Bayesian Structure Learners

• Bayesian structure learners are algorithms that infer posterior distributions over graphical models using observed data, integrating prior knowledge to model uncertainty.
• They utilize methods such as dynamic programming, stochastic sampling, and differentiable relaxations to navigate vast, combinatorial structure spaces.
• Recent advancements focus on scalability and expressiveness by extending these learners to models like probabilistic circuits and integrating nonparametric priors.

A Bayesian structure learner is an algorithm or statistical method that infers a posterior distribution over possible graphical structures (typically directed acyclic graphs, DAGs) underlying a probabilistic graphical model—most frequently, a Bayesian network—given observed data. In contrast to purely score-based or constraint-based approaches, Bayesian structure learning quantifies epistemic uncertainty by marginals on edges or features, formally encodes prior knowledge (including hierarchical and nonparametric priors), and supports coherent model averaging. Recent research has extended Bayesian structure learning to a wide variety of model classes and inference algorithms, targeting both scalability and expressiveness, including structure learning in BNs, sum-product networks, probabilistic circuits, and at the level of variable instantiations.

1. Formal Foundations of Bayesian Structure Learning

Let $D$ denote data samples and $G$ a candidate graph (e.g., a DAG over variables $X_1,\ldots,X_n$). The core Bayesian structure learning objective is the posterior

$p(G|D) \propto p(D|G) \cdot p(G)$

where $p(G)$ is a structural prior and $p(D|G)$ is the marginal likelihood, i.e., the likelihood integrated over parameters $\theta$ with their prior $p(\theta|G)$:

$p(D|G) = \int p(D|G,\theta) p(\theta|G) d\theta$

The support of $p(G|D)$ is over the combinatorially large (super-exponential in $n$) set of graphs obeying model-specific constraints (acyclicity for BNs, decomposability for PCs/SPNs, etc). The prior $p(G)$ can reflect sparsity, symmetry (block models), or nonparametric structure (e.g., CRP-based partitions).

Bayesian structure learners do not settle for a single “best” $G$, but rather perform inference over $p(G|D)$ (or in some settings jointly $p(G,\theta|D)$), supporting queries about posterior edge marginals, feature posteriors, and Bayesian model averaging.

2. Methodologies and Inference Techniques

Bayesian structure learning encompasses algorithms for exact marginalization, stochastic sampling, variational or differentiable optimization, and hybridizations. The principal approaches include:

• Dynamic Programming-Based Marginalization: For moderate $n$ (typically $n\leq 16$ for BNs), DP algorithms exploit modular decomposability of scores to exhaustively compute all node-local parent set scores, aggregate them across orderings or parent-sets, and enable exact computation of edge posterior probabilities and sampling of DAGs or CPDAGs under order-modular priors (He et al., 2015).
• Stochastic Sampling and Model Averaging: MCMC schemes such as MC³, birth-death MCMC, and order-MCMC draw samples from $p(G|D)$ or $p(G,\theta|D)$, allowing Monte Carlo estimates of feature posteriors. Methods like DDS/IW-DDS leverage DP tables for efficient, unbiased DAG sampling (Kuipers et al., 2018He et al., 2015).
• Probabilistic Circuits and Sum-Product Networks: For structure spaces beyond DAGs, Bayesian learners are defined over tractable probabilistic circuit classes—e.g., Bayesian SPNs or deterministic PCs—where structural parameters (scope functions, cutset splits) are assigned priors, and structure learning proceeds via Bayesian marginal likelihood or fully generative models (Trapp et al., 2019Yang et al., 2023).
• Bootstrap and Recursive Model Averaging: The recursive-bootstrap-based Graph Generative Tree (B-RAI) builds a tree of scored CPDAGs, integrating CI-test uncertainty and supporting posterior sampling or MAP selection (Rohekar et al., 2018).
• Variational and Differentiable Relaxations: DiBS leverages a differentiable, continuous relaxation of $G$ via embeddings $Z$, augmented with differentiable acyclicity constraints, enabling fully differentiable approximate inference (e.g., via SVGD) over a soft graph posterior $p(G|Z)$ that can be annealed to hard samples (Lorch et al., 2021).
• Hierarchical and Nonparametric Priors: Nonparametric stochastic blockmodel priors (e.g., CRP-based partitions) and hierarchical Bayesian frameworks encode structured prior knowledge about classes of variables and their edge probabilities, typically inferred via MCMC over partitions and graphs (Mansinghka et al., 2012).

3. Computational Strategies and Scalability

Bayesian structure learners face severe combinatorial scaling challenges. Innovations for tractable inference include:

Approach	Bottleneck	Scalability Solution
DP/Marginalization	$O(2^n)$ local tables	Parent set size restriction; PCs for marginals (Zhao et al., 18 Nov 2025)
MCMC sampling	Slow mixing	Order-space reduction; skeleton restriction (Kuipers et al., 2018)
Bootstrap recursion	Curse of dimensionality in CI-tests	Recursive bootstrapping; test reuse (Rohekar et al., 2018)
Probabilistic circuits	Handling large parent sets	PC-based marginalization (Zhao et al., 18 Nov 2025)
Differentiable relax.	Discrete graph constraints	Gumbel-softmax reparameterization (Lorch et al., 2021)

Probabilistic circuits offer polynomial-time, exact marginalization for all $n-1$ candidate parents (removing the exponential DP bottleneck), enabling Bayesian order-based structure search where all possible parent-sets are supported, not just those with bounded size. Posterior inference can thus be amortized over arbitrarily many queries at inference time, reversing the typical trade-off in classic score-based or DP-based BNs (Zhao et al., 18 Nov 2025).

DP-based learners remain restricted to moderate $n$, but exact sample-based methods (e.g., DDS, IW-DDS) provide unbiased structure posterior estimates and support non-modular features (He et al., 2015). For high-dimensional settings (hundreds to thousands of variables), recursive bootstrap algorithms (B-RAI), scalable Bayesian circuit learners, and hybrid approaches (e.g., BiDAG) dominate (Rohekar et al., 2018Yang et al., 2023Kuipers et al., 2018).

4. Extensions: Priors, Nonparametric Models, and Local/Hybrid Schemes

The Bayesian formalism enables a range of modeling extensions:

• Structured and Nonparametric Priors: Hierarchical blockmodels exploit a CRP over classes and Beta priors on class-class edge probabilities, supporting automatic discovery of "roles" and edge patterns, and yielding improved sample efficiency in small data regimes (Mansinghka et al., 2012).
• Modeling Feature Uncertainty: By supporting model averaging over CPDAGs, Markov-blanket features, and path-queries, Bayesian learners provide credible intervals and posterior feature probabilities, in contrast to point-estimates from MAP learners or constraint-based algorithms (Rohekar et al., 2018Kuipers et al., 2018).
• Bootstrap and Test-Level Model Averaging: B-RAI directly integrates uncertainty in CI-test outcomes via recursive, order-level bootstrapping, producing a Graph Generative Tree (GGT) whose leaves are scored CPDAGs. Sampling from this tree yields approximate posterior draws, encompassing both independence-test and structural uncertainty (Rohekar et al., 2018).
• Local and Hybrid Bayesian Learners: Local Bayesian learners, such as score-based local learning (SLL), focus on Markov blanket identification for target nodes using locally consistent, decomposable Bayesian scores, scaling structure learning to large domains via local optimization plus heuristic merges (Niinimaki et al., 2012).
• Structure Learning Beyond BNs: Bayesian approaches have been developed for sum-product networks (fully Bayesian SPN inference over computational graph, scope function, and parameters (Trapp et al., 2019)), deterministic probabilistic circuits (Bayesian marginal likelihood objective, cutset learning (Yang et al., 2023)), and instantiation-level knowledge bases (context-specific fusion of minimal-entropy per-world DAGs (Yakaboski et al., 2023)).

5. Empirical Performance, Use Cases, and Limitations

Empirical assessment of Bayesian structure learners emphasizes:

• Accuracy and Calibration: Bayesian model averaging via DP, MCMC, recursive boostrap, or stochastic variational optimization yields improved calibration of edge probabilities, AUROC, SHD, and predictive log-likelihood relative to greedy or classical hybrid schemes, especially in small-sample or under-determined settings (Rohekar et al., 2018Lorch et al., 2021).
• Uncertainty Quantification: Edge and feature marginal posteriors derived from $p(G|D)$ enable uncertainty-aware causal inference, robust predictive modeling, and principled selection of interventions (Lorch et al., 2021).
• Scalability: Recursive bootstrap (B-RAI) achieves posterior sampling and MAP model recovery up to $n\sim 700$ nodes; circuit-based Bayesian learners surpass parent-set restrictions even at $n\sim 20-30$ (Zhao et al., 18 Nov 2025Rohekar et al., 2018).
• Limitations: Exponential-time algorithms (DP-based, order enumeration) remain infeasible for large $n$; some PC-based or differentiable Bayesian learners are specialized to particular classes of scoring functions (e.g., linear-Gaussian BGe); empirical performance can degrade for highly dense or cyclic domains unless circuit/inference architecture is adapted.

6. Outlook and Recent Advances

Recent advances focus on expanded modeling expressivity, computational tractability, and realistic data challenges:

• Continuous and Nonlinear Structure Learning: Differentiable relaxations map discrete graph structure to continuous latent codes, enabling end-to-end SVGD-based approximation of $p(G,\theta|D)$ for general conditional distributions, including nonparametric neural CPDs (Lorch et al., 2021).
• Bayesian Circuits and Mixtures: Mixture-of-circuit models enable structural EM over Bayesian circuit learners, attaining state-of-the-art generative modeling with tractable posteriors (Yang et al., 2023).
• Scalable Handling of Nonparametric and Context-Specific Uncertainty: Instantiation-level structure learning via Bayesian knowledge bases decomposes the learning problem over observed exemplars, handling under-determined domains (e.g., genomics) and recovering both latent cycles and incompletely specified dependencies (Yakaboski et al., 2023).
• Structured Prior Induction and Transfer Learning: Blockmodel and transfer-focused priors encode type-level modularity, enabling improved sample efficiency and discovery of latent functional classes in small or transfer learning regimes (Mansinghka et al., 2012).
• Amortized and Tractable Marginalization: Probabilistic circuits trained to mimic node-local scores enable exact-on-the-fly marginalization in Bayesian structure search, overcoming longstanding parent-set restrictions (Zhao et al., 18 Nov 2025).

Bayesian structure learners continue to be foundational in the causal discovery and probabilistic modeling literature, with modern research unifying diverse graphical model classes, scalable inference techniques, and sophisticated priors to yield credible, tractable structure learning at scale.