While a directed graph structure is suitable for a set of random variables where there is clear causality, an undirected graph structure is better fit for a situation where dependence upon random variables flows in multiple directions.

For example, a model showing the health status (probability of having a cold) of you, your roommate, and a coworker could be represented with an undirected graph. Your health status would be a node with undirected vertices connected to your coworker's and roommate's health status respectively, as you can get either of them sick, either one of them could get you sick, but they cannot *directly get each other* sick.

When to Use Undirected Graph Models

For each clique $C$ in an undirected graph, a factor $\phi(\mathcal{C})$ called the **Clique Potential** measures the affinity of the variables in that clique. A clique is a subset of nodes in the graph that are all connected by vertices directly or indirectly.

Together, these factors form the **unnormalized probability distribution** $\tilde{p}(\bold{\text{x}}) = \prod_{\mathcal{C}\in \mathcal{G}}\phi(\mathcal{C})$

Clique Potential in Undirected Networks

Undirected Models (AKA Markov Random Fields/MRF/Markov Networks) have typical graph structure but the vertices are undirected.

University of Connecticut

Probabilistic models use graphs to show interactions between random variables. A graph is a set of nodes connected by vertices. A probabilistic model represents random variables with nodes and their relationships with vertices between them.

Graph Model Structure

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

Undirected Models

Separation is used to describe which variables interact indirectly. D-Separation simply refers to separation with a directed graph.

If a path exists between two variables with only unobserved variables, then the variables are separated. Paths like this are considered active, while paths with an observed variable are considered inactive.

Separation and D-Separation

A Bayesian network or Belief Model is a probabilistic graphical model for representing knowledge about an uncertain domain where each node corresponds to a random variable and each edge represents the conditional probability for the corresponding random variables.

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