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Learning Directed Acyclic Graphs from Partial Orderings

Ali Shojaie, Wenyu Chen

TL;DR

The paper tackles learning DAG structure from partial causal orderings, focusing on a two-layer bipartite setting where $\mathcal{X}\prec\mathcal{Y}$. It introduces PODAG, a framework that combines an intersection of naive cross-layer edge candidates with a search restricted to conditional Markov blankets, enabling efficient and accurate recovery of cross-layer edges under layering-adjacency-faithfulness. It establishes theoretical guarantees for both low- and high-dimensional regimes, demonstrates improved faithfulness requirements over traditional PC-based methods, and shows practical gains in applications such as eQTL mapping. The work extends naturally to multiple layers and nonparametric models, offering a versatile approach for causal graph learning with partial order information and broad applicability in biology and genomics.

Abstract

Directed acyclic graphs (DAGs) are commonly used to model causal relationships among random variables. In general, learning the DAG structure is both computationally and statistically challenging. Moreover, without additional information, the direction of edges may not be estimable from observational data. In contrast, given a complete causal ordering of the variables, the problem can be solved efficiently, even in high dimensions. In this paper, we consider the intermediate problem of learning DAGs when a partial causal ordering of variables is available. We propose a general estimation framework for leveraging the partial ordering and present efficient estimation algorithms for low- and high-dimensional problems. The advantages of the proposed framework are illustrated via numerical studies.

Learning Directed Acyclic Graphs from Partial Orderings

TL;DR

The paper tackles learning DAG structure from partial causal orderings, focusing on a two-layer bipartite setting where . It introduces PODAG, a framework that combines an intersection of naive cross-layer edge candidates with a search restricted to conditional Markov blankets, enabling efficient and accurate recovery of cross-layer edges under layering-adjacency-faithfulness. It establishes theoretical guarantees for both low- and high-dimensional regimes, demonstrates improved faithfulness requirements over traditional PC-based methods, and shows practical gains in applications such as eQTL mapping. The work extends naturally to multiple layers and nonparametric models, offering a versatile approach for causal graph learning with partial order information and broad applicability in biology and genomics.

Abstract

Directed acyclic graphs (DAGs) are commonly used to model causal relationships among random variables. In general, learning the DAG structure is both computationally and statistically challenging. Moreover, without additional information, the direction of edges may not be estimable from observational data. In contrast, given a complete causal ordering of the variables, the problem can be solved efficiently, even in high dimensions. In this paper, we consider the intermediate problem of learning DAGs when a partial causal ordering of variables is available. We propose a general estimation framework for leveraging the partial ordering and present efficient estimation algorithms for low- and high-dimensional problems. The advantages of the proposed framework are illustrated via numerical studies.
Paper Structure (27 sections, 18 theorems, 46 equations, 5 figures, 2 algorithms)

This paper contains 27 sections, 18 theorems, 46 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Learning Directed Graphs from Partial Orderings
  3. Problem Formulation
  4. Failure of Simple Algorithms
  5. Incorporating Partial Orderings into DAG Learning
  6. A modified PC Algorithm
  7. A new framework
  8. Graph Identification using the Conditional Markov Blanket
  9. Learning High-Dimensional DAGs from Partial Orderings
  10. Screening in low dimensions
  11. Screening in high dimensions
  12. Other Approaches
  13. Extensions and Other Considerations
  14. Learning Edges within Layers
  15. Directed Graphs with Multiple Layers
  16. ...and 12 more sections

Key Result

Lemma 2.1

Assume that $\mathcal{P}$ is faithful with respect to $G$. Let $H^{(0)}$ be the directed bipartite graph defined in eq:H0. Then, if $\{ X_1, \ldots X_p \} \prec \{ Y_1, \ldots Y_q \}$,

Figures (5)

  • Figure 1: A directed graph with four nodes.
  • Figure 2: Toy example illustrating estimation of DAGs from partial orderings in a two-layer network with ${{\mathcal{X}}} \prec {{\mathcal{Y}}}$, ${{\mathcal{X}}}=\{X_1,X_2\}$ and ${{\mathcal{Y}}} = \{Y_1, Y_2\}$. a) The full DAG $G$ with the edges between layers drawn in blue and edges within each layer shown in gray. Here, the true causal relations are linear and the goal is to estimate the bipartite graph $H$ defined by edges $X_1 \rightarrow Y_1$ and $X_2 \rightarrow Y_2$; b) Estimate of $H$ using $H^{(0)}$ in \ref{['eq:H0']}, obtained by regressing each $Y_j$, $j = 1, 2$ on $\{X_1, X_2\}$ using a linear model with $n=1,000$ observations, and drawing an edge if the corresponding coefficient is significant at $\alpha = 0.05$; the graph contains a false positive edge shown by an orange dashed arrow; c) estimate of $H$ using $H^{(-j)}$ in \ref{['eq:Hminusj']} obtained by regressing each $Y_j$, $j = 1, 2$ on $\{X_1, X_2, Y_{-j}\}$ using a linear regression similar to (b); d) estimate of $H$ obtained using the proposed approach.
  • Figure 3: Comparison of $\rho_{\min}^*(\mathcal{L}_{\text{PC}}), \rho_{\min}^*(\mathcal{L}_{\text{PC+}})$ and $\rho_{\min}^*(\mathcal{L}_{\text{PODAG}})$ for recovering the skeleton (left) and the entire DAG (middle). The number of conditional independence tests is shown on the right panel.
  • Figure 4: Average false positive and true positive rates (FPR, TPR) for learning DAGs from partial orderings. The edges are generated from random ER graphs with $50$ (left), $100$ (middle), and $150$ (right) nodes and 3 expected edges per node. Samples are drawn from Gaussian SEM with sample size $n=500$. Partial ordering information supplied to the algorithms in the form of two layers ( first row) or five layers ( second row).
  • Figure 5: Estimated quantitative trait mappings for yeast for $p=50$ randomly selected DNA loci and $q=50$ randomly selected gene expression levels. The plots show the estimated bipartite graph using the proposed PODAG algorithm, $\widehat{H}$; the $\widehat{H}^0$ estimate commonly obtained in eQTL analyses, by regressing the expression levels (nodes in the second layer) on the DNA loci; the $\widehat{H}^{(-j)}$ estimate defined in \ref{['eq:Hminusj']} and the intersection of edges in $\widehat{H}^0$ and $\widehat{H}^{(-j)}$, which is the starting point of PODAG.

Theorems & Definitions (34)

  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 3.1
  • Definition 3.2: Conditional Markov Blanket
  • Lemma 3.3
  • Definition 3.4: Layering-adjacency-faithfulness
  • Theorem 3.5
  • Corollary 3.6
  • Lemma 3.7
  • ...and 24 more