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Mathematics
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Computational Methods and Machine Learning for Causal Inference
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Computational Methods and Machine Learning for Causal Inference

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E1: Mathematics and Computer Science".

Deadline for manuscript submissions: 31 October 2025 | Viewed by 5617

Special Issue Editor

Prof. Dr. Bumba Mukherjee

E-Mail Website
Guest Editor
Department of Political Science, Pennsylvania State University, State College, PA 16802, USA
Interests: spatial statistics; statistical methodology; financial crisis

Special Issue Information

Dear Colleagues,

Assessing causality is challenging in the natural and social sciences. Yet, in recent years, causal inference has become vital for empirical evaluation across several fields such as computer science, economics, epidemiology, medical studies, political science, and sociology. Analyzing causal relationships is also critical for artificial intelligence (AI), as causality is necessary for overcoming limitations of predictions and assessment of correlations by machine learning. Evaluating causality in the context of AI is important as machine learning algorithms are widely used for decision making in key policymaking areas such as child welfare, criminal justice, public health, consumer lending, and medical trials.

In this Special Issue of Mathematics, we introduce readers to recent developments in causal inference across the natural and social sciences. To this end, the Special Issue pursues three goals. The first is to provide a comprehensive introduction to the computational implementation of different causal inference estimators from a historical perspective, where new estimators were developed to overcome the limitations of previous estimators. The second goal is to present original empirical research on computational causal inference and causal machine learning across a variety of fields. The third is to focus on advances in causal machine learning that address causal effect estimation for unstructured data, such as text and images.

Prof. Dr. Bumba Mukherjee
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • computational causal inference
  • machine learning
  • unstructured data

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Published Papers (4 papers)

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Research

18 pages, 1718 KiB  
Article
RCDi: Robust Causal Direction Inference Using INUS-Inspired Asymmetry with the Solomonoff Prior
by Ling Zhao, Zhe Chen, Qinyao Luo, Silu He and Haifeng Li
Mathematics 2025, 13(3), 544; https://doi.org/10.3390/math13030544 - 6 Feb 2025
Viewed by 818
Abstract
Investigating causal interactions between entities is a crucial task across various scientific domains. The traditional causal discovery methods often assume a predetermined causal direction, which is problematic when prior knowledge is insufficient. Identifying causal directions from observational data remains a key challenge. Causal [...] Read more.
Investigating causal interactions between entities is a crucial task across various scientific domains. The traditional causal discovery methods often assume a predetermined causal direction, which is problematic when prior knowledge is insufficient. Identifying causal directions from observational data remains a key challenge. Causal discovery typically relies on two priors: the uniform prior and the Solomonoff prior. The Solomonoff prior theoretically outperforms the uniform prior in determining causal directions in bivariate scenarios by using the causal independence mechanism assumption. However, this approach has two main issues: it assumes that no unobserved variables affect the outcome, leading to method failure if violated, and it relies on the uncomputable Kolmogorov complexity (KC). In addition, we employ Kolmogorov’s structure function to analyze the use of the minimum description length (MDL) as an approximation for KC, which shows that the function class used for computing the MDL introduces prior biases, increasing the risk of misclassification. Inspired by the insufficient but necessary part of an unnecessary but sufficient condition (INUS condition), we propose an asymmetry where the expected complexity change in the cause, due to changes in the effect, is greater than the reverse. This criterion supplements the causal independence mechanism when its restrictive conditions are not met under the Solomonoff prior. To mitigate prior bias and reduce misclassification risk, we introduce a multilayer perceptron based on the universal approximation theorem as the backbone network, enhancing method stability. Our approach demonstrates a competitive performance against the SOTA methods on the TCEP real dataset. Additionally, the results on synthetic datasets show that our method maintains stability across various data generation mechanisms and noise distributions. This work advances causal direction determination research by addressing the limitations of the existing methods and offering a more robust and stable approach. Full article
(This article belongs to the Special Issue Computational Methods and Machine Learning for Causal Inference)
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21 pages, 4348 KiB  
Article
A Novel Ensemble Method of Divide-and-Conquer Markov Boundary Discovery for Causal Feature Selection
by Hao Li, Jianjun Zhan, Haosen Wang and Zipeng Zhao
Mathematics 2024, 12(18), 2927; https://doi.org/10.3390/math12182927 - 20 Sep 2024
Viewed by 897
Abstract
The discovery of Markov boundaries is highly effective at identifying features that are causally related to the target variable, providing strong interpretability and robustness. While there are numerous methods for discovering Markov boundaries in real-world applications, no single method is universally applicable to [...] Read more.
The discovery of Markov boundaries is highly effective at identifying features that are causally related to the target variable, providing strong interpretability and robustness. While there are numerous methods for discovering Markov boundaries in real-world applications, no single method is universally applicable to all datasets. Therefore, in order to balance precision and recall, we propose an ensemble framework of divide-and-conquer Markov boundary discovery algorithms based on U-I selection strategy. We put three divide-and-conquer Markov boundary methods into the framework to obtain an ensemble algorithm, focusing on judging controversial parent–child variables to further balance precision and recall. By combining multiple algorithms, the ensemble algorithm can leverage their respective strengths and more thoroughly analyze the cause-and-effect relationships of target variables through various perspectives. Furthermore, it can enhance the robustness of the algorithm and reduce dependence on a single algorithm. In the experiment, we select four advanced Markov boundary discovery algorithms as comparison algorithms and compare them on nine benchmark Bayesian networks and three real-world datasets. The results show that EDMB ranks first in the overall ranking, which illustrates the superiority of the integrated algorithm and the effectiveness of the adopted U-I selection strategy. The main contribution of this paper lies in proposing an ensemble framework for divide-and-conquer Markov boundary discovery algorithms, balancing precision and recall through the U-I selection strategy, and judging controversial parent–child variables to enhance algorithm performance and robustness. The advantage of the U-I selection strategy and its difference from existing methods is the ability to independently obtain the maximum precision and recall of multiple algorithms within the ensemble framework. By assessing controversial parent–child variables, it further balances precision and recall, leading to results that are closer to the true Markov boundary. Full article
(This article belongs to the Special Issue Computational Methods and Machine Learning for Causal Inference)
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17 pages, 861 KiB  
Article
Estimating the Individual Treatment Effect with Different Treatment Group Sizes
by Luyuan Song and Xiaojun Zhang
Mathematics 2024, 12(8), 1224; https://doi.org/10.3390/math12081224 - 18 Apr 2024
Viewed by 1372
Abstract
Machine learning for causal inference, particularly at the individual level, has attracted intense interest in many domains. Existing techniques focus on controlling differences in distribution between treatment groups in a data-driven manner, eliminating the effects of confounding factors. However, few of the current [...] Read more.
Machine learning for causal inference, particularly at the individual level, has attracted intense interest in many domains. Existing techniques focus on controlling differences in distribution between treatment groups in a data-driven manner, eliminating the effects of confounding factors. However, few of the current methods adequately discuss the difference in treatment group sizes. Two approaches, a direct and an indirect one, deal with potential missing data for estimating individual treatment with binary treatments and different treatment group sizes. We embed the two methods into certain frameworks based on the domain adaption and representation. We validate the performance of our method by two benchmarks in the causal inference community: simulated data and real-world data. Experiment results verify that our methods perform well. Full article
(This article belongs to the Special Issue Computational Methods and Machine Learning for Causal Inference)
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34 pages, 7124 KiB  
Article
Exploratory Matching Model Search Algorithm (EMMSA) for Causal Analysis: Application to the Cardboard Industry
by Richard Aviles-Lopez, Juan de Dios Luna del Castillo and Miguel Ángel Montero-Alonso
Mathematics 2023, 11(21), 4506; https://doi.org/10.3390/math11214506 - 31 Oct 2023
Viewed by 1685
Abstract
This paper aims to present a methodology for the application of matching methods in industry to measure causal effect size. Matching methods allow us to obtain treatment and control samples with their covariates as similar as possible. The matching techniques used are nearest, [...] Read more.
This paper aims to present a methodology for the application of matching methods in industry to measure causal effect size. Matching methods allow us to obtain treatment and control samples with their covariates as similar as possible. The matching techniques used are nearest, optimal, full, coarsened exact matching (CEM), and genetic. These methods have been widely used in medical, psychological, and economic sciences. The proposed methodology provides two algorithms to execute these methods and to conduct an exhaustive search for the best models. It uses three conditions to ensure, as far as possible, the balance of all covariates, the maximum number of units in the treatment and control groups, and the most significant causal effect sizes. These techniques are applied in the carton board industry, where the causal variable is downtime, and the outcome variable is waste generated. A dataset from the carton board industry is used, and the results are contrasted with an expert in this process. Meta-analysis techniques are used to integrate the results of different comparative studies, which could help to determine and prioritize where to reduce waste. Two machines were found to generate more waste in terms of standardized measures whose values are 0.52 and 0.53, representing 48.60 and 36.79 linear meters (LM) on average for each production order with a total downtime of more than 3000 s. In general, for all machines, the maximum average wastage for each production order is 24.98 LM and its confidence interval is [13.40;36.23] LM. The main contribution of this work is the use of causal methodology to estimate the effect of downtime on waste in an industry. Particularly relevant is the contribution of an algorithm that aims to obtain the best matching model for this application. Its advantages and disadvantages are evaluated, and future areas of research are outlined. We believe that this methodology can be applied to other industries and fields of knowledge. Full article
(This article belongs to the Special Issue Computational Methods and Machine Learning for Causal Inference)
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