BNs are graphical models that set probabilistic relationships among variables of interest. They depict the relationships between causes and effects. The BNs are strong knowledge representation and reasoning tool under conditions of uncertainty. The BNs are a directed acyclic graph having nodes and arcs with a conditional probability distribution linked for each node. Nodes stand for domain variables, and arcs between nodes stand for probabilistic dependencies. Set of nodes and a set of directed links between them must not form a cycle. Each node represents a random variable that can take discrete or continuous finite, mutually exclusive values. These values depend on a probability distribution, which can be different for each node. Each link states probabilistic cause-effect relations among the linked variables. A link is shown by an arc starting from the affecting variable (parent node) and ending on the affected variable (child node).

We will use BNs to represent risk. For example, Figure 3.1 shows BN for "Decreased profits" risk. By linking together different risks we can model multiple risks in a project and we will look at this property in Chapter 5.

Bayes' Theorem was developed after Rev. Thomas Bayes, an 18th century mathematician and theologian. Bayes set out his theory of probability in Essay towards solving a problem in the doctrine of chances published in the Philosophical Transactions of the Royal Society of London in 1764. Richard Price, a friend of Bayes' sent the paper to the Royal Society and wrote:

I now send you an essay which I have found among the papers of our deceased friend Mr Bayes, and which, in my opinion, has great merit... In an introduction which he has writ to this Essay, he says, that his design at first in thinking on the subject of it was, to find out a method by which we might judge concerning the probability that an event has to happen, in given circumstances, upon supposition that we know nothing concerning it but that, under the same circumstances, it has happened a certain number of times, and failed a certain other number of times.

(Hogben 1970)

Laplace accepted Bayes's results in a 1781 memoir and Condorcet rediscovered them (as Laplace mentions). They stayed accepted until Boole doubted them in the Laws of Thought . Mathematically Bayes theorem is stated as:

Where it is possible to update our belief in hypothesis H given the additional evidence E. The left-hand term, P(H|E) is known as the "posterior probability," or the probability of H after considering the effect of E. The term P(H) is called the "prior probability" of H. The term P(E|H) is called the "likelihood" and gives the probability of the evidence assuming the hypothesis H is true. Finally, the last term P(E) is free of H and can be viewed as a normalizing or scaling factor.

The power of Bayes' theorem is that in many situations where we actually want to calculate p(H|E) it turns out that it is hard to do so directly, yet we might have direct information about the likelihood, p(E|H). Bayes' theorem allows us to calculate p(H|E) in terms of p(E|H).

1.3 The Bayesian Approach to Probability and Statistics

Understanding of the Bayesian method to probability and statistics helps to know BNs and related learning techniques. The
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