two weighted sets of plans, c and d, and interpret8the weight of a plan as the number of times it 'occurs', e.g., wc(φ) = Mc[φ]. In the simple case that there is only a single plan in the intersection, {α} = c ∩ d, there is only one way to take a transitive closure - for all φ in c and φ
′ ∈ d \ c:

$$\hat{O}_{cd}(\phi,\phi^{\prime})=\hat{O}_{c}(\phi,\alpha)\cdot\hat{O}_{d}(\alpha,\phi^{\prime}),$$ $$=\frac{M_{c}[\phi]}{M_{c}[\alpha]}\cdot\frac{M_{d}[\alpha]}{M_{d}[\phi^{\prime}]},$$ $$=\frac{M_{c}[\phi]}{s\cdot M_{d}[\phi^{\prime}]},\text{with}s=\frac{M_{c}[\alpha]}{M_{d}[\alpha]},$$

so in particular we can merge d into c by first rescaling d:

$$M_{c d}[\phi]=\left\{\begin{array}{l l}{{M_{c}[\phi]}}&{{\mathrm{if~}\phi\in c}}\\ {{s\cdot M_{d}[\phi]}}&{{\mathrm{otherwise}}}\end{array}\right..$$

In general let s cd α =Mc[α]
Md[α]
for any α ∈ c ∩ d be the *scale factor* of c and d w.r.t. α. Then in the case that there are multiple plans in the intersection we are faced with multiple ways of performing a transitive closure, i.e., a set of scale factors. These will normally be different from one another, but, in the limit of data, assuming preferences and feasibility constraints are actually independent of one another, every scale factor between two clusters will be equal. So, then, we take the average:

$$s^{cd}:=\frac{1}{|c\cap d|}\cdot\sum_{\alpha\in\cap d}s^{cd}_{\alpha},\text{and,}\tag{12}$$  $$M_{cd}[\phi]:=\left\{\begin{array}{ll}M_{c}[\phi]&\text{if}\phi\in c\\ s^{cd}\cdot M_{d}[\phi]&\text{otherwise}\end{array}\right..\tag{13}$$

In short, if all the assumptions are met, and enough data is given, the described process will reproduce the correct prior distribution on preferences. Algorithm 2 provides the remaining details in pseudocode, and in the following we discuss these details and the result of the rescaling process operating in the Travel domain.