+1
m
j =2 1≤h1 ≤j −1
+ (−1)
P(Ah1 ∩ Aj ) −
j −1
P(Ah1 ∩ Ah2 ∩ Aj ) + · · ·
1≤h1
