Dan Naiman
John Hopkins University
Baltimore, Maryland
Abstract Tubes, Inclusion-Exclusion Inequalities, and Reliability Bounds
Probabilists and statisticians use inclusion-exclusion inequalities to
bound 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 this context. This notion will be reviewed and key
properties of abstract tubes will be described. 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 related to
network reliability due to Klaus Dohmen will be described, and some
recent related work of Naiman and Wynn will be presented.
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