# Bayesian Networks for Uncertainty Management

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## Reasoning with BNs

A BN can be considered as a probabilistic knowledge base, in which the process of querying is called inference. In fact, inference is the main task behind constructing a BN. The general form of the probabilistic inference is P(Q|E=e), which reflects the following human reasoning question: suppose we observe e on a subset of E of the domain, what are the probability values that another subset Q can take? Q is called the query or target and E=e is called an observation or evidence.

### Querying the distribution

We present here the most common types of probabilistic queries: Marginal distribution can be viewed as the projection of P on a smaller subset of variables by marginalizing out over the other variables. When Q,E 6= ;, the query is called the posterior marginal of Q given the knowledge on observed  variables E. It is called also the conditional probability query. In the sequel, x will be used to denote the assignment X = x. Answering the marginal distribution query involves the computation of both the joint distributions P(Q, e) and P(e) as : P(Q|e) = P(Q, e) P(e) (3.3).
Both distributions can be computed by marginalizing out the variables in W = V \ Q [ E. In this case, the numerator can be computed by summing out over W,P(Q, e) = P w2WP(Q, e,w), while the denominator, which is called the probability of evidence, can be deduced from the previous equation by P(e) = X q2Q P(q, e) When E = ;, the distribution P(Q) is called prior marginal.

#### Probabilistic Relational Models

When dealing with complex and large systems, BNs do not seem to be well adapted. Indeed, the cost of designing and maintaining such systems becomes high [73, 84] and the inference might be intractable. A complex system may induce a big number of probabilistic dependencies that can, not only lead to intractable modeling costs, but also implies important update costs when changes occur in the underlying structure. Actually, every time an entity and/or a relation in the domain changes, a new BN should be defined. Such shortcomings are essentially due to the fact that BNs completely ignore the concept of objects and their interactions. To cope with the aforementioned problems several extensions were proposed in the literature, among which, one can find the following:
• Object-oriented extension, such as Object-Oriented BNs [12, 73, 84].
• First Order logic extension, such as Markov Logic Network [66, 68].
• Relational models such as Entity-Relationship [60, 76].
In this thesis, we are particularly interested in Probabilistic Relational Models (PRMs) [12, 106], an extension of the BNs attribute-based representation that combines the OO paradigm with the relational representation in database theory.
Hence, PRMs specify a generic template defining a probabilistic model, a set of dependencies, that hold in a relational domain, which can be represented by a relational database. The OO paradigm helps to abstract and capture general properties about similar entities in the domain in terms of classes having similar properties and can be instantiated for specific contexts as many times as suited. While the relational model helps to describe interactions or relationships between classes and their instantiated objects. To understand these concepts and give more formal definitions, we use the PRMs version described in .
For the illustration purpose, let us consider Fig. 3.8, which depicts a BN. In such a configuration, one can identify repeated patterns and abstract them as a generic type or class of objects. For instance, in Fig. 3.8, random variables Xi, Yi  form a pattern. A PRM class also abstracts all the interactions between its random variables by drawing arcs expressing the probabilistic dependencies. Class’ random variables, which describe and characterize the pattern represented by the class, are called attributes. Basically, a class attribute can only be visible to attributes of the same class. However, one might want to refer to attributes outside the class.
That is why PRMs define the notion of reference slots allowing communications between different classes and hence between attributes of different classes. Now let us give formal definitions for PRM concepts Definition 3.3.1 (Class). A class C is defined by a DAG over a set of attributes A(C) and a set of reference slots R(C) (see Definition 3.3.2).

Uncertain OO-BRs Principles

From Chapter 2 we recall that BRs are rules in the form “If <condition> Then <action>” that are exploited for reasoning by forward chaining inference engines. OO-BRMSs execute BRs against an OM that describes the application objects based on a data model. In this latter we can differentiate two components, which include an executable data model that can be implemented in JAVA or XML
schema, and a business-oriented model, which uses specific vocabulary and terms familiar to business users and is expressed in a domain-specific language. Hence, WMEs (or facts) are objects in the sense of the OO paradigm. It follows that an elementary action or condition operates on tuples rather than simple values.
A single action affects WMEs by inserting a new object, removing an existing object or updating an existing object. Each of these actions potentially leads to a re-evaluation of any rule that matches the object in question.
It turns out that there exists a natural mapping between model elements in BRMSs and PRMs. We illustrate this through a simplified example from an insurance application. This example is not intended to be exhaustive but rather to illustrate necessary concepts.
Example 5.2.1 (Simple insurance application). An insurance organism allows its subscribers to request reimbursements depending on invoice types. A request must be validated by a health care professional and each subscriber can have many reimbursement requests.

Application to IBM ODM

The aim of this section is to provide a practical view of the foundations introduced in the previous section. Our implementation is based on IBM ODM  as an OO-BRMS and a Graphical Universal Modeler (aGrUM)  as a probabilistic engine. It is important to emphasize that the methodology we applied can be easily generalized to any OO-BRMS as we showed previously. This section gives an overview of the ODM functionalities and is mainly built from material in [19, 63], where a more detailed presentation is given.

1 Introduction
1.1 General Context
1.2 Motivation
1.3 Contributions and Outline
2 IT for Business Rules Management
2.1 Introduction
2.3.1 Features of a BRMS
2.3.2 Rule-based Expert Systems
2.4 Discussion and Conclusion
3 Bayesian Networks for Uncertainty Management
3.1 Introduction
3.2 Bayesian Networks
3.2.1 Definition and Design
3.2.2 Graphical Semantics
3.2.3 Reasoning with BNs
3.3 Probabilistic Relational Models
3.4 Conclusion
4 Incremental Junction Tree Inference
4.1 Introduction
4.2 Junction Tree algorithm
4.3 Incremental Junction Tree Inference
4.3.1 Optimal Roots
4.3.2 A new Incremental Inference
4.4 Evaluation
4.4.1 Messages Optimization
4.4.2 Time Optimization
4.5 Conclusion
5.1 Introduction
5.2 Coupling BRs with PRMs
5.2.1 Uncertain OO-BRs Principles
5.2.2 Model Extension
5.3 Application to IBM ODM
5.3.1 Overview
5.3.2 A Complex Compilation Process
5.3.3 A Loosely Coupling-based Execution