# Difference and relation between dependency graph and graphical model?

1. Let $A_1 , A_2 , \dots, A_n$ be $n$ events on a probability space $Ω$. The dependency graph is a directed graph $D = (V, E)$ on the set of vertices $V =\{1, \dots, n\}$ (corresponding to $A_1 , A_2 , \dots, A_n$ ) if for each $1 ≤ i ≤ n$, $A_i$ is mutually independent of all the events $\{A_j : (i, j) \notin E\}$.

2. In a graphical model, each vertex represents a random variable. Any two vertices are conditionally independent given the values of their parents. In general, any two sets of nodes are conditionally independent given a third set if a criterion called d-separation holds in the graph.

Dependency graphs and graphical models are very similar in that both represent random variables represented as vertices (an event can be seen as a random variable by taking the indicator function of the event), and encode their dependency relation in edges.

So I wonder if they are related somehow, and can be converted to each other?

Thanks!

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## 1 Answer

A dependency graph represents part of the PGM. In PGMs there is additional information such as the structure of the conditional probability distributions of each node given its parents.

Remember that a PGM is essentially a condensed form of a joint probability distribution over all variables, so that you don't have to explicitly represent the whole joint probability distribution as a big table. The dependency graph just talks about which variables are dependent on each other.

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Thanks! Does "PGM" mean probabilitic graphical model? –  Tim Oct 12 '12 at 2:27
@Tim Yes, PGM means probabilistic graphical model. –  Bitwise Oct 12 '12 at 13:52
Thanks, Bitwise! (1) Can a graphical model capture independence besides conditional independence? Also see here (2)Can a dependency graph capture conditional independence besides independence? –  Tim Mar 29 '14 at 18:09