# Student projects on Probabilistic Networks

A probabilistic network, or (Bayesian) belief network, is a concise representation of a joint probability distribution on a set of stochastic variables. A probabilistic network combines a directed acyclic graph, coding the independence relation on the set of nodes representing the variables, with (conditional) probability parameters specified locally for each node in the graph.

An independence relation can be defined as a set of triplets (X,Y|Z), the so-called independence statements. Any independence relation can be characterised by a small basic set of statemens, the kernel. In order to have this relation represent a notion of independence, the set of triplets must obey a number of axioms. Any independence statement can then be derived by (repeated) application of the axioms to the kernel.

Several open questions remain concerning, for example,

• algorithms for determining the axiomatic closure of a given kernel, and vice versa
• which properties of independence relations affect the complexity of these algorithms
Different experimentation projects can defined, related to this topic.
Further information:

The directed acyclic graph of a probabilistic network is constructed with the help of domain experts, learned from data, or a combination of the two approaches is used. Traditionally, the stochastic variables are discrete, although generalisations to continuous variables exist. The probability parameters are also assessed from domain experts, from data, or from a combination thereof.

Continuous domain variables need to be discretised before being modelled in a probabilistic network. Several methods exist for doing this. An alternative is to use a continuous variable, and assume that its distribution is gaussian, or some other simple distribution.

Several open questions remain concerning, for example,

• How do the different discretisation methods affect performance of the network?
• How realistic is the gaussian assumption in pratice?
Different experimentation projects can defined, related to this topic.
Further information:

For ease of construction of a probabilistic network, often different assumptions are made concerning the structure or the parameters of a network. One assumption can be that the network models a so-called naive Bayesian classifier. Another assumption can be that interactions among variables can be described parametrically, for example with a noisy-or gate. The performance of a probabilistic network is known to be quite robust to the structure of the network, but can be sensitive to inaccuracies in the probability parameters. Sensitivity of network performance to parameter inaccuracies can be studied by means of an uncertainty analysis or a sensitivity analysis.

Several, very different, open questions remain concerning, for example,

• Do uncertainty analyses and sensitivity analyses give different insights into the sensitivity of network performance to parameter inaccuracies?

• How sensitive is the performance of the network to noisy-or parameters?
• Which interactions found in applications of probabilistic networks are actually noisy-or?

• What are interesting properties of (two-way) sensitivity functions, describing the relation between a probability of interest and a probability parameter?
• How can we summarise and/or visualise the results of two-way sensitivity analyses?
• What can we say in general about n-way sensitivity analysis in naive Bayesian classifiers?
• What is the effect of chosing different co-variation schemes in a sensitivity analysis?
Different graduation and experimentation projects can defined, related to these topics.
Further information:

Different algorithms exist for computing probabilities from a probabilistic network and for supporting decision making.

Decision support can basically be offered in two different ways, either by adding a control on top of the probabilistic network, or by extending the probabilistic network into an influence diagram.

Several, very different, open questions remain concerning, for example,

• How can costs be efficiently taken into account upon decision making?
• How can the different assumptions underlying influence diagrams be relaxed?
• How can sensitivity analysis be performed in influence diagrams?
Different graduation and experimentation projects can defined, related to these topics.
Further information:

The DSS group is and has been involved in building a number of applications for real-life domains:

• the domain of oesophageal cancer, in cooperation with the Netherlands Cancer Institute, Antonie van Leeuwenhoekhuis, Amsterdam;
• the domain of classical swine fever, in cooperation with several European partners and IDC Lelystad;
• the domain of pediatric healthcare, in cooperation with the Wilhelmina Kinder Ziekenhuis, Utrecht;
External contacts in addition include contacts at
• the Faculty of Veterinary Disease, UU
• UMC Utrecht
• Philips Medical Systems, Best