Title: Axioms for independence
Student: vacancy
Supervisor: Peter de Waal
ECTS: 7.5 or 15
Related courses: Probabilistic Reasoning
Description:

For a set of variables we can define an independence relation as a set
of triplets (X,Y|Z), the so-called independence statements. The
statement (X,Y|Z) can be interpreted as "variables X and Y are
independent given Z". In order to have this relation represent a
notion of independence that corresponds to, for instance, independence
as we know from random variables, the set of triplets must obey a
number of axioms, for instance symmetry.

Any independence relation can be characterised by a small basic set
of statemens, the kernel. Any independence statement can then be
derived by (repeated) application of the axioms to the kernel.

There exist algorithms to determine the axiomatic closure of a given
kernel. Two algorithms have also been developed to tackle the
representation problem the other way round. These algorithms determine
the kernel set given a set of independence statements. We have no good
insight, however, into the complexity of the full set of independence
statements, given only the kernel statements and vice versa. We also
would like to investigate the complexities of the mentioned algorithms
and we would like to obtain insight into which properties of
independence relations influence these complexities.

A possible topic for this experimentation project is to implement some
of these algorithms and to investigate the complexity of these
algorithms. Another topic is to investigate which properties of an
independence relation determine the size of the kernel, and which
properties of the kernel determine the size of the axiomatic closure.

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Title: Complexity in Multi-dimensional Bayesian Classifiers
Student: vacancy
Supervisor: Peter de Waal
ECTS: 7,5 or 15
Related courses: Probabilistic Reasoning, Advanced Data Mining
Description

Bayesian network classifiers are popular tools in machine learning for
classification of data instances. A classifier tries to determine the
best guess for the class of a given data instance, given observations
of other features in this data. A Bayesian spam filter, for instance,
tries to compute the possibility that an email message is Spam, based
on the occurence of words or other features in the body of the
message.

Up to now Bayesian network classifiers usually deal with one
dimensional classification: the class of a data instance can be
represented as a one-dimensional variable.  Recently we have developed
an extension to Bayesian classifiers, where the variable to be
classified is multi-dimensional. An example is a classifier in the
medical domain for oesophageal cancer, which tries to classify the
size of the tumor, and the extent of the tumor's invasion of other
organs. It is expected that multi-dimensional classifiers use fewer
parameters in the models than their one-dimensional counterparts. This
hopefully will lead to better estimation of these parameters and to
improved classification performance.

The topic of this research project is to investigate whether or when
multi-dimensional classifiers indeed do need fewer parameters, and for
which types of networks this occurs. For this an analysis of the
classification algorithms has to be made on existing data sets, and
possibly on artificial data sets.

For this project the software tool Dazzle will be used. Dazzle has
been developed at the ICS departement to model and analyse Bayesian
networks, including classifiers.

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Title: Sensitivity in Dynamic Bayesian Networks
Student: vacancy
Supervisor: Peter de Waal:
ECTS: 7.5 or 15
Related courses: Probabilistic Reasoning
Description

Dynamic Bayesian networks (DBN) have become quite popular as extension
to classical Bayesian networks for modelling processes that evolve
over time. It is common in modelling with DBN's to assume that the
unknown parameters of the DBN are homogeneous, i.e. they do not change
over time. Recently research has started on sensitivity analysis for
this type of networks. We conjecture that sensitivity analysis can
also be used to study the effect of relaxing the homogeneity
assumption.

In this experimentation project you would investigate different
approaches to using sensitivity analysis for this purpose. This would
be done on benchmark data sets, using our homemade analysis tool
Dazzle.