This paper proposes a novel analytical framework: Transition Network Analysis (TNA), an approach that integrates Stochastic Process Mining and probabilistic graph representation to model, visualize, and identify transition patterns in the learning process data. Combining the relational and temporal aspects into a single lens offers capabilities beyond either framework, including centralities to capture important learning events, community finding to identify patterns of behavior, and clustering to reveal temporal patterns. This paper introduces the theoretical and mathematical foundations of TNA. To demonstrate the functionalities of TNA, we present a case study with students (n=191) engaged in small-group collaboration to map patterns of group dynamics using the theories of co-regulation and socially-shared regulated learning. The analysis revealed that TNA could reveal the regulatory processes and identify important events, temporal patterns and clusters. Bootstrap validation established the significant transitions and eliminated spurious transitions. In doing so, we showcase TNA's utility to capture learning dynamics and provide a robust framework for investigating the temporal evolution of learning processes. Future directions include advancing estimation methods, expanding reliability assessment, exploring longitudinal TNA, and comparing TNA networks using permutation tests.

A counterfactual distribution is the probability distribution of a random variable under a hypothetical scenario that differs from the observed reality. "What would have been the outcome for this individual if they had received a different treatment?" is an example of a counterfactual question. Here the personal data of the individual constitute the evidence that specifies the observed reality, and the interest lies in the distribution of the outcome under a hypothetical treatment. Counterfactual questions belong to the third and highest level in the causal hierarchy (Shpitser and Pearl, 2008) and are in general more difficult than associational (first level) or interventional (second level) questions. Algorithms for checking the identifiability of counterfactual queries from observational and experimental data have been developed (Shpitser and Pearl, 2007; Shpitser and Sherman, 2018; Correa et al., 2021) and implemented (Tikka, 2022).

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.

A causal effect is defined as the distribution P (Y do(X), Z) where variables Y are observed, variables X are intervened upon (forced to values irrespective of their natural causes) and variables Z are conditioned on. Instead of placing various parametric restrictions based on background knowledge, we are interested in this paper in the question of identifiability: can the causal effect be uniquely determined from the distributions (data) we have and a graph representing our structural knowledge on the generating causal system. 1 In the most basic setting we are identifying causal effects from a single observational input distribution, corresponding to passively observed data. To solve such problems more generally than what is possible with the backdoor adjustment (Spirtes et al., 1993; Pearl, 2009; Greenland et al., 1999), Pearl (1995) introduced do-calculus, a set of three rules that together with probability theory enable the manipulation of interventional distributions. Shpitser and Pearl (2006a) and Huang and Valtorta (2006) showed that do-calculus is complete by presenting polynomial-time algorithms whose each step can be seen as a rule of do-calculus or as an operation based on basic probability theory. The algorithms have a high practical value because the rules of do-calculus do not by themselves provide an indication on the order in which they should be applied.

Identification of causal effects is one of the most fundamental tasks of causal inference. We consider a variant of the identifiability problem where a causal effect of interest is not identifiable from observational data alone but some experimental data is available for the identification task. This corresponds to a real-world setting where experiments were conducted on a set of variables, which we call surrogate outcomes, but the variables of interest were not measured. This problem is a generalization of identifiability using surrogate experiments and we label it as surrogate outcome identifiability and show that the concept of transportability provides a sufficient criteria for determining surrogate outcome identifiability for a large class of queries.

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.

Causal models communicate our assumptions about causes and effects in real-world phe- nomena. Often the interest lies in the identification of the effect of an action which means deriving an expression from the observed probability distribution for the interventional distribution resulting from the action. In many cases an identifiability algorithm may return a complicated expression that contains variables that are in fact unnecessary. In practice this can lead to additional computational burden and increased bias or inefficiency of estimates when dealing with measurement error or missing data. We present graphical criteria to detect variables which are redundant in identifying causal effects. We also provide an improved version of a well-known identifiability algorithm that implements these criteria.