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?
 A: 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.
A: 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

*

*it can represent directed and undirected graphical model. Many inferences are agnostic to both models.

*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.
