--- title: Equivalence Score in Bayesian Networks url: https://www.emergentmind.com/topics/equivalence-score type: topic --- # Equivalence Score in Bayesian Networks “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 [1302.3566]. 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 [1506.02113, 2605.20396, 2206.15322, 2106.01254, 2510.06296]. ## 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 [1302.3566]. 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 [1301.0553]. 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 [1302.3566]. ## 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 [1506.02113]. 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 \(1\), yielding a fully polynomial regime [1506.02113]. 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: $$ \Delta S = s(Y, pa^G_Y \cup \{X\}) - s(Y, pa^G_Y). $$ 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 \(10\)-fold speed-up and reduced structural error relative to GES [2506.22331]. ## 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 $$ L = CL + E_L, \qquad X = BL + E_X, $$ the marginal covariance over the observed variables is $$ \Sigma_X = B(I - C)^{-1}\Omega_L (I - C)^{-\top} B^\top + \Omega_X. $$ The distribution set \(\mathcal{M}(\mathcal{G})\) is the set of such marginal covariances allowed by a graph \(\mathcal{G}\), and algebraic equivalence means that two structures induce the same set of polynomial equality constraints on \(\Sigma_X\). The proposed scoring functions are $$ \mathrm{score}_{\dim}(\mathcal{G}, \mathbf{D}) = \begin{cases} \dim(\mathcal{G}) & \text{if } \mathcal{G} \text{ can generate sample covariance } S \\ \infty & \text{otherwise} \end{cases} $$ and $$ \mathrm{score}_{\mathrm{BIC}}(\mathcal{G}, \mathbf{D}) = \mathcal{L}^*(\mathcal{G}, \mathbf{D}) + \frac{\log T}{2}\dim(\mathcal{G}). $$ 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 [2605.20396]. A related development for partially observed linear causal models defines a maximum-likelihood score over the observed covariance: $$ \mathrm{score}_{\mathrm{ML}}(G, \mathcal{D}) = \max_{(F,\Omega):\, \mathrm{supp}(F)\subseteq \mathrm{supp}(F_G),\ \Omega \in \mathrm{diag}(\mathbb{R}_{>0}^{n+m})} \mathcal{L}(F,\Omega). $$ 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 [2510.04378]. For cyclic and acyclic linear Gaussian directed graphs, the literature also defines distribution equivalence directly by equality of distribution sets: $$ G_1 \equiv G_2 \iff \Theta(G_1)=\Theta(G_2), $$ where \(\Theta(G)\) is the set of precision matrices generated by graph \(G\). 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 \(\ell_0\)-regularized negative log-likelihood, and the global minimizer identifies the quasi-equivalence class asymptotically under generalized faithfulness [1910.12993]. ## 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 [2206.15322]. 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 [2206.15322]. In Bayesian-network polyhedral theory, score equivalence can be expressed as a linear-algebraic property of objectives over family-variable encodings. An objective vector \(\mathrm{o}\) is score equivalent if $$ G \sim H \implies \langle \mathrm{o}, n_G \rangle = \langle \mathrm{o}, n_H \rangle . $$ The paper characterizes score-equivalent objectives through the representation $$ \mathrm{o}(a \mid B) = m(\{a\}\cup B) - m(B), $$ where \(m\) 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 [1503.00829]. ## 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 \(c_0,c_1,\ldots,c_{K-1}\), where \(c_k\) is the expected score obtained by combining \(k\) raters, and compares the classifier’s score \(h_{\text{score}}\) against that curve; if \(c_{k-1}