# What makes Bayesian Networks decomposable into joint trees?

Given a Bayesian Network $N$, one can build a junction/joint tree $JT$ over $N$ by applying series of steps (namely, moralisation,triangulation..etc). Then we can use $JT$ to answer queries over $N$.

My question is: what makes BN decomposable into $JT$? The structure (along with the CPTs) must exhibit certain conditions otherwise any graphical model is decomposable.

I found the answer in valuation algebra more specifically in this paper .
They assume the set of functions (i.e. tables/relations/potentials/probability distributions) form a commutative semigroup. There are six axioms that the graphical model/representation should obey in order to use local computations (and thus joint trees). The valuation algebra has three operations:

1. Labeling: Mapping each potential $\phi$ to its scope $d(\phi)$ (variables that defined it).
2. Combination: Two potentials $\phi$ and $\psi$ are combined through combination operator $\phi \otimes \psi$ (in BNs $\otimes$ is the multiplication of CPTs).
3. Marginalization: The usual projection operator (this corresponds to summing up irrelevant variables in BNs).

where the axioms are:

1. Commutative Semigroup: it is clear that the set of CPTs represent a commutative semigroup under the combination operator.
2. Labeling: for two potentials $\phi$ and $\psi$, $d(\phi \otimes \psi)=d(\phi) \cup d(\psi)$
3. Marginalization: $d(\phi^{\downarrow x})=x$ where $x\subseteq d(\phi)$
4. Transitivity: $(\phi^{{\downarrow y}})^{\downarrow x}=\phi^{\downarrow x}$ where $x\subseteq y\subseteq d(\phi)$
5. Combination: $(\phi \otimes \psi)^{\downarrow z}=\phi \otimes \psi^{\downarrow{z\cap y}}$ where $d(\phi)=x,d(\psi)=y$ and $x\subseteq z\subseteq x\cup y$
6. Domain: $\phi^{\downarrow x}=\phi$

The axioms are clearly satisfied by the nature of multiplication and marginalization over BN CPTs. Really do not know why axiom 6 is there. Sorry for abusing some of the notations.