# Factor graph vs Factor graphical model

In inference we use the terms undirected graphical models and directed graphical models. Why do we say factor graph instead of factor graphical models?

You say factor graph when referring to the graph. When referring to the model, you could indeed say factor graphical model. That Wikipedia page contradicts itself by saying that a factor graph is both a graph and a model.

• It seems to be the norm in the inference field to talk about "Factor graph". Googling "Factor graphical model" yields only 20 results... Oct 14, 2014 at 18:04

First you need to know the definition of probability graphical model (PGM).

Given a graph G, a distribution p.
a PGM is a pair (G,p), where p factorizes over G.[^1]

The keypoint is where G is a graph , not a factor graph. Well ,I think there is nothing wrong to say factor graphical model (factor graph G, a distribution p), though there's no such use in literature, because factor graph has redundant information for this definition.

The graph's main advantage to representation is I-map property (i.e. conditional independence ). Even so, it is not very fined-grained representation. That's why factor graph or other representations comes in. (e.g. many factor graphs may represent the same Markov Network [^2] )

In UGM case, we usually have three representation : Markov Network, Factor Graph, Log-linear Model. The latter is finer-grained than previous one.[^3]

Why inference use factor graph? My opinion is

1. it can represent directed and undirected graphical model. Many inferences are agnostic to both models.
2. it can reveal more finer structure than Markov Network that help the message passing algorithm. [^4] Though I am not very familiar to Belief propagation ,I guess it may like reveal tree structure that is ambiguity (to a clique) in Markov Network.

[^1]: Kollar & Friedman 2009 (definition 3.5)
[^2]: Murphy 2012 (Figure 22.2) [^3]: Kollar & Friedman 2009 (4.4)
[^4]: Murphy 2012 (22.2.3.1)

### reference:

Kollar & Friedman 2009. Probabilistic Graphical Model.
Murphy 2012. Machine Learning, a probabilistic perspective.