# Why does a belief network need to be represented using a directed acyclic graph (DAG)?

I would have thought that it's because DAGs preserve the dependency relationships between the variables, but I am currently unsure.

Thanks

Yes, we use DAGs to represent dependency relationships.

We need directed graph because condition probability P(A|B) is not same as P(B|A).

We assume it to be acyclic to get certain properties and ease calculations. You can create cyclic graph and you are going to get a lot of contradictions.

If you ease both the restriction(directed and acyclic), we get more general model called Markov Network.

A Belief (Bayesian) network is "defined" to be a DAG. A better question would be

Why a distribution needs to be represented by a Belief network, i.e. a DAG?

Dependency relationships can be modeled with both directed-acyclic and undirected-cyclic representations. Therefore, a distribution may be represented in either ways.

Why acyclic and directed?

Every distribution that can be sequentially factorized into conditional probabilities as follows can be represented by a DAG:

$$P(X_1,...,X_n)=P(X_n|X_1,...,X_{n-1})\color{blue}{P(X_1,...,X_{n-1})}$$,
$$\color{blue}{P(X_1,...,X_{n-1})}=P(X_{n-1}|X_1,...,X_{n-2})P(X_1,...,X_{n-2})$$,
...,
$$P(X_1, X_2, X_3)=P(X_3|X_1, X_2)\color{brown}{P(X_1, X_2)}$$,
$$\color{brown}{P(X_1, X_2)}=P(X_2|X_1)P(X_1)$$.

Belief network $$G$$ is built as follows: $$X_n$$ has no outlink, $$X_{n-1}$$ links to $$X_n$$, $$X_{n-2}$$ links to $$X_{n-1}$$ and $$X_{n}$$, and finally $$X_1$$ links to all $$X_2$$ to $$X_n$$.

Generally, there is no unique order for factorization, thus multiple networks can represent the same distribution. For example $$P(A, B)$$ can be factorized as $$P(A|B)P(B)$$ or $$P(B|A)P(A)$$ which are represented with two different Bayesian networks.

Every Bayesian network represents a sub-graph of $$G$$. Some directed links from $$G$$ are removed to introduce "independence assumptions between variables".

For example, $$\phi_1(A, B)=A.B/2=B.A/2$$ is a factor from factorization $$P_{\theta}(A, B, C, D) = \frac{1}{Z(\theta)}\phi_1(A, B)\phi_2(B, C, D).$$ Note that we cannot unambiguously assign a direction to relation $$(A, B)$$, since both directions are justified for $$\phi_1(A, B)$$.
As an example for cyclic relations, factor $$\phi(B, C, D) = B.(C+D)$$ would be represented with a triangle among $$B$$, $$C$$, and $$D$$.