Equivalence Score in Bayesian Networks

• Equivalence score is a method that assigns equal scores to all directed acyclic graphs in the same Markov equivalence class, defined by identical skeletons and v-structures.
• It underlies efficient structure learning algorithms like Greedy Equivalence Search and its variants by reducing redundancy in the search space through decomposable scoring.
• Extensions of the concept apply to latent, partially observed, and cyclic causal models, as well as staged trees and formal code verification, highlighting its broad applicability.

“Equivalence score” denotes a family of scoring constructions in which a score is attached to an equivalence relation rather than to a single syntactic representation. In the canonical Bayesian-network setting, a score-equivalent scoring function assigns the same value to all directed acyclic graphs (DAGs) in the same Markov equivalence class, where equivalence is characterized by identical skeletons and identical v-structures and is commonly represented by a completed partially directed acyclic graph (CPDAG) or essential graph (Chickering, 2013). Closely related notions appear in selective search over Bayesian-network equivalence classes, latent-variable and partially observed causal discovery, staged trees, survey-based classifier evaluation, and formal verification of code-specification pairs (Chickering et al., 2015, Ng et al., 19 May 2026, Hughes et al., 2022, Resnick et al., 2021, Zeng et al., 7 Oct 2025).

1. Core meaning in structure learning

In Bayesian network structure learning, two DAGs are equivalent if they represent the same set of conditional independence relationships; equivalently, they have the same skeleton and the same set of v-structures. When a scoring function is score equivalent, it assigns the same score to all DAGs in such a class, so the natural search object is the equivalence class rather than an individual DAG (Chickering, 2013).

This viewpoint leads to the representation of equivalence classes by completed PDAGs or essential graphs. Directed edges in a completed PDAG are compelled, undirected edges are reversible, and the representation is unique for the class. In this setting, the “equivalence score” of a class is simply the score of any DAG belonging to that class:

$S(C) := S(G), \quad \forall G \in C .$

For decomposable Bayesian-network scores, the score of a DAG factors over local parent sets, and the score of an essential graph or CPDAG is therefore inherited from any consistent extension in its class (Auvray et al., 2012).

A central consequence is algorithmic. If a scoring function already returns the same value for all members of an equivalence class, then search over individual DAGs introduces redundancy. Search over equivalence classes removes this redundancy and aligns the search space with the statistical indistinguishability induced by observational data (Chickering, 2013).

2. Equivalence scores in greedy search over Bayesian-network classes

Greedy Equivalence Search (GES) and its variants operationalize score equivalence by treating CPDAGs as search states and evaluating local insertions and deletions through score differences. The scoring function is assumed to satisfy three properties: score equivalence, local consistency, and decomposability. In the decomposable case,

$$\mathrm{Score}(\mathcal{G}, \mathcal{D}) = \sum_i \mathrm{Score}(X_i, \mathrm{Pa}_i).$$

The standard search uses a forward phase that greedily adds edges and a backward phase that greedily removes edges, always choosing the optimal single-step transformation according to score difference (Chickering et al., 2015).

Selective Greedy Equivalence Search (SGES) restricts the set of deletion operators considered in the backward phase while retaining the large-sample optimality guarantees of GES. The restriction is defined by a hereditary and equivalence-invariant property $\Pi$: only $\Pi$-consistent deletions are considered, meaning deletions for which $\Pi$ holds in the induced subgraph over the affected endpoints and their common descendants. Under data sampled independently from a distribution that is perfect with respect to a DAG $\mathcal{G}$, SGES identifies $\mathcal{G}$’s equivalence class after a number of score evaluations that is polynomial in the number of nodes and exponential in complexity measures such as maximum-number-of-parents, maximum-clique-size, and v-width. The paper introduces v-width as the maximum size, over all pairs of non-adjacent nodes $X,Y$, of the largest clique among their common children; it is at least as small as, and potentially much smaller than, the other two measures. For property $\Pi^k_{VW}$, SGES runs in $poly(n)\cdot exp(k)$ time, and for polytrees the v-width is $$\mathrm{Score}(\mathcal{G}, \mathcal{D}) = \sum_i \mathrm{Score}(X_i, \mathrm{Pa}_i).$$0, yielding a fully polynomial regime (Chickering et al., 2015).

Less Greedy Equivalence Search (LGES) modifies the insertion step rather than the deletion step. Instead of always applying the highest-scoring insertion, it avoids edge insertions between variables for which the score implies some conditional independence. With a decomposable score, the insertion score difference is local:

$$\mathrm{Score}(\mathcal{G}, \mathcal{D}) = \sum_i \mathrm{Score}(X_i, \mathrm{Pa}_i).$$1

LGES retains sample-limit recovery of the true equivalence class from observational and interventional data, even with misspecified prior assumptions, while experiments report up to a $$\mathrm{Score}(\mathcal{G}, \mathcal{D}) = \sum_i \mathrm{Score}(X_i, \mathrm{Pa}_i).$$2-fold speed-up and reduced structural error relative to GES (Ejaz et al., 27 Jun 2025).

3. Extensions to latent, partially observed, and cyclic causal models

For latent-variable causal models, score equivalence requires a broader notion of indistinguishability than ordinary Markov equivalence on observed DAGs. In linear latent variable causal models of the form

$$\mathrm{Score}(\mathcal{G}, \mathcal{D}) = \sum_i \mathrm{Score}(X_i, \mathrm{Pa}_i).$$3

the marginal covariance over the observed variables is

$$\mathrm{Score}(\mathcal{G}, \mathcal{D}) = \sum_i \mathrm{Score}(X_i, \mathrm{Pa}_i).$$4

The distribution set $$\mathrm{Score}(\mathcal{G}, \mathcal{D}) = \sum_i \mathrm{Score}(X_i, \mathrm{Pa}_i).$$5 is the set of such marginal covariances allowed by a graph $$\mathrm{Score}(\mathcal{G}, \mathcal{D}) = \sum_i \mathrm{Score}(X_i, \mathrm{Pa}_i).$$6, and algebraic equivalence means that two structures induce the same set of polynomial equality constraints on $$\mathrm{Score}(\mathcal{G}, \mathcal{D}) = \sum_i \mathrm{Score}(X_i, \mathrm{Pa}_i).$$7. The proposed scoring functions are

$$\mathrm{Score}(\mathcal{G}, \mathcal{D}) = \sum_i \mathrm{Score}(X_i, \mathrm{Pa}_i).$$8

and

$$\mathrm{Score}(\mathcal{G}, \mathcal{D}) = \sum_i \mathrm{Score}(X_i, \mathrm{Pa}_i).$$9

For Markov equivalent structures, both scores are equal. Under generalized faithfulness, minimizing the dimension score is consistent for identifying the true structure up to algebraic equivalence; under stricter structural assumptions, the method identifies the structure up to Markov equivalence (Ng et al., 19 May 2026).

A related development for partially observed linear causal models defines a maximum-likelihood score over the observed covariance:

$\Pi$0

In that framework, graphs maximizing likelihood and minimizing dimension are algebraically equivalent to the truth in the large-sample limit, and under the Generalized N Factor Model this algebraic equivalence coincides with Markov equivalence. The resulting Latent variable Greedy Equivalence Search searches over equivalence classes with well-defined operators and is globally consistent for the true Markov equivalence class under the stated assumptions (Dong et al., 5 Oct 2025).

For cyclic and acyclic linear Gaussian directed graphs, the literature also defines distribution equivalence directly by equality of distribution sets:

$\Pi$1

where $\Pi$2 is the set of precision matrices generated by graph $\Pi$3. In this setting, the paper explicitly states that the equivalence score is not a single scalar; it is determined by whether equivalence or quasi-equivalence holds. Structure learning is then based on an $\Pi$4-regularized negative log-likelihood, and the global minimizer identifies the quasi-equivalence class asymptotically under generalized faithfulness (Ghassami et al., 2019).

4. Staged trees and polyhedral formulations of score equivalence

Staged trees provide a distinct probabilistic graphical setting in which equivalence is defined by equality of represented distributions. Two staged trees are statistically equivalent if and only if they encode the same set of probability distributions. Because equivalent staged trees can encode different causal hypotheses while remaining observationally indistinguishable, a suitable score should assign the same value to all members of the equivalence class (Hughes et al., 2022).

The Bayesian Dirichlet metric for staged trees depends on Dirichlet hyperparameters assigned to stages and outgoing edges. The paper shows that the previously used CS-BDeu construction is not score-equivalent for non-stratified staged trees, and introduces BDepu, a Bayesian Dirichlet scoring function based on path uniformity and mass conservation. BDepu assigns equal prior mass to each root-to-leaf path and propagates this backward through the staged tree. The main theorem proves that BDepu is invariant under the swap and resize operators that connect statistically equivalent trees, yielding the first scoring metric that is fully score-equivalent for staged trees regardless of stratification; on stratified staged trees it coincides with CS-BDeu (Hughes et al., 2022).

In Bayesian-network polyhedral theory, score equivalence can be expressed as a linear-algebraic property of objectives over family-variable encodings. An objective vector $\Pi$5 is score equivalent if

$\Pi$6

The paper characterizes score-equivalent objectives through the representation

$\Pi$7

where $\Pi$8 is a standardized set function, and establishes a one-to-one correspondence between score-equivalent faces of the family-variable polytope, faces of the characteristic-imset polytope, and standardized supermodular functions. A facet-defining score-equivalent inequality corresponds to an extreme ray of the cone of standardized supermodular functions. The same analysis also shows that when maximizing a score-equivalent objective, non-score-equivalent family-variable facets can be eliminated, but score-equivalent facets alone are not sufficient because some necessary inequalities come from facets of the characteristic-imset polytope that do not define facets in the family-variable model (Cussens et al., 2015).

5. Human-calibrated and formally verified equivalence scores

Outside graphical-model structure learning, “equivalence score” can denote a calibrated evaluation target rather than invariance over model classes. In survey settings with noisy or subjective labels, Survey Equivalence defines the score of a classifier as the minimum number of human raters needed to produce the same expected score as the classifier when both are evaluated against a held-out human rater. The procedure computes a human survey power curve $\Pi$9, where $\Pi$0 is the expected score obtained by combining $\Pi$1 raters, and compares the classifier’s score $\Pi$2 against that curve; if $\Pi$3, the survey equivalence is interpolated linearly. The result is a human-interpretable quantity: a classifier can be described as being “as good as” some number of raters (Resnick et al., 2021).

In formal verification of LLM-generated code, VeriEquivBench defines the equivalence score as a formally grounded, ground-truth-free metric for code-specification pairs. For a method $\Pi$4 with arguments $\Pi$5, output $\Pi$6, and specification $\Pi$7, equivalence requires the bidirectional implication

$\Pi$8

The first direction checks that the implementation satisfies the specification; the second checks that the specification uniquely and exhaustively describes the implementation. If both directions are mechanically verified, the pair receives an equivalence score of $\Pi$9; otherwise it receives $\Pi$0. This construction is used in a benchmark of $\Pi$1 complex algorithmic problems (Zeng et al., 7 Oct 2025).

Setting	Equivalence notion	Score output
Bayesian networks	Same skeleton and v-structures within a Markov equivalence class	Same score for all DAGs in the class
Survey settings	Same expected predictive score as a survey of $\Pi$2 raters	Minimum equivalent number of raters
Formally verifiable code	Bidirectional implication between code and specification	Binary $\Pi$3 success

These formulations differ in codomain and operational meaning, but each replaces literal representation matching with a criterion grounded in observational, semantic, or formal indistinguishability (Resnick et al., 2021, Zeng et al., 7 Oct 2025).

6. Scope, limitations, and deliberate violations of score equivalence

Score equivalence is often treated as a desideratum because observational data typically identify only an equivalence class. However, the literature also contains deliberate departures from score equivalence when stronger structural assumptions make finer identification possible. In an empirical Bayes approach to order-based DAG learning, the prior imposes equal error variances, leading to a non-decomposable posterior score

$\Pi$4

Because this score is non-decomposable, Markov equivalent DAGs can receive different scores. The method is therefore explicitly “without score equivalence,” and under the minimal trace or equal-variance identifiability condition it concentrates on the unique true DAG rather than merely on its equivalence class; the same paper reports improved MCMC mixing and stronger directional recovery, while also noting bias when error variances are highly heterogeneous (Chang et al., 2022).

A related misconception is that every equivalence-based evaluation defines a specialized scalar called “equivalence score.” Some benchmarks instead use ordinary task accuracy on an equivalence-checking problem. EquiBench evaluates whether two programs produce identical outputs for all possible inputs and reports accuracy over balanced equivalent and inequivalent program pairs; in the most challenging categories, the best accuracies are $\Pi$5 and $\Pi$6, only modestly above the $\Pi$7 random baseline (Wei et al., 18 Feb 2025). This suggests that “equivalence score” is best understood as a family of constructions whose common feature is invariance or calibration with respect to an equivalence relation, not a single universal metric.