Chapter 3 A Chain Event Graph
3.2 CEG Modelling and Reasoning
The representation of a process using a CEG model is obtained in three major steps: the elicitation of the event tree T that describes the qualitative structure supporting the process; its subsequent embellishment with colours to obtain the staged tree; and eventually its transformation into the CEG graph according to some simple graphical rules (Smith and Anderson, 2008, Thwaites et al., 2008, Smith, 2010, Freeman and Smith, 2011a). The CEG model procedure is also discussed in Barclay et al. (2013) and in Cowell and Smith (2014).
Figure 3.1 depicts an event tree corresponding to the multiple paths that a tourist can take before booking a train as described in Section 3.1. In this tree, there are three distinct types of random variables: a binary variable P Ci for theith, i=
a, b, c, site visited; a binary variable Fi, i = a, b, c, (b- Booked; s- Searching)
representing if a tourist booked a train at P Ci or carried on searching at other
PC; and the variable trainT indicating the train option.
Figure 3.1: The event tree associated with a train ticket search for the last three points of contact visited by a tourism. Variables: - PC; - F;♦- Train.
An event tree provides a flexible graphical framework that facilitates the visualisa- tion and understanding the whole process. For these purposes, vertices associated with the same type of random variable are depicted in the event tree using an identical geometric shape. Edges associated with events with probability zero are omitted. For example, situations corresponding to variables F and PC that are given a dashed geometric shape. Since all tourists included in the sample booked a train, the variable F3 has necessarily value equal to Booked.
since this movement is interpreted as a single visit to a PC Ship by the sample design. So, P Ci+1, i = 1,2 cannot assume value Ship if P Ci is equal to Ship.
In contrast, this logical exclusion does not hold within the PC category Others
because this category merges 16different types of PCs.
For instance, a tourist in the initial situation s0 can unfold into the situation s1
where he visits a point of contact associated with the cruiser company. He might then decide to book a train, situations3, or carry on searching a train ticket at an
other point of contact, situation s4.
The situation s3 represents the state of a tourist who opts for booking a train in
his visit to a PC that is linked to a cruiser company in this case. Analogously, the situation s25 represents an individual that books the train ticket after visiting
three point of contacts that are not managed by the cruiser company.
Depicting the different ways that each unit can follow along the process under analysis, an event tree (Shafer, 1996) provides an intuitive graphical interface to translate the system dynamic into a mathematical model using plain language. Formally, the non-leaf vertices of the tree characterises a particular situation s
that a unit can be at a given point during the process. It represents a transitional state from the root node to a potential end of unfolding process. Its emanating edges are associated with the events that may happen when a unit is at situations. In contrast, a leaf vertex l, or simply a leaf l, represents a possible terminating state of the process. So, both types of vertices, a situation s and a leaf l, are defined by the consecutive events that occur along its root-to-sor root-to-lpaths. An important graphical structure corresponding to a situationsis the star subgraph of T called floret F(s) = (V(s), E(s)), where the vertex set V(s) is constituted by the situation s and its child situations, and the edge set includes all outgoing edges of s. For example, the floret associated with situation s1 in Figure 3.1 is
depicted in bold. Now a floretF(s)enables us to associate every sitution s in an event tree with a random variable X(s) whose state space X(si) ={eij} is given
This random variable X(si) fully represents how a process develops given that a
unit is atsi. In this way, we can implicitly obtain a probability measure yielded by
the primitive conditional probabilities defined at each situation si as follows:
p(X(si) =eij|s) =πij, eij ∈X(si). (3.1)
Embedding the set of conditional probabilities given by Equation 3.1 within an event tree enables us to obtain a probability tree. In this case, the set of situations whose state spaces are equivalent and whose conditional probabilities are identical constitute a stage. Thus, the probabilities corresponding to the emanating edges of florets rooted at any two situations that are at the same stage are hypothesised to be equal. Each stage is associated with a unique colour. When the situations in an event tree are embellished with these colours we obtain a staged tree. For formal detail about this construction, see e.g. Freeman and Smith (2011a). Two situationssaandsb are said to be in the same positionwif and only if they are
at the same stageuand their whole subsequent unfolding processes develop under a completely analogous probability law. Thus, there is a probabilistic and graphical isomorphism between the subtrees unfolding from sa and sb and the subsequent
evolutions of units at these situations are undistinguishable. The vertices of a CEG corresponds exactly to these positions and so the set of positions is said to be the position structure W of a CEG.
Return to the train booking example. The demographic model aims at identifying heterogeneous population of tourists according their predisposition to choose a public or cruiser-organised train using the three demographic variables: nation- ality, age and number of prior cruise travels. Note that in the first model the corresponding process of visiting a PC and booking a train has a very well-known and rigid sequence of possible events.
In distinction to the PC sequence model this does not happen in our second model where only a partial variable order is defined a priori. In this case, each leaf node of the demographic event tree should immediately unfolds from an event corresponding to the state space of the variable Train. This is justified because
our focus is to understand whether and how the demographic variables can explain a tourist’s option for a public or cruiser-organised train. In the absence of any further domain hypotheses that enable us to elicit a complete variable order, we run a CEG model search algorithm to define it using the data available and the partial order already known.
Figure 3.2: A hypothesised staged tree associated with the demographic variables and the variable Train. Variable order: C ≻ V ≻ A ≻ T. The stage struc- ture is given by: u0 ={s0}, u1 = {s1}, u2 = {s2}, u3 ={s3, . . . , s7}, u4 ={s8},
u5 ={s9, s11, . . . , s14, s17},u6={s10, s15, s16, s18},u7={s19, s20}.
Figure 3.2 shows a staged tree for the demographic variables associated with the train booking process under the assumed variable orderC ≻V ≻A≻T and the hypothesised stage structure: u0 ={s0},u1 ={s1},u2 ={s2},u3 ={s3, . . . , s7}, u4 ={s8}, u5 = {s9, s11, . . . , s14, s17}, u6 ={s10, s15, s16, s18}, u7 = {s19, s20}.
Consider the situations s5 and s7 whose state spaces are identical,
As these situations are coloured the same we can conclude that the probabilities of two passengers to be young given that one of them comes from local regions and has a strong tendency to take a cruise journey and the other is an overseas tourist with a moderate propensity for taking cruise trip are identical. However, these situations are not at the same position because the corresponding child situations s14 and s18 are embellished with different colours. So, the unfolding
subtrees rooted at situations s5 and s7 are not the same since mature passengers
have their train booking processes governed by quite different probability rules. On the other hand, it is easy to observe that situations s4 and s5 are not only in
the same stage but also at the same position.
Transforming a staged tree into a CEG requires us two simple steps: to merge all situations at the same position w into a single vertex w and to divert all leaf nodes into a single sink vertex w∞. A CEG is a compact representation of
its corresponding staged tree. Therefore, it facilitates the interpretation and the readability of hypotheses embodied within the probability model by domain experts and decision makers.
The CEG model depicted in Figure 3.3 corresponds to a graphical transformation of the staged tree showed in Figure 3.2. It summarises the information depicts in the staged tree without implying any loss of information or requiring further assump- tions. In doing this, the CEG graphs communicates more easily the hypothesised conditional independences structures to laypeople.
For example, the positions w3 ={s4, s5}, w4 ={s3, s7} and w5 ={s6} in Fig-
ure 3.3 are coloured red because the situations si, i= 3, . . . ,7, gathered by them
are all at the same stage u3. So, the variable Age associated with a passenger
is context-specific conditional independent of his corresponding variables Country and Visit given that this tourist is not at position w6: he is not from overseas
countries and does not have a strong preference for cruise trips. In contrast to BN models this kind of asymmetric conditional independences is directly and easily depicted by a CEG. Observe that all other stages in this CEG coincide with a single
position.
Figure 3.3: The CEG corresponding to the stage tree depicted in Figure 3.2. It represents the train booking process when the demographic variables are taken into consideration in the following order: C ≻V ≻A≻T. The stage structure is given by: u0 ={w0},
u1 = {w1}, u2 = {w2}, u3={w3, w4, w5}, u4 ={w6}, u5 = {w7}, u6 = {w8},
u7 ={w9}.
The pairG= (T, U)completely characterises the graphical structure of a CEGC, where T is an event tree and U is the set of stages or the stage structure. So, the topology of a CEG is fully defined by its underlying stage tree. A triple
C= (T, U,P)formally defines a CEG model, whereP is the elicited probabilistic
measure corresponding to G.