Equivalence Classes
Context: Equivalence Classes, Essential Graphs, and Edges
An Equivalence Class refers to a set of network graphs that represent exactly the same joint probability distribution, although arc directions may differ between the networks.
For instance, the following six graphs can represent the same joint probability distribution. When learning these networks from data, it would not be possible to distinguish between them, as they all belong to the Equivalence Class.
In the above example, all arcs can be inverted (within the constraint of acyclicality), and the network still remains in the Equivalence Class.
Essential Graph
In Validation Mode, BayesiaLab can visualize the Equivalence Class by displaying the Essential Graph for any given network.
The Essential Graph of a network consists of:
- Directed arcs that cannot be inverted. If such directed arcs were inverted, the joint probability distribution of the network graph would change, and that graph would no longer belong to the Equivalence Class.
- Edges, i.e., undirected arcs, which can be inverted. For an Edge, either arc direction maintains the same joint probability distribution, meaning the network graph remains in the Equivalence Class.
For the example above, the Essential Graph is straightforward, as all arcs can be inverted. The following Essential Graph only features Edges:
Equivalence Class
In non-trivial cases, recognizing the Essential Graph is much less obvious.
To have BayesiaLab display the Essential Graph, go into Validation Mode and select Main Menu > Analysis > Visual > Graphs > Equivalence Classes. Alternatively, use the shortcut E to activate this function.
With the Essential Graph on display, you can see the Edges that allow you to choose an orientation within the Equivalence Class.
Among the Edges, you can assign an arc direction. However, assigning one arc direction will immediately update the Essential Graph and potentially provide a new set of Edges.
Class Instantiation
Returning to a directed graph means instantiating one network graph from the Equivalence Class.
To do this, select Main Menu > Analysis > Visual > Graph > Class Instantiation or use the shortcut O. Unless you changed an arc direction while the Equivalence Class was active, BayesiaLab will return to the original graph.
Workflow Animation
The following animation illustrates the Equivalence Class and Class Instantiation functions. It also shows arc inversions, both while the Equivalence Class is active and after performing Class Instantiation.