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                Department of Mathematical Sciences                       
                   The Johns Hopkins University                           
                                                                        
                              SEMINAR                                
                                                                        
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Professor Daniel Q. Naiman                        January 27, 2000
Department of Mathematical Sciences               304 Whitehead Hall
The Johns Hopkins University                      Preseminar: 3:00 p.m.
                                                  Refreshments: 3:30 p.m.
                                                  Seminar: 4:00 p.m.

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 ABSTRACT TUBES, INCLUSION-EXCLUSION INEQUALITIES, AND RELIABILITY BOUNDS

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                             ABSTRACT

Probabilists and statisticians use inclusion-exclusion inequalities to
bound or estimate probabilities of unions of events.  Examples from
statistics and network reliability in which the evaluation of such
probabilities is desired will be presented.  In 1992, Naiman and Wynn
introduced the notion of an abstract tube and described why it is relevant
and useful in this context.  This notion will be reviewed and key properties
of abstract tubes will be presented.

In particular, associated with any abstract tube is an inclusion-exclusion
identity and corresponding truncation inequalities.  Classical
inclusion-exclusion arises as a special case, but there are theorems to the
effect that the corresponding truncation inequalities are typically weaker
than can be obtained when geometric information about the underlying events
is available.  New abstract tubes leading to improved inequalities for
network reliability due to Klaus Dohmen will be described, and some recent
related work of Naiman and Wynn will be presented.  Also, applications of
abstract tubes to estimating p-values via importance sampling, as
developed in recent work by Naiman and Priebe, will be presented.

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