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Improved Bonferroni Inequalities and Binomially Bounded Functions
Electronic Notes in Discrete Mathematics 28 (2007) 91–93 www.elsevier.com/locate/endm
Improved Bonferroni Inequalities and Binomially Bounded Functions Klaus Dohmen and Peter Tittmann Hochschule Mittweida, Technikumplatz 17, 09648 Mittweida, Germany Abstract We introduce the concept of a binomially bounded function and associate with each such function an improved Bonferroni-type inequality.
1
Introduction
Let Av , v ∈ V , be finitely many events in some probability space (Ω, A, P ). The classical Bonferroni inequalities state that for any r ∈ N0 , (1) Av ≥ (−1)r (−1)|I|−1P Ai (−1)r P v∈V
I⊆V 0<|I|≤r
i∈I
where N0 := {0, 1, 2, . . . }. There are a lot of improvements and applications of these inequalities, see e.g., [1,2]. In this note, we establish an improvement of (1) by introducing an additional term on the right-hand side, which involves the (r + 1)-subsets of V and a so-called binomially bounded function. By choosing this function appropriately, several well-known and new results are obtained.
2
Binomially bounded functions
Definition 2.1 For any finite set V and any k ∈ N we use [V ]k to denote the set of k-subsets of V . A function f : [V ]k → R is called binomially bounded if |W | − 1 f (I) ≤ k−1 I⊆W |I|=k
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.01.013
92
K. Dohmen, P. Tittmann / Electronic Notes in Discrete Mathematics 28 (2007) 91–93
for any non-empty subset W of V . Example 2.2 Let V be a finite set and Q a probability distribution on V . Then, for any k ∈ N, the restriction of Q to [V ]k is binomially bounded. In particular, I → |I|/|V | for any I ∈ [V ]k defines a binomially bounded function. Example 2.3 Let G = (V, E) be a finite tree and f : [V ]2 → {0, 1} denote the edge indicator function of G. Then, for any non-empty subset W of V ,
f (I) = m(G[W ]) ≤ |W | − 1 ,
I⊆W |I|=2
where m(G[W ]) denotes the number of edges in the subgraph induced by W . Hence, the edge indicator function of any finite tree is binomially bounded.
3
Main result and consequences
Theorem 3.1 Let Av , v ∈ V , be finitely many events in some probability space (Ω, A, P ). Then, for any r ∈ N0 and any binomially bounded function f : [V ]r+1 → R we have (−1)r P
v∈V
Av
≥ (−1)r
(−1)|I|−1P
I⊆V 0<|I|≤r
i∈I
Ai +
P
I⊆V |I|=r+1
Ai
f (I).
i∈I
Remark 3.2 It can be shown that if the preceding inequality holds for any probability measure and any finite family of events, then f is binomially bounded. Thus, the requirement on f in Theorem 3.1 cannot be weakened. By choosing f appropriately, several known and new results are obtained. Corollary 3.3 (Galambos [3]) Let Av , v ∈ V , be events in some probability space (Ω, A, P ), where V is assumed to be finite and non-empty. Then, for any r ∈ N0 we have r + 1 Av ≥ (−1)r (−1)|I|−1P Ai + P Ai . (−1)r P |V | I⊆V I⊆V v∈V i∈I i∈I 0<|I|≤r
|I|=r+1
Corollary 3.4 (Hunter [5]) Let Av , v ∈ V , be finitely many events in some probability space (Ω, A, P ). Then, for any tree G = (V, E) on the index-set of
K. Dohmen, P. Tittmann / Electronic Notes in Discrete Mathematics 28 (2007) 91–93
93
these events we have P Av ≤ P (Av ) − P (Ai ∩ Aj ) . v∈V
v∈V
{i,j}∈E
Corollary 3.5 If, in addition to the requirements of Corollary 3.3, P (Av ) > 0 for at least one v ∈ V , then (−1)r P Av ≥ (−1)r (−1)|I|−1 P Ai v∈V
I⊆V 0<|I|≤r
+
P
I⊆V |I|=r+1
i∈I
Ai
i∈I
i∈I
P (Ai )
P (Av ).
v∈V
Remark 3.6 Hunter’s inequality (Corollary 3.4) has been generalized to uniform hypertrees by Tomescu [6] and sparse uniform hypergraphs by Grable [4]. Both generalizations can be deduced using binomially bounded functions. Examples where Theorem 3.1 outperforms Hunter’s inequality can be given by choosing f in such a way that some function values of f are negative. For details and proofs the reader is referred to the extended version of this note.
References [1] Dohmen K., “Improved Bonferroni Inequalities via Abstract Tubes, Inequalities and Identities of Inclusion-Exclusion Type,” Lecture Notes in Mathematics, No. 1826, Springer-Verlag, Berlin Heidelberg, 2003. [2] Galambos J., and I. Simonelli, “Bonferroni-type Inequalities with Applications,” Springer Series in Statistics, Probability and Its Applications, Springer-Verlag, New York, 1996. [3] Galambos J., Methods for proving Bonferroni-type inequalities, J. London Math. Soc. 2 (1975), 561–564. [4] Grable D.A., Sharpened Bonferroni inequalities, J. Comb. Theory Ser. B 57 (1993), 131–137. [5] Hunter D., An upper bound for the probability of a union, J. Appl. Prob. 13 (1976), 597–603. [6] Tomescu I., Hypertrees and Bonferroni Inequalities, J. Combin. Theory Ser. B 41 (1986), 209–217.
Improved Bonferroni Inequalities and Binomially Bounded Functions Klaus Dohmen and Peter Tittmann Hochschule Mittweida, Technikumplatz 17, 09648 Mittweida, Germany Abstract We introduce the concept of a binomially bounded function and associate with each such function an improved Bonferroni-type inequality.
1
Introduction
Let Av , v ∈ V , be finitely many events in some probability space (Ω, A, P ). The classical Bonferroni inequalities state that for any r ∈ N0 , (1) Av ≥ (−1)r (−1)|I|−1P Ai (−1)r P v∈V
I⊆V 0<|I|≤r
i∈I
where N0 := {0, 1, 2, . . . }. There are a lot of improvements and applications of these inequalities, see e.g., [1,2]. In this note, we establish an improvement of (1) by introducing an additional term on the right-hand side, which involves the (r + 1)-subsets of V and a so-called binomially bounded function. By choosing this function appropriately, several well-known and new results are obtained.
2
Binomially bounded functions
Definition 2.1 For any finite set V and any k ∈ N we use [V ]k to denote the set of k-subsets of V . A function f : [V ]k → R is called binomially bounded if |W | − 1 f (I) ≤ k−1 I⊆W |I|=k
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.01.013
92
K. Dohmen, P. Tittmann / Electronic Notes in Discrete Mathematics 28 (2007) 91–93
for any non-empty subset W of V . Example 2.2 Let V be a finite set and Q a probability distribution on V . Then, for any k ∈ N, the restriction of Q to [V ]k is binomially bounded. In particular, I → |I|/|V | for any I ∈ [V ]k defines a binomially bounded function. Example 2.3 Let G = (V, E) be a finite tree and f : [V ]2 → {0, 1} denote the edge indicator function of G. Then, for any non-empty subset W of V ,
f (I) = m(G[W ]) ≤ |W | − 1 ,
I⊆W |I|=2
where m(G[W ]) denotes the number of edges in the subgraph induced by W . Hence, the edge indicator function of any finite tree is binomially bounded.
3
Main result and consequences
Theorem 3.1 Let Av , v ∈ V , be finitely many events in some probability space (Ω, A, P ). Then, for any r ∈ N0 and any binomially bounded function f : [V ]r+1 → R we have (−1)r P
v∈V
Av
≥ (−1)r
(−1)|I|−1P
I⊆V 0<|I|≤r
i∈I
Ai +
P
I⊆V |I|=r+1
Ai
f (I).
i∈I
Remark 3.2 It can be shown that if the preceding inequality holds for any probability measure and any finite family of events, then f is binomially bounded. Thus, the requirement on f in Theorem 3.1 cannot be weakened. By choosing f appropriately, several known and new results are obtained. Corollary 3.3 (Galambos [3]) Let Av , v ∈ V , be events in some probability space (Ω, A, P ), where V is assumed to be finite and non-empty. Then, for any r ∈ N0 we have r + 1 Av ≥ (−1)r (−1)|I|−1P Ai + P Ai . (−1)r P |V | I⊆V I⊆V v∈V i∈I i∈I 0<|I|≤r
|I|=r+1
Corollary 3.4 (Hunter [5]) Let Av , v ∈ V , be finitely many events in some probability space (Ω, A, P ). Then, for any tree G = (V, E) on the index-set of
K. Dohmen, P. Tittmann / Electronic Notes in Discrete Mathematics 28 (2007) 91–93
93
these events we have P Av ≤ P (Av ) − P (Ai ∩ Aj ) . v∈V
v∈V
{i,j}∈E
Corollary 3.5 If, in addition to the requirements of Corollary 3.3, P (Av ) > 0 for at least one v ∈ V , then (−1)r P Av ≥ (−1)r (−1)|I|−1 P Ai v∈V
I⊆V 0<|I|≤r
+
P
I⊆V |I|=r+1
i∈I
Ai
i∈I
i∈I
P (Ai )
P (Av ).
v∈V
Remark 3.6 Hunter’s inequality (Corollary 3.4) has been generalized to uniform hypertrees by Tomescu [6] and sparse uniform hypergraphs by Grable [4]. Both generalizations can be deduced using binomially bounded functions. Examples where Theorem 3.1 outperforms Hunter’s inequality can be given by choosing f in such a way that some function values of f are negative. For details and proofs the reader is referred to the extended version of this note.
References [1] Dohmen K., “Improved Bonferroni Inequalities via Abstract Tubes, Inequalities and Identities of Inclusion-Exclusion Type,” Lecture Notes in Mathematics, No. 1826, Springer-Verlag, Berlin Heidelberg, 2003. [2] Galambos J., and I. Simonelli, “Bonferroni-type Inequalities with Applications,” Springer Series in Statistics, Probability and Its Applications, Springer-Verlag, New York, 1996. [3] Galambos J., Methods for proving Bonferroni-type inequalities, J. London Math. Soc. 2 (1975), 561–564. [4] Grable D.A., Sharpened Bonferroni inequalities, J. Comb. Theory Ser. B 57 (1993), 131–137. [5] Hunter D., An upper bound for the probability of a union, J. Appl. Prob. 13 (1976), 597–603. [6] Tomescu I., Hypertrees and Bonferroni Inequalities, J. Combin. Theory Ser. B 41 (1986), 209–217.