Abstract Sepsis is a life-threatening and serious global health issue. This study combines knowledge with available hospital data to investigate the potential causes of Sepsis that can be affected by policy decisions. We investigate the underlying causal structure of this problem by combining clinical expertise with score-based, constraint-based, and hybrid structure learning algorithms. A novel approach to model averaging and knowledge-based constraints was implemented to arrive at a consensus structure for causal inference. The structure learning process highlighted the importance of exploring data-driven approaches alongside clinical expertise. This includes discovering unexpected, although reasonable, relationships from a clinical perspective. Hypothetical interventions on Chronic Obstructive Pulmonary Disease, Alcohol dependence, and Diabetes suggest that the presence of any of these risk factors in patients increases the likelihood of Sepsis. This finding, alongside measuring the effect of these risk factors on Sepsis, has potential policy implications. Recognising the importance of prediction in improving health outcomes related to Sepsis, the model is also assessed in its ability to predict Sepsis by evaluating accuracy, sensitivity, and specificity. These three indicators all had results around 70%, and the AUC was 80%, which means the causal structure of the model is reasonably accurate given that the models were trained on data available for commissioning purposes only.

Since its introduction to the public, ChatGPT has had an unprecedented impact. While some experts praised AI advancements and highlighted their potential risks, others have been critical about the accuracy and usefulness of Large Language Models (LLMs). In this paper, we are interested in the ability of LLMs to identify causal relationships. We focus on the well-established GPT-4 (Turbo) and evaluate its performance under the most restrictive conditions, by isolating its ability to infer causal relationships based solely on the variable labels without being given any other context by humans, demonstrating the minimum level of effectiveness one can expect when it is provided with label-only information. We show that questionnaire participants judge the GPT-4 graphs as the most accurate in the evaluated categories, closely followed by knowledge graphs constructed by domain experts, with causal Machine Learning (ML) far behind. We use these results to highlight the important limitation of causal ML, which often produces causal graphs that violate common sense, affecting trust in them. However, we show that pairing GPT-4 with causal ML overcomes this limitation, resulting in graphical structures learnt from real data that align more closely with those identified by domain experts, compared to structures learnt by causal ML alone. Overall, our findings suggest that despite GPT-4 not being explicitly designed to reason causally, it can still be a valuable tool for causal representation, as it improves the causal discovery process of causal ML algorithms that are designed to do just that.

Many Bayesian Network structure learning algorithms are unstable, with the learned graph sensitive to arbitrary dataset artifacts, such as the ordering of columns (i.e., variable order). PC-Stable [1] attempts to address this issue for the widely-used PC algorithm, prompting researchers to use the ‘stable’ version instead. However, this problem seems to have been overlooked for score-based algorithms. In this study, we show that some widely-used score-based algorithms, as well as hybrid and constraint-based algorithms, including PC-Stable, suffer from the same issue. We propose a novel solution for score-based greedy hill-climbing that eliminates instability by determining a stable node order, leading to consistent results regardless of variable ordering. The new Tabu-Stable algorithms achieve the highest overall performance in terms of mean BIC score, log-likelihood, and structural accuracy across networks. These results highlight the importance of addressing instability in structure learning and provide a robust and practical approach for future applications. This paper extends the scope and impact of our previous work presented at Probabilistic Graphical Models 2024 [2] by incorporating continuous variables, implementing new stable orders that improve performance further, and demonstrating that the approach remains effective in the presence of sampling noise. The implementations, along with usage instructions, are freely available on GitHub at https://github.com/causal-iq/discovery .

OBJECTIVES: Establishing causal dependencies is crucial in applied domains, such as medicine and healthcare, where decision-making must be explainable. In these settings, small sample sizes and missing data call for federated approaches to maximise the amount of information we can use. METHODS: We propose a novel federated causal discovery algorithm capable of pooling information from multiple sources with heterogeneous missing data to learn a graph representing cause-effect relationships. In particular, we learn a causal graph on a centralised server while taking into account both prior knowledge and missingness mechanism specific to each client. RESULTS: We applied the proposed algorithm to synthetic data and real-world data from a multicentric study on endometrial cancer, validating the obtained causal graph through quantitative analyses and a clinical literature review. CONCLUSION: Our approach learns an accurate model despite data missing not-at-random.

Causal Bayesian Networks (CBNs) are an important tool for reasoning under uncertainty in complex real-world systems. Determining the graphical structure of a CBN remains a key challenge and is undertaken either by eliciting it from humans, using machine learning to learn it from data, or using a combination of these two approaches. In the latter case, human knowledge is generally provided to the algorithm before it starts, but here we investigate a novel approach where the structure learning algorithm itself dynamically identifies and requests knowledge for relationships that the algorithm identifies as “uncertain” during structure learning. We integrate this approach into the Tabu structure learning algorithm and show that it offers considerable gains in structural accuracy, which are generally larger than those offered by existing approaches for integrating knowledge. We suggest that a variant which requests only arc orientation information may be particularly useful where the practitioner has little preexisting knowledge of the causal relationships. As well as offering improved accuracy, the approach can use human expertise more effectively and contributes to making the structure learning process more transparent. • Novel Tabu-AL algorithm requests human input as it learns a causal structure. • Algorithm mainly asks about arc orientations which it is “uncertain” about. • Produces more accurate graphs than other forms of knowledge fusion. • Achieves similar accuracy gains to increasing sample size by 100 times. • Provides an exploratory and efficient approach to causal discovery.

Diabetes, a pervasive and enduring health challenge, imposes significant global implications on health, financial healthcare systems, and societal well-being. This study undertakes a comprehensive exploration of various structural learning algorithms to discern causal pathways amongst potential risk factors influencing diabetes progression. This study evaluates a diverse set of structure learning algorithms to discern causal pathways amongst potential risk factors influencing diabetes progression. The methodology involves the application of these algorithms to relevant diabetes data, followed by the conversion of their output graphs into Causal Bayesian Networks (CBNs), enabling predictive analysis and the evaluation of discrepancies in the effect of hypothetical interventions within our context-specific case study. This study highlights the substantial impact of algorithm selection on intervention outcomes. To consolidate insights from diverse algorithms, we employ a model-averaging technique that helps us obtain a unique causal model for diabetes derived from a varied set of structural learning algorithms.We also investigate how each of those individual graphs, as well as the average graph, compare to the structures elicited by a domain expert who categorised graph edges into high confidence, moderate, and low confidence types, leading into three individual graphs corresponding to the three levels of confidence. The resulting causal model and data are made available online, and serve as a valuable resource and a guide for informed decision-making by healthcare practitioners. Our applied work integrates and evaluates existing causal structure learning methods for decision support in patients with diabetes. It offers a comprehensive understanding of the interactions between relevant risk factors for intervention, and enables us to simulate the effect of hypothetical interventions before implementation. Therefore, this research not only contributes to the academic discussion on diabetes, but also provides practical guidance for healthcare professionals in developing efficient intervention and risk management strategies.

Abstract Causal Bayesian Networks (CBNs) provide an important tool for reasoning under uncertainty with potential application to many complex causal systems. Structure learning algorithms that can tell us something about the causal structure of these systems are becoming increasingly important. In the literature, the validity of these algorithms is often tested for sensitivity over varying sample sizes, hyper-parameters, and occasionally objective functions, but the effect of the order in which the variables are read from data is rarely quantified. We show that many commonly-used algorithms, both established and state-of-the-art, are more sensitive to variable ordering than these other factors when learning CBNs from discrete variables. This effect is strongest in hill-climbing and its variants where we explain how it arises, but extends to hybrid, and to a lesser-extent, constraint-based algorithms. Because the variable ordering is arbitrary, any significant effect it has on learnt graph accuracy is concerning, and raises questions about the validity of both many older and more recent results produced by these algorithms in practical applications and their rankings in performance evaluations.

Abstract One of the challenges practitioners face when applying structure learning algorithms to their data involves determining a set of hyperparameters; otherwise, a set of hyperparameter defaults is assumed. The optimal hyperparameter configuration often depends on multiple factors, including the size and density of the usually unknown underlying true graph, the sample size of the input data, and the structure learning algorithm. We propose a novel hyperparameter tuning method, called the Out-of-sample Tuning for Structure Learning (OTSL), that employs out-of-sample and resampling strategies to estimate the optimal hyperparameter configuration for structure learning, given the input dataset and structure learning algorithm. Synthetic experiments show that employing OTSL to tune the hyperparameters of hybrid and score-based structure learning algorithms leads to improvements in graphical accuracy compared to the state-of-the-art. We also illustrate the applicability of this approach to real datasets from different disciplines.

Abstract In Bayesian Networks (BNs), the direction of edges is crucial for causal reasoning and inference. However, Markov equivalence class considerations mean it is not always possible to establish edge orientations, which is why many BN structure learning algorithms cannot orientate all edges from purely observational data. Moreover, latent confounders can lead to false positive edges. Relatively few methods have been proposed to address these issues. In this work, we present the hybrid mFGS-BS (majority rule and Fast Greedy equivalence Search with Bayesian Scoring) algorithm for structure learning from discrete data that involves an observational data set and one or more interventional data sets. The algorithm assumes causal insufficiency in the presence of latent variables and produces a Partial Ancestral Graph (PAG). Structure learning relies on a hybrid approach and a novel Bayesian scoring paradigm that calculates the posterior probability of each directed edge being added to the learnt graph. Experimental results based on well-known networks of up to 109 variables and 10 k sample size show that mFGS-BS improves structure learning accuracy relative to the state-of-the-art and it is computationally efficient.

Causal machine learning (ML) algorithms recover graphical structures that tell us something about cause-and-effect relationships. The causal representation provided by these algorithms enables transparency and explainability, which is necessary for decision making in critical real-world problems. Yet, causal ML has had limited impact in practice compared to associational ML. This paper investigates the challenges of causal ML with application to COVID-19 UK pandemic data. We collate data from various public sources and investigate what the various structure learning algorithms learn from these data. We explore the impact of different data formats on algorithms spanning different classes of learning, and assess the results produced by each algorithm, and groups of algorithms, in terms of graphical structure, model dimensionality, sensitivity analysis, confounding variables, predictive and interventional inference. We use these results to highlight open problems in causal structure learning and directions for future research. To facilitate future work, we make all graphs, models, data sets, and source code publicly available online.

Abstract Causal Bayesian networks have become a powerful technology for reasoning under uncertainty in areas that require transparency and explainability, by relying on causal assumptions that enable us to simulate hypothetical interventions. The graphical structure of such models can be estimated by structure learning algorithms, domain knowledge, or a combination of both. Various knowledge approaches have been proposed in the literature that enables us to specify prior knowledge that constrains or guides these algorithms. This paper introduces some novel, and also describes some existing, knowledge-based approaches that enable us to combine structure learning with knowledge obtained from heterogeneous sources. We investigate the impact of these approaches on structure learning across different algorithms, case studies and settings that we might encounter in practice. Each approach is assessed in terms of effectiveness and efficiency, including graphical accuracy, model fitting, complexity, and runtime; making this the first paper that provides a comparative evaluation of a wide range of knowledge approaches for structure learning. Because the value of knowledge depends on what data are available, we illustrate the results both with limited and big data. While the overall results show that knowledge becomes less important with big data due to higher learning accuracy rendering knowledge less important, some of the knowledge approaches are found to be more important with big data. Amongst the main conclusions is the observation that reduced search space obtained from knowledge does not always imply reduced computational complexity, perhaps because the relationships implied by the data and knowledge are in tension.

Learning the structure of a Bayesian Network (BN) with score-based solutions involves exploring the search space of possible graphs and moving towards the graph that maximises a given objective function. Some algorithms offer exact solutions that guarantee to return the graph with the highest objective score, while others offer approximate solutions in exchange for reduced computational complexity. This paper describes an approximate BN structure learning algorithm, which we call Model Averaging Hill-Climbing (MAHC), that combines two novel strategies with hill-climbing search. The algorithm starts by pruning the search space of graphs, where the pruning strategy can be viewed as an aggressive version of the pruning strategies that are typically applied to combinatorial optimisation structure learning problems. It then performs model averaging in the hill-climbing search process and moves to the neighbouring graph that maximises the objective function, on average, for that neighbouring graph and over all its valid neighbouring graphs. Comparisons with other algorithms spanning different classes of learning suggest that the combination of aggressive pruning with model averaging is both effective and efficient, particularly in the presence of data noise.

Structure learning algorithms that learn the graph of a Bayesian network from observational data often do so by assuming the data correctly reflect the true distribution of the variables. However, this assumption does not hold in the presence of measurement error, which can lead to spurious edges. This is one of the reasons why the synthetic performance of these algorithms often overestimates real-world performance. This paper describes a heuristic algorithm that can be added as an additional learning phase at the end of any structure learning algorithm, and serves as a correction learning phase that removes potential false positive edges. The results show that the proposed correction algorithm successfully improves the graphical score of five well-established structure learning algorithms spanning different classes of learning in the presence of measurement error.

Abstract Learning from data that contain missing values represents a common phenomenon in many domains. Relatively few Bayesian Network structure learning algorithms account for missing data, and those that do tend to rely on standard approaches that assume missing data are missing at random, such as the Expectation-Maximisation algorithm. Because missing data are often systematic, there is a need for more pragmatic methods that can effectively deal with data sets containing missing values not missing at random. The absence of approaches that deal with systematic missing data impedes the application of BN structure learning methods to real-world problems where missingness are not random. This paper describes three variants of greedy search structure learning that utilise pairwise deletion and inverse probability weighting to maximally leverage the observed data and to limit potential bias caused by missing values. The first two of the variants can be viewed as sub-versions of the third and best performing variant, but are important in their own in illustrating the successive improvements in learning accuracy. The empirical investigations show that the proposed approach outperforms the commonly used and state-of-the-art Structural EM algorithm, both in terms of learning accuracy and efficiency, as well as both when data are missing at random and not at random.

Several algorithms have been proposed towards discovering the graphical structure of Bayesian networks. Most of these algorithms are restricted to observational data and some enable us to incorporate knowledge as constraints in terms of what can and cannot be discovered by an algorithm. A common type of such knowledge involves the temporal order of the variables in the data. For example, knowledge that event B occurs after observing A and hence, the constraint that B cannot cause A. This paper investigates real-world case studies that incorporate interesting properties of objective temporal variable order, and the impact these temporal constraints have on the learnt graph. The results show that most of the learnt graphs are subject to major modifications after incorporating incomplete temporal objective information. Because temporal information is widely viewed as a form of knowledge that is subjective, rather than as a form of data that tends to be objective, it is generally disregarded and reduced to an optional piece of information that only few of the structure learning algorithms may consider. The paper argues that objective temporal information should form part of observational data, to reduce the risk of disregarding such information when available and to encourage its reusability across related studies.

Numerous Bayesian Network (BN) structure learning algorithms have been proposed in the literature over the past few decades. Each publication makes an empirical or theoretical case for the algorithm proposed in that publication and results across studies are often inconsistent in their claims about which algorithm is ‘best’. This is partly because there is no agreed evaluation approach to determine their effectiveness. Moreover, each algorithm is based on a set of assumptions, such as complete data and causal sufficiency, and tend to be evaluated with data that conforms to these assumptions, however unrealistic these assumptions may be in the real world. As a result, it is widely accepted that synthetic performance overestimates real performance, although to what degree this may happen remains unknown. This paper investigates the performance of 15 state-of-the-art, well-established, or recent promising structure learning algorithms. We propose a methodology that applies the algorithms to data that incorporates synthetic noise, in an effort to better understand the performance of structure learning algorithms when applied to real data. Each algorithm is tested over multiple case studies, sample sizes, types of noise, and assessed with multiple evaluation criteria. This work involved learning approximately 10,000 graphs with a total structure learning runtime of seven months. In investigating the impact of data noise, we provide the first large scale empirical comparison of BN structure learning algorithms under different assumptions of data noise. The results suggest that traditional synthetic performance may overestimate real-world performance by anywhere between 10% and more than 50%. They also show that while score-based learning is generally superior to constraint-based learning, a higher fitting score does not necessarily imply a more accurate causal graph. The comparisons extend to other outcomes of interest, such as runtime, reliability, and resilience to noise, assessed over both small and large networks, and with both limited and big data. To facilitate comparisons with future studies, we have made all data, raw results, graphs and BN models freely available online.

Child mortality from preventable diseases such as pneumonia and diarrhoea in low and middle-income countries remains a serious global challenge. We combine knowledge with available Demographic and Health Survey (DHS) data from India, to construct Causal Bayesian Networks (CBNs) and investigate the factors associated with childhood diarrhoea. We make use of freeware tools to learn the graphical structure of the DHS data with score-based, constraint-based, and hybrid structure learning algorithms. We investigate the effect of missing values, sample size, and knowledge-based constraints on each of the structure learning algorithms and assess their accuracy with multiple scoring functions. Weaknesses in the survey methodology and data available, as well as the variability in the CBNs generated by the different algorithms, mean that it is not possible to learn a definitive CBN from data. However, knowledge-based constraints are found to be useful in reducing the variation in the graphs produced by the different algorithms, and produce graphs which are more reflective of the likely influential relationships in the data. Furthermore, valuable insights are gained into the performance and characteristics of the structure learning algorithms. Two score-based algorithms in particular, TABU and FGES, demonstrate many desirable qualities; (a) with sufficient data, they produce a graph which is similar to the reference graph, (b) they are relatively insensitive to missing values, and (c) behave well with knowledge-based constraints. The results provide a basis for further investigation of the DHS data and for a deeper understanding of the behaviour of the structure learning algorithms when applied to real-world settings.

Some structure learning algorithms have proven to be effective in reconstructing hypothetical Bayesian Network graphs from synthetic data. However, in their mission to maximise a scoring function, many become conservative and minimise edges discovered. While simplicity is desired, the output is often a graph that consists of multiple independent subgraphs that do not enable full propagation of evidence. While this is not a problem in theory, it can be a problem in practice. This article examines a novel unconventional associational heuristic called Saiyan, which returns a directed acyclic graph that enables full propagation of evidence. Associational heuristics are not expected to perform well relative to sophisticated constraint-based and score-based learning approaches. Moreover, forcing the algorithm to connect all data variables implies that the forced edges will not be correct at the rate of those identified unrestrictedly. Still, synthetic and real-world experiments suggest that such a heuristic can be competitive relative to some of the well-established constraint-based, score-based and hybrid learning algorithms.

We show that existing Bayesian network (BN) modelling techniques cannot capture the correct intuitive reasoning in the important case when a set of mutually exclusive events need to be modelled as separate nodes instead of states of a single node. A previously proposed ‘solution’, which introduces a simple constraint node that enforces mutual exclusivity, fails to preserve the prior probabilities of the events, while other proposed solutions involve major changes to the original model. We provide a novel and simple solution to this problem that works in all cases where the mutually exclusive nodes have no common ancestors. Our solution uses a special type of constraint and auxiliary node together with formulas for assigning their necessary conditional probability table values. The solution enforces mutual exclusivity between events and preserves their prior probabilities while leaving all original BN nodes unchanged.

We present a Bayesian network (BN) model for forecasting Association Football match outcomes. Both objective and subjective information are considered for prediction, and we demonstrate how probabilities transform at each level of model component, whereby predictive distributions follow hierarchical levels of Bayesian inference. The model was used to generate forecasts for each match of the 2011/2012 English Premier League (EPL) season, and forecasts were published online prior to the start of each match. Profitability, risk and uncertainty are evaluated by considering various unit-based betting procedures against published market odds. Compared to a previously published successful BN model, the model presented in this paper is less complex and is able to generate even more profitable returns.