A Stratified Chain Event Graph

A useful class of CEGs for model selection is the so-called Stratified CEGs (SCEGs) (Cowell and Smith, 2014). The SCEG class has the discrete BNs and context- specific BNs as particular subclasses of models. As in the BN framework, a SCEG is defined by a set of random variables Z _{= {Z1, Z2, . . . , ZN}, N ≥ 2, where}

each variableZn, n= 1, . . . , N, corresponds to a particular measurement on each

of the units observed in any target system. Let I be a permutation of the set

{1,2, . . . , N_} such that

{1,2, . . . , N_}7−→I (i1, i2, . . . , iN),

which is used to order the set of variablesZ _{as following} Z _7−→I _(Zi

1, Zi2, . . . , ZiN),Z(I),

where Z(I) is the ordered sequence of the variables in Z _{spanned by a permuta-}

tion I. Now let

Z(k)_{(I) =Z}

i1 ×Zi2 × · · · ×Zik

be the product space of the firstk variables in Z(I).

Each permutation I spans a different event tree T(Z(I)) called Z_{−compatible}

constitute the family of event trees denoted byTZ. This familyTZ supports the

SCEG class.

Definition 21 (Z_{−compatible Event Tree). An event treeT(Z(I))is said to be} Z_{−compatible if the following two conditions hold:}

1. Its vertex set V(_T(Z(I))) consists of a root vertex s0 together with a set

of vertices s(z(k)), one for each z(k) = (zi1, zi2, . . . , zik) in Z

(k)_{(I), and} 1_≤k _≤N. Note that eachs(z(N)) is a leaf node.

2. Its edge setE(_T(Z(I)))is formed by the set of labelled edges(s0, s(z(1)), zi1),

zi1 ∈Zi1, together with a set of labelled edges (s(z

(k)_{), s(z}(k+1)_{), z}

ik+1),

where z(k+1) = (z(k), zik+1) and zik+1 ∈Zik+1, one for each z

(k) _{in Z}(k)_(I),

and 1_≤k_≤N₋1.

Observe that in any event tree_T(Z(I)),_T(Z(I))_∈T_{Z, all of its non-root vertices}

s(z(k)),z(k) ∈Z(k)_{(I), are at the same distancekfrom the rootv}

0. Also note that

each edge (s(z(k)_{), s(z}(k+1)_{), z}

ik+1) is labelled by a possible value zik+1 ∈ Zik+1

of the variable Zik+1 on the ordered components of Z(I) determined by I. For

further discussion about Z_{−compatible Event Trees, see Example 5.}

Example 5 (Train Booking with two demographic variables). Recall the train booking process presented in Section 3.1. For simplicity, assume now that we would like to explore the interplay between the demographic variables Country (C) and Visit (V).

Two demographic variables yield only two possible Z_{−compatible Event Trees,}

where Z _{= {C, V}. These event trees are depicted in Figure 4.1. They corre-}

spond to the variables orders Z(I1) = (C, V) and Z(I2) = (V, C). Note that

T_{Z ={T(Z(I1)),T(Z(I2))}.}

Note that if a process is characterised by asymmetrical developments and so has a non-product event space, then its corresponding event tree will often be not

Z_{-compatible. We are now able to define the SCEG class.}

Definition 22 (Stratified Chain Event Graph). A CEG is called a Z_-Stratified

(a) Event TreeT(Z(I1)) (b) Event TreeT(Z(I2))

Figure 4.1: Z₋compatible event treesTZ ={T(Z(I1)),T(Z(I2))} yielded by the set

of demographic variablesZ =_{C, V_}, where Z(I1) = (C, V) andZ(I2) = (V, C).

tree _T(Z(I)) for some permutation I and its stage partition has the following properties:

1. Each stage only gathers situations that are at the same distance from the root node.

2. For any two situationss1(z(1k)) and s2(z(2k)) at the same stage the mapping

associating their floretsF(s1)andF(s2)always maps the edges so that their

labelszik+1 ∈ Zik+1 on the full tree coincide.

The first condition implies that each stage can only gather situations s(z(k)_),

z(k) _∈Z(k)_{(I), associated with the same variable Z}

k+1, k = 1, . . . , N −1, in Z.

Observe that in a SCEG the root situation s0 will always constitute the singleton

stage u0={s0} and the root position w0 = {s0}. The second condition simply

demands that when two situationss1(z(1k))ands2(z(2k))are at the same stage the

conditional probability distributions of variables X(s1)and X(s2)associated with

their florets are then the same for allzik+1 ∈Zik+1:

So the meaning of edges associated with the same variableZk cannot be permuted

to constitute a stage under this condition. The SCEG class is illustrated in the two examples below.

Example 5 (Train Booking with two demographic variables - cont.). In Example 5 the event tree T(Z(I1)) supports two Z-SCEGs whilst the event tree T(Z(I2)) supports five Z_{-SCEGs. To see this, take first the event tree T(Z(I1)). From}

condition 1 in Definition 22 we cannot merge the root situations0 with any other situation. In other words, we only merge situations associated with the same variable. Condition 2 in Definition 22 implies that there is only one way to merge situationss1 ands2: the probability of a particular event to happen should be the same whether a tourist is at situation s1 or s2.

For instance, we are not allowed to gather situations s1 and s2 if the probability vector on the edges (w, m, s) of s1 is identical to the probability vector on the edges (m, s, w) of s2 because the probability matching between these situations are associated with a permutation of edges. To merge these situations into a single position the probability vector on the edges (w, m, s) of s1 has to be identical to the probability vector on the edges (w, m, s)of s2. In light of these conditions we only have the following two possible stage structures:

1. Ua(I1) ={u0 ={s0}, u1 ={s1}, u2 ={s2}}, and 2. Ub(I2) ={u0 ={s0}, u1 ={s1, s2}}.

For analogous reasons, it is straightforward to verify that the event treeT(Z(I2)) only supports five possible Z_{-SCEGs whose stage structures are:}

1. Ua(I1) ={u0 ={s0}, u1 ={s1}, u2 ={s2}, u3 ={s3}}, 2. Ub(I2) ={u0 ={s0}, u1 ={s1, s2}, u2 ={s3}}, 3. Uc(I3) ={u0 ={s0}, u1 ={s1, s3}, u2 ={s2}}, 4. Ud(I3) ={u0 ={s0}, u1 ={s1}, u2 ={s2, s3}}, and 5. Ue(I5) ={u0 ={s0}, u1 ={s1, s2, s3}}.

described in Section 3.1. Any event tree yielded by the set of demographic vari- ables Z _{= {C, V, A, T} is a} Z_{−compatible event tree and so the event tree}

corresponding to the staged depicted in Figure 3.2 is a Z_{−compatible event tree.}

If we only merge situations into a single stage associated with the same variable every CEG supported by one of thisZ_{−compatible event tree will be a}Z_−SCEG.

This is exactly the case with the CEG showed in Figure 3.3.