The ability to sustain engagement and recover from setbacks (i.e., resilience)—is fundamental for learning. When resilience weakens, students are at risk of disengagement and may drop out and miss on opportunities. Therefore, predicting disengagement long before it happens during the window of hope is important. In this article, we test whether early warning signals of resilience loss, grounded in the concept of critical slowing down (CSD) can forecast disengagement before dropping out. CSD has been widely observed across ecological, climate, and neural systems, where it precedes tipping points into catastrophic failure (dropping out in our case). Using 1.67 million practice attempts from 9,401 students who used a digital math learning environment, we computed CSD indicators: autocorrelation, return rate, variance, skewness, kurtosis, and coefficient of variation. We found that 88.2% of students exhibited CSD signals prior to disengagement, with warnings clustering late in activity and before practice ceased (dropping out). Our results provide the first evidence of CSD in education, suggesting that universal resilience dynamics also govern social systems such as human learning. These findings offer a practical indicator for early detection of vulnerability and supporting learners across different applications and contexts long before critical events happen. Most importantly, CSD indicators arise universally, independent of the mechanisms that generate the data, offering new opportunities for portability across contexts, data types, and learning environments.

Counterfactual inference considers a hypothetical intervention in a parallel world that shares some evidence with the factual world. If the evidence specifies a conditional distribution on a manifold, counterfactuals may be analytically intractable. We present an algorithm for simulating values from a counterfactual distribution where conditions can be set on both discrete and continuous variables. We show that the proposed algorithm can be presented as a particle filter leading to asymptotically valid inference. The algorithm is applied to fairness analysis in credit-scoring.

Graphs are commonly used to represent and visualize causal relations. For a small number of variables, this approach provides a succinct and clear view of the scenario at hand. As the number of variables under study increases, the graphical approach may become impractical, and the clarity of the representation is lost. Clustering of variables is a natural way to reduce the size of the causal diagram, but it may erroneously change the essential properties of the causal relations if implemented arbitrarily. We define a specific type of cluster, called transit cluster, that is guaranteed to preserve the identifiability properties of causal effects under certain conditions. We provide a sound and complete algorithm for finding all transit clusters in a given graph and demonstrate how clustering can simplify the identification of causal effects. We also study the inverse problem, where one starts with a clustered graph and looks for extended graphs where the identifiability properties of causal effects remain unchanged. We show that this kind of structural robustness is closely related to transit clusters.

Causal effect identification considers whether an interventional probability distribution can be uniquely determined from a passively observed distribution in a given causal structure. If the generating system induces context-specific independence (CSI) relations, the existing identification procedures and criteria based on do-calculus are inherently incomplete. We show that deciding causal effect non-identifiability is NP-hard in the presence of CSIs. Motivated by this, we design a calculus and an automated search procedure for identifying causal effects in the presence of CSIs. The approach is provably sound and it includes standard do-calculus as a special case. With the approach we can obtain identifying formulas that were unobtainable previously, and demonstrate that a small number of CSI-relations may be sufficient to turn a previously non-identifiable instance to identifiable.

Obtaining a non-parametric expression for an interventional distribution is one of the most fundamental tasks in causal inference. Such an expression can be obtained for an identifiable causal effect by an algorithm or by manual application of do-calculus. Often we are left with a complicated expression which can lead to biased or inefficient estimates when missing data or measurement errors are involved. We present an automatic simplification algorithm that seeks to eliminate symbolically unnecessary variables from these expressions by taking advantage of the structure of the underlying graphical model. Our method is applicable to all causal effect formulas and is readily available in the R package causaleffect.