Multivariate Bonferroni-Type Inequalities: Theory and Applications
presents a systematic account of research discoveries on
multivariate Bonferroni-type inequalities published in the past
decade. The emergence of new bounding approaches pushes the
conventional definitions of optimal inequalities and demands new
insights into linear and Frechet optimality. The book explores
these advances in bounding techniques with corresponding innovative
applications. It presents the method of linear programming for
multivariate bounds, multivariate hybrid bounds, sub-Markovian
bounds, and bounds using Hamilton circuits. The first half of the
book describes basic concepts and methods in probability
inequalities. The author introduces the classification of
univariate and multivariate bounds with optimality, discusses
multivariate bounds using indicator functions, and explores linear
programming for bivariate upper and lower bounds. The second half
addresses bounding results and applications of multivariate
Bonferroni-type inequalities. The book shows how to construct new
multiple testing procedures with probability upper bounds and goes
beyond bivariate upper bounds by considering vectorized upper and
hybrid bounds. It presents an optimization algorithm for bivariate
and multivariate lower bounds and covers vectorized
high-dimensional lower bounds with refinements, such as
Hamilton-type circuits and sub-Markovian events. The book concludes
with applications of probability inequalities in molecular cancer
therapy, big data analysis, and more.
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