Mutual-information-inspired heuristics for constraint-based causal structure learning

Abstract

In constraint-based approaches to Bayesian network structure learning, when the assumption of orientation-faithfulness is violated, not only the correctness of edge orientation can be greatly degraded, the soaring cost of conditional independence testing also limits their applicability in learning very large causal networks. Inspired by the strong connection between the degree of mutual information shared by two variables and their conditional independence, we extend the PC-MI algorithm in two ways: (a) the Weakest Edge-First (WEF) strategy implemented in PC-MI is further integrated with Markov-chain consistency to reduce the number of independence testing and sustain the number of false positive edges in skeletal learning; (b) the Smaller Adjacency-Set (SAS) strategy is proposed and we prove that the Smaller Adjacency-Set captures sufficient information for determining whether an unshielded triple forms a v-structure. We have conducted experiments with both low-dimensional and high-dimensional data sets, and the results indicate that our MIIPC approach outperforms the state-of-the-art approaches in both the quality of learning and the execution time.

Introduction

Bayesian network is a kind of probabilistic graph model. Due to its compact representation and powerful reasoning ability under uncertain environment, it has attracted an increasing number of attention from researchers, and it has been widely used in the fields of neuroscience [1], computer science [2], industrial applications [3], dependability and risk analysis [4].

A Bayesian network is mainly composed of two parts: a directed acyclic graph and a probability distribution table, where the nodes represent random variables, and the edges represent direct dependencies between two variables. The probability distribution table describes the dependence of each variable on its parent nodes. The Bayesian network structure learning techniques in the literature can be roughly grouped into three categories: approaches based on structural equation models, score-based approaches, and constraint-based approaches.

Utilizing an appropriately defined structural equation model to learn causal structures is essentially a “data-driven” approach where the observed data distribution is used to discover the underlying causal relations [5]. Some representative models include the classic Linear Non-Gaussian Acyclic Model, Nonlinear additive noise model, and Post-nonlinear causal model [6], where independence testing is used to identify the causal direction between variables. Others have attempted to formalize causal structure learning as a continuous optimization problem. Such methods include DAG-GNN learning method based on graph neural network [7] and RL-BIC2 learning method based on reinforcement learning technology [8].

In the score-based approaches, a scoring function is used to evaluate how well each candidate structure can fit the observation data, and the final output is one or more directed acyclic graphs with the best score. Since the space of the candidate structures increases exponentially with the number of variables, researchers have investigated various search strategies such as structural space search, equivalence class space search, and variable order space search. There are exact learning approaches that attempt to find the global optimal solution, such as integer linear programming methods, dynamic programming methods, branch and bound methods [9], as well as approximate approaches such as evolutionary methods [10], heuristic algorithms [11], Local–global learning methods [12], [13], surrogate model [14], and bounded tree-width structure optimization [15].

The constraint-based approaches use variable independence testing and edge orientation rules to learn the Bayesian network structure that can best explain the observed dependencies. As shown in Fig. 1, a constraint-based approach typically has three steps.

• Step 1: Skeleton learning (adjacency search). Some conditional independence test, such as $G^2$ test, mutual information and Fisher’s Z-test, is used to identify whether there exists an edge between any two variables.

• Step 2: V-structure recognition. The conditional-separation sets generated in Step 1 are used to identify potential v-structures.

• Step 3: The v-structures identified in Step 2, Meek’s orientation rules[16], together with the observational data, are used to transform the undirected graph produced in Step 1 into a directed acyclic graph.

The classic constraint-based learning approach is the PC algorithm introduced by Spirtes et al. [17]. Researchers have recently tried to extend the constraint-based algorithm to learn from time series data [18] and nonstationary data [19]. Because the PC algorithm is order-dependent, researchers have successively proposed order-independent methods such as PC-stable algorithm [20] and PC-MI algorithm [21].

Most constraint-based approaches take the orientation-faithfulness assumption, which, however, is hard to satisfy in real world situations. Researchers have proposed CPC algorithm [22], MPC algorithm, CPC-stable algorithm and MPC-stable algorithm [20] for learning Bayesian network structures when the orientation-faithfulness assumption is violated. Although these methods can improve the quality of structure learning, they often suffer from high computational cost and tend to generate graphs with many unoriented edges.

In this study, we mainly focus on steps 1 and 2, investigating effective strategies for learning high-quality Bayesian network structures when the orientation-faithfulness assumption is partially violated. Our contribution is threefold:

• We propose an algorithm named MIIPC for learning Bayesian network structures. The adjacency search step of MIIPC is an extension of our PC-MI algorithm [21], where we integrate the WEF heuristic strategy with the notion of Markov-chain consistency, in the hope to effectively reduce the number of conditional independence testing, and to sustain the number of false positive edges.

• We propose the Smaller Adjacency-Set heuristics for v-structure recognition. We also prove that the Smaller Adjacency-Set is surprisingly powerful in the sense that it can capture sufficient information for determining whether an unshielded triple forms a v-structure.

• We experimentally show that the proposed algorithm MIIPC, empowered by the WEF and SAS strategies, outperforms the state-of-the-art approaches in both the quality of causal structure learning and the execution time.

The remainder of the paper is organized as follows. Relevant work and terminology are given in Section 2. In Section 3, we propose the notion of Markov-chain consistency and the Smaller Adjacency-Set (SAS) heuristics. Markov-chain consistency is implemented in the adjacency search step of the MIIPC algorithm to learn skeletons, and the SAS heuristics is utilized in the orientation step of MIIPC for determining v-structures. Experiments and comparisons are covered in Section 4, and Section 5 concludes the paper.