## Imprecise Probabilistic Graphical Models: Equivalent Representations, Inference Algorithms and Applications

**Author:** Alessandro Antonucci

**Abstract:** Credal networks are probabilistic graphical models that extend Bayesian
nets to deal with imprecision in probability, and can actually be
regarded as sets of Bayesian nets. Credal nets appear to be powerful
means to represent and deal with many important and challenging problems
in uncertain reasoning. The counterpart of having more freedom in
the modeling phase is an increased inferential complexity of inferences,
e.g., the so-called belief updating becomes a hard task even on relatively
simple topologies. In this thesis, I start my investigation on credal
networks by considering equivalent representations of those models.
More specifically, I first deliver a new graphical language, which
is called decision-theoretic being inspired by the formalism of decision
graphs, for a unified representation of credal networks of any kind.
I also present another representation, which is called binarization,
being in fact a reformulation of a credal network solely based on
binary variables. Remarkably, I prove that if a credal net is first
reformulated by its decision-theoretic representation and then by
the corresponding binarization, the resulting representation is completely
equivalent. An equivalence relation between Bayesian and credal nets,
when the reason for the missingness of some of the variables in the
Bayesian nets is unknown, is also provided. The developed equivalent
representations are applied to inference problems. First, I show
that, by a decision-theoretic formulation, the algorithms that have
been already designed for credal networks, which are mostly referred
to a specific class of models, called separately specified nets,
can be generalized to credal networks of any kind. Similar formalisms
are also employed to solve inference and classification problems
with missing observations. I also present a state-ofthe- art updating
algorithm which is based on the equivalent binary representation.
This algorithm, called GL2U, offers an efficient procedure for approximate
updating of general credal nets. The quality of the overall approximation
is investigated by promising numerical experiments. As a further
theoretical investigation, I consider a classification problem for
Bayesian networks for which a hardness proof together with a fast
algorithm for a subclass of models is provided. Finally, two real-world
applications of credal networks are presented. First, I consider
a military identification problem, consisting in the detection of
the goal of an intruder entering a no-fly area. The problem, together
with the necessary fusion of the information gathered by the sensors
is mapped by our techniques into a credal network updating task.
The solution is then obtained by the GL2U algorithm. The second application
is an environmental model for hazard assessment of debris flows by
credal networks. A credal network evaluates the level of risk, corresponding
to the observed values of the triggering factors, for this specific
natural hazard. For some factors, whose observations are more difficult,
the corresponding soft evidential information is embedded by our
formalism into the structure of the network. This model is employed
for extensive numerical analysis on the Swiss territory.

**Year:** 2008.

**Details:** Ph.D. thesis, Università della Svizzera Italiana.

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