Learning Equivalence Classes of Bayesian Networks Structures (1302.3566v1)

Abstract: Approaches to learning Bayesian networks from data typically combine a scoring function with a heuristic search procedure. Given a Bayesian network structure, many of the scoring functions derived in the literature return a score for the entire equivalence class to which the structure belongs. When using such a scoring function, it is appropriate for the heuristic search algorithm to search over equivalence classes of Bayesian networks as opposed to individual structures. We present the general formulation of a search space for which the states of the search correspond to equivalence classes of structures. Using this space, any one of a number of heuristic search algorithms can easily be applied. We compare greedy search performance in the proposed search space to greedy search performance in a search space for which the states correspond to individual Bayesian network structures.

• The paper introduces a novel E-space search method using CPDAGs to uniquely represent equivalence classes of Bayesian networks.
• It demonstrates that E-space yields higher scoring and structurally more accurate networks compared to traditional search methods.
• The methodology employs key operators, including edge deletion, reversal, addition, and v-structure insertion, to systematically traverse the search space.

Learning Equivalence Classes of Bayesian Network Structures

David Maxwell Chickering's paper "Learning Equivalence Classes of Bayesian Network Structures" focuses on the theoretical underpinnings and practical application of learning Bayesian networks from data by leveraging equivalence classes. Bayesian networks (BNs) are powerful tools for representing probabilistic relationships among a set of variables, and learning these networks from data involves combining scoring functions with heuristic search procedures to identify optimal network structures.

Background and Motivation

Traditionally, learning Bayesian networks involves defining a scoring function that evaluates how well a candidate network structure fits observed data, thus reducing the task to a combinatorial search problem. Given that search space for Bayesian network structures, represented as directed acyclic graphs (DAGs), is NP-hard, heuristic search algorithms become necessary for practical applications.

A noteworthy feature of many scoring functions derived in the literature is their ability to return a score for entire equivalence classes of network structures rather than individual structures. This property, called score equivalence, implies that networks capable of representing the same set of probability distributions share the same score.

Equivalence Classes and Completed PDAGs

Two Bayesian networks are considered equivalent if they define the same set of probability distributions, categorized into equivalence classes characterized by same skeletons and v-structures. Chickering introduces the concept of the “completed partially directed acyclic graph” (CPDAG) to uniquely represent these equivalence classes.

A CPDAG includes directed edges for compelled directions (consistent across all DAGs in the equivalence class) and undirected edges for reversible directions. This representation resolves the issue of multiple DAGs representing the same equivalence class in traditional search spaces, providing a more efficient and unique state representation for the search algorithms.

Defining the Search Space: E-space

To optimize the search process, Chickering proposes using a search space where the states correspond to equivalence classes of Bayesian networks, termed "E-space”. States in this space are represented by CPDAGs, and transitions between states are defined by a set of operators:

1. Edge Deletion: Removing an undirected or directed edge.
2. Edge Reversal: Reversing a directed edge.
3. Edge Addition: Adding an undirected or directed edge.
4. V-structure Insertion: Introducing a v-structure involving triads of nodes.

The effectiveness and completeness of these operators are crucial, ensuring that all necessary transformations can be applied to traverse the space systematically without creating cycles or inconsistent structures.

Experimental Results

Chickering compares the performance between greedy search algorithms applied to B-space (traditional network structure search space) and E-space. The experiments, including varying the number of nodes in the gold standard, database sizes, and real-world data from the Alarm network, demonstrated consistent outperformance of the E-space approach in terms of higher scoring networks and closer structural accuracy to the gold standard.

Key experimental findings:

• Score Improvement: E-space consistently yields structures with higher scores compared to B-space.
• Structural Accuracy: Search in E-space results in networks with significantly lower structural differences from the ground-truth networks.
• Runtime Efficiency: Despite longer runtime due to additional operators and consistency checks, the quality of results justifies the computational expense, particularly as data size increases.

Implications and Future Directions

The paper's theoretical and empirical findings significantly enhance the methodology for learning Bayesian networks, highlighting the benefits of leveraging equivalence classes in search algorithms. By using E-space and CPDAGs, researchers and practitioners can obtain more accurate network structures, leading to better probabilistic models.

Future developments could explore integrating both E-space and B-space within sophisticated search algorithms, balancing runtime efficiency with the quality of results. Additionally, approaches could be devised to adaptively switch between these spaces during the search process, optimizing performance further.

Conclusion

David Maxwell Chickering's work on learning equivalence classes of Bayesian network structures introduces a robust framework for addressing the combinatorial complexity of network learning. By defining E-space and utilizing CPDAGs, the paper provides a concrete solution to improve accuracy and efficiency in identifying high-scoring Bayesian networks, thereby contributing to both theoretical advancements and practical applications in probabilistic modeling.