Hybrid Moment-Ratio Scoring (H-MRS)

• The paper introduces H-MRS, a two-stage causal discovery method that uses a log-linear SEM and a raw-scale moment-ratio score to recover causal order in strictly positive data.
• It employs a greedy ordering phase based on moment-ratio minimization followed by Elastic Net sparsity for precise parent selection in the DAG construction.
• Empirical results show H-MRS outperforms classical methods like PC and GES on synthetic and real-world positive-valued systems, offering higher precision and lower structural error.

Searching arXiv for the target paper and closely related methods mentioned in the supplied data. I’m looking up the cited and related arXiv records to anchor the article in the current arXiv literature. Hybrid Moment-Ratio Scoring (H-MRS) is a two-stage causal discovery method for learning directed acyclic graphs (DAGs) from strictly positive-valued variables whose relationships are more naturally multiplicative than additive. Introduced in "A novel hybrid approach for positive-valued DAG learning" (Zhao, 10 Apr 2026), it assumes a log-linear structural equation model, uses a raw-scale moment-ratio score to recover a causal ordering, and then applies a separate sparse regression step to identify parent sets. The method is motivated by settings such as gene expression, finance, economics, and population counts, where learning a DAG on the original scale with ordinary additive models is misspecified and where positivity constraints are intrinsic rather than incidental.

1. Problem setting and structural assumptions

H-MRS is designed for a DAG over variables $X_1,\dots,X_p$, all strictly positive. Its structural equation model is

$$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$

or equivalently,

$$X_j = \exp\!\left(\theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j\right).$$

This form guarantees positivity and is meant to reflect systems where the log of a quantity responds linearly to its causes on the original scale (Zhao, 10 Apr 2026).

The paper positions this model against standard causal discovery tools that are not ideal for positive-valued data. PC and GES are described as commonly grounded in Gaussian or additive assumptions, while LiNGAM and DirectLiNGAM can identify full causal direction under non-Gaussianity but still assume an additive structural equation model. The central modeling claim is therefore not merely that the variables are positive, but that their interactions are often multiplicative; positive data may behave multiplicatively, for example if one variable scales another exponentially.

The noise terms satisfy four conditions: $\mathbb{E}[\epsilon_j]=0$, $\mathrm{Var}(\epsilon_j)=\sigma_j^2>0$, $|\epsilon_j|\le B$ almost surely, and $\epsilon_j \perp X_k$ for all $k\in \mathrm{Pa}(j)$. The bounded-noise assumption is emphasized because it ensures finite moments of the positive variables, which the theory relies on. For finite-sample analysis, the paper additionally assumes sub-exponential tails for $X_j$ and $\mathbb{E}[X_j\mid S]$, together with a well-behaved design covariance matrix for regressors.

2. Moment-ratio criterion and the ordering principle

The central population score is the moment ratio

$$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$0

where $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$1 is a candidate conditioning set, typically a set of already-ordered variables. The score can be rewritten as

$$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$2

This expression implies that the score is at least $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$3, and it becomes smaller when conditioning explains more of the variation in $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$4 (Zhao, 10 Apr 2026).

The key theoretical feature is the plateau property. If $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$5 contains all true parents of $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$6 and contains no descendants of $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$7, then $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$8 reaches its minimum. Any superset of the true parents consisting only of nondescendants attains the same minimum score. If $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$9 is missing at least one parent, the score is strictly larger. This yields an ordering mechanism: the variable with the smallest score when conditioned on the already-selected set is the one whose parents are already represented in the ordering.

The proof is based on the law of total variance, conditional variance decomposition, conditional Jensen’s inequality, and local Markov properties of DAGs. Intuitively, adding true parents reduces unexplained conditional variance, while adding irrelevant nondescendants does not change the conditional mean, so the score remains on a flat minimum plateau. A frequent misconception is that the score identifies the minimal parent set directly. The paper explicitly rejects that interpretation: the plateau property implies that supersets of the true parent set can share the same minimum, so ordering and parent recovery must be separated.

3. Hybrid estimation architecture

The method is "hybrid" because it uses different scales for different tasks: log scale to estimate conditional expectations stably via Ridge regression, and raw scale to compute the moment-ratio score that drives ordering. This separation is presented as central to the design (Zhao, 10 Apr 2026).

For each node $$X_j = \exp\!\left(\theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j\right).$$0 and candidate set $$X_j = \exp\!\left(\theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j\right).$$1, H-MRS fits

$$X_j = \exp\!\left(\theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j\right).$$2

by Ridge regression:

$$X_j = \exp\!\left(\theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j\right).$$3

The conditional mean on the original scale is then approximated by exponentiating the fitted log prediction:

$$X_j = \exp\!\left(\theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j\right).$$4

The paper emphasizes that Ridge is used because it gives stable predictions and avoids selection bias from sparsity penalties. This matters because the moment-ratio ordering requires consistent conditional expectation estimates across many candidate sets. For numerical stability, near-zero values are clipped before taking logs:

$$X_j = \exp\!\left(\theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j\right).$$5

The empirical score is

$$X_j = \exp\!\left(\theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j\right).$$6

This score is explicitly described as scale-invariant: multiplying $$X_j = \exp\!\left(\theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j\right).$$7 by a positive constant multiplies numerator and denominator by the same factor $$X_j = \exp\!\left(\theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j\right).$$8, leaving the ratio unchanged.

4. Two-stage algorithmic procedure

H-MRS operates in two phases. In the greedy causal ordering phase, the procedure starts with no ordered variables. At step $$X_j = \exp\!\left(\theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j\right).$$9, for every remaining variable $\mathbb{E}[\epsilon_j]=0$0, the candidate set $\mathbb{E}[\epsilon_j]=0$1 is the set of previously ordered variables. Ridge is fitted on $\mathbb{E}[\epsilon_j]=0$2 using predictors $\mathbb{E}[\epsilon_j]=0$3, $\mathbb{E}[\epsilon_j]=0$4 is computed, and the variable with the smallest score is appended to the ordering (Zhao, 10 Apr 2026).

For the first variable, $\mathbb{E}[\epsilon_j]=0$5, and the score reduces to

$\mathbb{E}[\epsilon_j]=0$6

which the paper describes as essentially a coefficient-of-variation-type quantity. The greedy logic is that a node can be ordered early if its parents are already available among earlier nodes, making its moment ratio small.

Once the ordering is known, parent selection is performed by Elastic Net rather than Ridge. For each node $\mathbb{E}[\epsilon_j]=0$7, the regression uses only predecessors:

$\mathbb{E}[\epsilon_j]=0$8

This second stage is necessary because moment-ratio scoring identifies an ordering but not the exact minimal parent set. Elastic Net is chosen because the $\mathbb{E}[\epsilon_j]=0$9 component promotes sparsity and the $\mathrm{Var}(\epsilon_j)=\sigma_j^2>0$0 component stabilizes selection when predictors are correlated. After fitting Elastic Net, coefficients are thresholded at $\mathrm{Var}(\epsilon_j)=\sigma_j^2>0$1, and only the top-$\mathrm{Var}(\epsilon_j)=\sigma_j^2>0$2 largest remaining coefficients are retained as parents.

5. Theoretical properties

The paper states three main theoretical results. Proposition 1 formalizes the plateau property. Under the log-linear model and bounded independent noise, it establishes that $\mathrm{Var}(\epsilon_j)=\sigma_j^2>0$3, that $\mathrm{Var}(\epsilon_j)=\sigma_j^2>0$4 whenever $\mathrm{Var}(\epsilon_j)=\sigma_j^2>0$5, and that the minimum is attained exactly when

$\mathrm{Var}(\epsilon_j)=\sigma_j^2>0$6

The interpretation given is that true parents are the minimal information needed to explain $\mathrm{Var}(\epsilon_j)=\sigma_j^2>0$7, while adding nondescendants does not improve the conditional mean, so the score plateaus (Zhao, 10 Apr 2026).

Proposition 2 gives finite-sample concentration for the empirical score. For fixed $\mathrm{Var}(\epsilon_j)=\sigma_j^2>0$8, the estimate concentrates around the population ratio at rate $\mathrm{Var}(\epsilon_j)=\sigma_j^2>0$9. More precisely, for large enough $|\epsilon_j|\le B$0,

$|\epsilon_j|\le B$1

The proof decomposes the ratio error into numerator and denominator errors and uses Bernstein concentration for sub-exponential variables, Ridge consistency for $|\epsilon_j|\le B$2, and control of the prediction error in $|\epsilon_j|\le B$3. This supports the ordering step when the population gap between true and false candidate sets is sufficiently large.

Proposition 3 concerns computational complexity. The paper states a time complexity of

$|\epsilon_j|\le B$4

with $|\epsilon_j|\le B$5 and $|\epsilon_j|\le B$6 for $|\epsilon_j|\le B$7 predictors and $|\epsilon_j|\le B$8 coordinate-descent iterations, together with space complexity $|\epsilon_j|\le B$9. For asymptotic consistency, the stated hyperparameter conditions are $\epsilon_j \perp X_k$0 and $\epsilon_j \perp X_k$1, $\epsilon_j \perp X_k$2 and $\epsilon_j \perp X_k$3 for Elastic Net, and $\epsilon_j \perp X_k$4.

6. Empirical behavior, scope, and limitations

The experimental study uses synthetic data generated from the same log-linear model. For each simulation, a random topological order is sampled, edges are added from earlier to later nodes with maximum in-degree $\epsilon_j \perp X_k$5, parameters are drawn as $\epsilon_j \perp X_k$6, $\epsilon_j \perp X_k$7, and $\epsilon_j \perp X_k$8, and samples are generated by

$\epsilon_j \perp X_k$9

The experiments test $k\in \mathrm{Pa}(j)$0 with $k\in \mathrm{Pa}(j)$1, and report Structural Hamming Distance (SHD), Precision, Recall, and F1 score. PC, GES, and DirectLiNGAM serve as baselines; for PC and GES, which return CPDAGs rather than fully directed DAGs, undirected edges are treated as bidirected in the adjacency matrix for comparison (Zhao, 10 Apr 2026).

H-MRS performs best overall on these synthetic positive multiplicative systems. For $k\in \mathrm{Pa}(j)$2, the reported results are: $k\in \mathrm{Pa}(j)$3, SHD $k\in \mathrm{Pa}(j)$4, Precision $k\in \mathrm{Pa}(j)$5, Recall $k\in \mathrm{Pa}(j)$6, F1 $k\in \mathrm{Pa}(j)$7; $k\in \mathrm{Pa}(j)$8, SHD $k\in \mathrm{Pa}(j)$9, Precision $X_j$0, Recall $X_j$1, F1 $X_j$2; $X_j$3, SHD $X_j$4, Precision $X_j$5, Recall $X_j$6, F1 $X_j$7. For $X_j$8, the reported results are: $X_j$9, SHD $\mathbb{E}[X_j\mid S]$0, Precision $\mathbb{E}[X_j\mid S]$1, Recall $\mathbb{E}[X_j\mid S]$2, F1 $\mathbb{E}[X_j\mid S]$3; $\mathbb{E}[X_j\mid S]$4, SHD $\mathbb{E}[X_j\mid S]$5, Precision $\mathbb{E}[X_j\mid S]$6, Recall $\mathbb{E}[X_j\mid S]$7, F1 $\mathbb{E}[X_j\mid S]$8; $\mathbb{E}[X_j\mid S]$9, SHD $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$00, Precision $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$01, Recall $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$02, F1 $$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$03. Compared to baselines, H-MRS generally has much higher precision than PC and GES, better or comparable recall, much lower SHD, and stronger performance than DirectLiNGAM on these multiplicative positive-valued systems. The paper also notes that performance decreases somewhat with more nodes, as expected.

The real-data application uses financial data from 2,223 companies and 19 positive-valued variables. The learned DAG is interpreted as showing Equity Capital as an upstream source and Interest Expense as a major system-wide driver, with economically sensible pathways through operating profit, leverage, working capital, and valuation. This is presented as evidence that the method can recover interpretable structures in real positive-valued systems.

The paper is explicit about its domain of applicability and its failure modes. H-MRS is suggested to be most suitable when all variables are strictly positive, causal effects are multiplicative or exponential in nature, the true system is reasonably well approximated by a DAG, the log-linear model is plausible, there are no many zero values or feedback cycles, and the graph is sparse enough for the greedy ordering plus sparse selection to work well. Its limitations include zero-inflated data, which can break the log transform; cyclic systems, which violate acyclicity; strong model misspecification, under which the moment-ratio ordering may fail; higher in-degree or denser graphs, where the greedy ordering plus thresholded Elastic Net may struggle; and finite-sample estimation error when the sample is too small relative to graph size or the moment-ratio separation is weak. The paper suggests hurdle or zero-inflated extensions as future work.

Within prior work, the closest method discussed is Moments Ratio Scoring (MRS) for Poisson structural equation models. H-MRS adapts that philosophy to continuous positive-valued data by using the ratio

$$\log X_j = \theta_j + \sum_{k \in \mathrm{Pa}(j)} \beta_{kj} X_k + \epsilon_j,$$04

rather than Poisson mean-variance structure. Both methods share the high-level idea that the score decreases as the conditioning set becomes sufficient for the true parents. The paper also contrasts H-MRS with post-nonlinear causal models, describing its semi-log structure as a special, more computationally tractable case with an explicit moment-ratio plateau. A plausible implication is that the distinctive contribution of H-MRS is not positivity alone, but the combination of a positivity-preserving log-linear SEM, raw-scale moment-ratio ordering, and a separate sparse parent-selection stage.