Bayesian DAG Selection via RPDAG Methods

• Bayesian DAG Selection Method is a framework that infers network structures by maximizing decomposable scores while balancing statistical identifiability with computational efficiency.
• RPDAG representation reduces the search space by grouping equivalent DAGs and postponing edge orientation decisions, thereby avoiding premature commitments.
• Local operators and constant-time score updates enable faster evaluations in high-dimensional domains, leading to improved efficiency and structural recovery compared to traditional methods.

A Bayesian Directed Acyclic Graph (DAG) selection method refers to any algorithmic or statistical framework designed to identify the structure of a Bayesian network: a DAG where nodes correspond to random variables and directed edges encode conditional dependences. The selection is performed under a Bayesian paradigm, typically by maximizing a decomposable score function or inferring the posterior probability over network structures, given observed data. Central to Bayesian DAG selection is the trade-off between computational tractability and statistical identifiability, especially in high-dimensional and complex domains.

1. RPDAG Representation and Its Motivation

Restricted Acyclic Partially Directed Graphs (RPDAGs) are introduced as an alternative representation for exploring Bayesian network structures. Unlike standard DAGs, which enforce a full specification of every edge direction, and CPDAGs (Completed PDAGs), which provide a unique representation for each Markov equivalence class, RPDAGs relax some constraints:

• Definition: An RPDAG encodes certain edge orientations (those involved in v-structures or h-h patterns) but allows other edges to remain undirected when the data do not force a unique orientation.
• Non-uniqueness: Multiple RPDAGs may correspond to the same Markov equivalence class, trading unique representation for ease of search.
• Restriction properties: RPDAGs prohibit both directed and undirected cycles, and only enforce orientation for arcs involved in h-h patterns; otherwise, undirected edges are retained.
• Structural efficiency: By “keeping some arc directions undetermined,” the RPDAG space groups together multiple DAGs differing only in undetermined orientations.

This representation is designed to postpone irreversible orientation commitments, thereby smoothing the search space and facilitating a more efficient structural exploration (Acid et al., 2011).

2. Search Space, Operators, and Score Decomposability

The RPDAG-based method transforms the typical search over all possible DAGs into a more operable subspace:

• Reduced size: Many DAGs are subsumed into one RPDAG, effectively reducing the combinatorial search burden.
• Local operators: The method introduces nuanced edge addition and deletion operators (e.g., A_arc, A_link, D_arc, D_link, A_hh for h-h pattern creation) whose allowable usage depends on the local neighborhood conditions (number of parents, children, and undirected neighbors).
• Operator application: For a nonadjacent pair (x, y), the operator (addition or deletion as arc, link, or h-h structure) is selected via decision trees and tables mapping the local graph state to available moves.
• Constant-time evaluation: With a decomposable score (i.e., a sum of local scores, e.g., BDeu or BIC), each local move requires updating only 1–2 local terms:

$g(H : D) = \sum_{x} g_D(x, Pa_H(x))$

When an operator is applied,

$$g(G' : D) = g(G : D) - g_D(y, \text{OldParents}) + g_D(y, \text{NewParents})$$

No global rescoring is needed.

These choices provide significant computational benefits, particularly for large node sets (Acid et al., 2011).

3. Equivalence, Topology, and Avoidance of Premature Orientation

• Equivalence class navigation: All DAGs in the same Markov equivalence class (i.e., identical skeleton and v-structures) can be represented by RPDAGs, thereby avoiding redundant evaluation of structurally equivalent models.
• Smoother topology: Because RPDAGs preserve undetermined edges, the search landscape is less “rugged”—suboptimal traps due to early, unjustified directional assignments are less likely.
• Completing/undoing operators: After an operator is applied, a closure step ensures the RPDAG still satisfies required structural properties.
• No premature commitment: By delaying edge direction resolution unless forced by data (through h-h patterns), the algorithm can find models closer to the global optimum, evidenced by improved scores compared to conventional DAG searches (Acid et al., 2011).

4. Empirical Evaluation, Scoring, and Performance

The efficacy of the RPDAG approach is systematically evaluated:

Dataset	Method	Scoring Function	Relative Score	Hamming Distance	Iterations/Time
Alarm	RPDAG	BDeu	Higher	Lower	Fewer/faster
Insurance/Hailfinder	RPDAG	BDeu, BIC	Higher/similar	Lower/similar	Fewer/faster
UCI datasets	RPDAG	(various)	Competitive	Competitive	Fewer/faster

• Score comparison: Across multiple benchmarks (Alarm, Insurance, Hailfinder, UCI data), RPDAG search typically finds networks with higher (better) decomposable scores and/or lower Hamming distance to the gold-standard.
• Efficiency: RPDAG-based methods require fewer iterations and compute less statistics per move than traditional DAG search.
• Comparative performance: When set against CPDAG-based (equivalence-class) methods, Tabu Search, K2, and independence-based algorithms such as PC and BNPC, RPDAGs are consistently competitive—often faster and occasionally more accurate (Acid et al., 2011).

5. Mathematical Formulation and Operator Logic

Several formal aspects underlie RPDAG-based search:

• Decomposable scores as sums over local parent configurations.
• Operator application conditions: Explicit formalism for when and how to apply A_arc/A_link/D_arc/D_link/A_hh, often set forth via set notation:
  • $Pa_G(x)$: parents of $x$ in $G$
  • $n_G(x)$: number of undirected neighbors of $x$ in $G$
  • $c_G(x)$: number of children of $x$ in $G$
• Decision tree mappings (see Figure 1/Table 1 of the source), linking possible local states (no parents, no neighbors, etc.) to permissible moves and their structural consequences.
• Score update expressions enable constant-time evaluation for each candidate move, decoupling structural search from global recomputation (Acid et al., 2011).

6. Broader Implications and Future Directions

• Smoother search landscape: By utilizing undirected edges and delaying orientation, RPDAGs can, in practice, avoid many suboptimal local minima that stymie DAG searches.
• Extensibility: The approach is amenable to integration with advanced heuristics, including hybrid operators (arc reversals, h-h pattern destructions), more global search strategies, and stochastic metaheuristics (ant colony optimization, variable neighborhood, Bayesian model averaging).
• Implementation trade-off: While RPDAGs are not unique for each equivalence class like CPDAGs, their structural simplicity and scoring efficiency make them favorable in large-scale problems.
• Scalability and real-world use: The method’s computational properties suggest applicability to large domains in classification and reasoning, where standard approaches may be computationally infeasible (Acid et al., 2011).

7. Summary and Position in the Literature

The RPDAG selection methodology represents a principled, efficient local search for Bayesian network structures, leveraging a restricted and structurally-motivated search space to mitigate premature direction assignments and reduce the cost of model evaluation. Empirical results establish both improved efficiency and, in many cases, superior structural recovery relative to classic DAG-based and equivalence-class methods. The approach’s mathematical rigour, operator definitions, and applicability to decomposable scoring functions suggest a wide potential for integration with advanced search and inference strategies in the construction of high-dimensional probabilistic graphical models.