ProCause: Diverse Causal Analysis

• ProCause is a polysemous term defining distinct causal-analysis frameworks that attribute, predict, and evaluate counterfactuals across various domains.
• In protein dynamics, ProCause employs information imbalance and imbalance gain to infer directional causal links among principal components extracted from MD trajectories.
• Other implementations include influence-function estimators for probability of causation, formal identification of program action causes, and generative counterfactual evaluators in process mining.

Searching arXiv for "ProCause" and closely related titles to ground the article in current preprints. “ProCause” is a reused name for several distinct causal-analysis frameworks spanning molecular biophysics, semiparametric causal inference, process mining, formal methods for accountability, and probability-of-causation theory. In the protein-dynamics literature, ProCause denotes a pipeline for inferring probabilistic causal links among principal components (PCs) extracted from molecular-dynamics (MD) trajectories, using information imbalance and imbalance gain to detect directional asymmetries between collective modes (Banerjee et al., 23 Jun 2026). In other literatures, the same name has been used for an influence-function-based estimator for the probability of causation (Cuellar et al., 2018), a formal framework for identifying program actions as actual causes of violations in concurrent protocols (Datta et al., 2015), and a generative evaluator for prescriptive process monitoring based on counterfactual outcome generation (Moor et al., 31 Aug 2025). The term therefore does not denote a single unified theory; rather, it labels multiple domain-specific approaches centered on causal attribution, causal explanation, or counterfactual evaluation.

1. ProCause in protein dynamics

In “Investigating causality between principal components in protein dynamics,” ProCause is the name given to a pipeline that infers probabilistic causal links among high-dimensional motion variables, specifically PCs obtained from PCA of aligned protein trajectories (Banerjee et al., 23 Jun 2026). The workflow begins from an MD trajectory of length $T$ sampled at times $t=1,\dots,T$. After alignment removing overall translation and rotation, the covariance matrix of the $C_\alpha$ atom positions is diagonalized to obtain eigenvectors $v_i$ and eigenvalues $\lambda_i$. The $i$-th PC time series is then defined by

$X_i(t) = v_i \cdot [ r(t) - \langle r \rangle ],$

where $r(t)\in\mathbb{R}^{3N}$ is the instantaneous coordinate vector of the $N$ $C_\alpha$ atoms and $t=1,\dots,T$0 is its time average (Banerjee et al., 23 Jun 2026).

The causal question addressed by this ProCause formulation is whether $t=1,\dots,T$1 “causes” $t=1,\dots,T$2 in the operational sense that knowledge of $t=1,\dots,T$3 improves prediction of $t=1,\dots,T$4 beyond what is achievable from $t=1,\dots,T$5 alone. This shifts the analysis from variance-dominant or decorrelation-dominant descriptions toward directional predictability across lags. The paper explicitly states that such directional relationships are a necessary condition for the presence of a causal link, not by themselves a sufficient condition (Banerjee et al., 23 Jun 2026).

This use of ProCause is positioned as complementary to PCA and TICA rather than a replacement. PCA identifies dominant fluctuation modes, and TICA captures slow decorrelating coordinates, but neither directly resolves driver–responder asymmetry among PCs. ProCause addresses precisely that asymmetry via nonparametric rank-based comparison of predictive information across time lags (Banerjee et al., 23 Jun 2026).

2. Information imbalance, imbalance gain, and directed PC networks

The central technical object in the protein-dynamics ProCause pipeline is the Information Imbalance (II) introduced by Del Tatto et al., defined between two representations $t=1,\dots,T$6 and $t=1,\dots,T$7 of the same ensemble of configurations. If $t=1,\dots,T$8 is the rank of point $t=1,\dots,T$9 among the $C_\alpha$0 nearest neighbors of point $C_\alpha$1 in $C_\alpha$2-space, then

$C_\alpha$3

The quantity $C_\alpha$4 is small when neighborhoods in $C_\alpha$5 remain neighborhoods in $C_\alpha$6, meaning that $C_\alpha$7 carries information about $C_\alpha$8 (Banerjee et al., 23 Jun 2026).

To quantify causal asymmetry between scalar time series $C_\alpha$9 and $v_i$0, ProCause compares $v_i$1 with the best achievable imbalance after augmenting $v_i$2 by $v_i$3 scaled by $v_i$4. The joint distance is

$v_i$5

and the Imbalance Gain (IG) is

$v_i$6

By construction, $v_i$7 means that adding $v_i$8 strictly improves prediction of $v_i$9; the paper interprets this as a necessary condition for saying that $\lambda_i$0 drives $\lambda_i$1 at lag $\lambda_i$2 (Banerjee et al., 23 Jun 2026).

Time integration yields

$\lambda_i$3

with the practical implementation replacing the integral by a sum over a lag grid. Directional asymmetry is then summarized either pointwise,

$\lambda_i$4

or in integrated form,

$\lambda_i$5

These quantities support construction of a directed network whose nodes are PCs and whose edge weights are normalized integrated gains $\lambda_i$6 (Banerjee et al., 23 Jun 2026).

A notable feature is the statistical testing scheme. ProCause uses a permutation test in which the driver series $\lambda_i$7 is randomly reshuffled across trajectory chunks, recomputing null gains $\lambda_i$8 over $\lambda_i$9 repetitions, for example $i$0. The $i$1-value is the fraction of null gains greater than or equal to the observed gain, and significance is declared if $i$2, for example $i$3, for at least one $i$4 or for $i$5 (Banerjee et al., 23 Jun 2026).

3. Algorithmic workflow and empirical behavior on proteins

The ProCause algorithm for protein dynamics is presented as a six-step workflow. The input is an MD trajectory and a set of $i$6 PCs of interest, for example $i$7, and the output is a directed PC network with edge strengths $i$8 (Banerjee et al., 23 Jun 2026).

Trajectory preparation includes frame alignment to a reference, for example by RMSD fit of all $i$9, and subdivision into $X_i(t) = v_i \cdot [ r(t) - \langle r \rangle ],$0 non-overlapping chunks of equal length $X_i(t) = v_i \cdot [ r(t) - \langle r \rangle ],$1, for example $X_i(t) = v_i \cdot [ r(t) - \langle r \rangle ],$2 chunks. PCA extraction then computes the mean structure and covariance of aligned coordinates, diagonalizes the covariance, projects frames onto the leading PCs, and assembles chunk-wise time series $X_i(t) = v_i \cdot [ r(t) - \langle r \rangle ],$3. For each ordered PC pair $X_i(t) = v_i \cdot [ r(t) - \langle r \rangle ],$4 and each lag on a grid such as $X_i(t) = v_i \cdot [ r(t) - \langle r \rangle ],$5 ns, the method collects the $X_i(t) = v_i \cdot [ r(t) - \langle r \rangle ],$6 chunk starts and corresponding futures, computes $X_i(t) = v_i \cdot [ r(t) - \langle r \rangle ],$7 with $X_i(t) = v_i \cdot [ r(t) - \langle r \rangle ],$8, chooses a neighborhood size $X_i(t) = v_i \cdot [ r(t) - \langle r \rangle ],$9 such as $r(t)\in\mathbb{R}^{3N}$0, searches over $r(t)\in\mathbb{R}^{3N}$1, and evaluates $r(t)\in\mathbb{R}^{3N}$2. Integration over $r(t)\in\mathbb{R}^{3N}$3 produces $r(t)\in\mathbb{R}^{3N}$4, and edge selection uses both asymmetry and significance, with a directional link $r(t)\in\mathbb{R}^{3N}$5 declared when $r(t)\in\mathbb{R}^{3N}$6 and $r(t)\in\mathbb{R}^{3N}$7 is below threshold (Banerjee et al., 23 Jun 2026).

Two applications are summarized. For ubiquitin (PDB 1UBQ), the analysis used a 1 ms Anton simulation truncated to the first 650 $r(t)\in\mathbb{R}^{3N}$8s. Among the leading $r(t)\in\mathbb{R}^{3N}$9 PCs, PC$N$0 was described as most variant, slowest, and highly collective, whereas PC$N$1–PC$N$2 were faster and more localized. Reported integrated gains include $N$3 versus $N$4, $N$5 versus $N$6, $N$7 versus $N$8, and $N$9 versus $C_\alpha$0 (Banerjee et al., 23 Jun 2026). The stated biological interpretation is that the most collective, slow, protein-wide fluctuation conditions the future of the next two modes more strongly than it does higher-order ones, while PC$C_\alpha$1 also drives PC$C_\alpha$2 and PC$C_\alpha$3 modestly.

For NTL9 (PDB 2HBA), the trajectory was a 2.9 ms folding simulation from which a folded segment of 230 $C_\alpha$4s was selected. PC$C_\alpha$5 was reported as the slowest with $C_\alpha$6s and relatively localized on the $C_\alpha$7-turn, whereas PC$C_\alpha$8–PC$C_\alpha$9 relaxed in approximately 200 ns. Reported values include $t=1,\dots,T$00 versus $t=1,\dots,T$01, $t=1,\dots,T$02 versus $t=1,\dots,T$03, $t=1,\dots,T$04 versus $t=1,\dots,T$05, and $t=1,\dots,T$06 versus $t=1,\dots,T$07 (Banerjee et al., 23 Jun 2026). The paper interprets this as a localized slow fluctuation around the $t=1,\dots,T$08-turn exerting directional influence on both localized and more delocalized faster PCs.

A central contrast is drawn with TICA. Because TICA’s lagged covariance $t=1,\dots,T$09 is symmetric under detailed balance, it cannot distinguish PC$t=1,\dots,T$10PC$t=1,\dots,T$11 from PC$t=1,\dots,T$12PC$t=1,\dots,T$13. ProCause is presented as revealing directional asymmetries that covariance-based and symmetric lagged analyses do not capture (Banerjee et al., 23 Jun 2026).

4. Other established meanings of “ProCause”

The name ProCause has also been used in several unrelated causal frameworks.

In semiparametric causal inference, ProCause denotes an estimator for the probability of causation $t=1,\dots,T$14 under positivity, consistency, no unobserved confounding, and monotonicity, where

$t=1,\dots,T$15

Instead of assuming $t=1,\dots,T$16 follows a finite-dimensional model exactly, the method projects it onto a working model $t=1,\dots,T$17 by minimizing an $t=1,\dots,T$18-type criterion and estimates the projection coefficient with an efficient influence-function-based estimating equation (Cuellar et al., 2018). The framework is explicitly described as nonparametric, cross-fitting compatible, and asymptotically normal under stated regularity conditions. In the Kenya spring-protection application, the IFB projection estimator reported $t=1,\dots,T$19 with 95% confidence interval $t=1,\dots,T$20 for an “average” child, contrasting with higher parametric and nonparametric plugin estimates (Cuellar et al., 2018).

In formal methods and security, ProCause refers to “Program Actions as Actual Causes,” a framework in which actual causes of safety-property violations are identified in interacting concurrent programs by a two-phase procedure. The first phase finds a minimal projected-prefix “Lamport cause” sufficient for the violation, and the second phase dummifies nonessential actions to eliminate mere progress enablers, yielding a minimal projected sublog as the actual cause (Datta et al., 2015). The framework proves existence of causes for every reordering-closed safety violation and analyzes a public-key certification attack in which compromised notaries contribute to unauthorized access (Datta et al., 2015).

In process mining, “ProCause: Generating Counterfactual Outcomes to Evaluate Prescriptive Process Monitoring Methods” defines a generative evaluator for prescriptive process monitoring. It models $t=1,\dots,T$21 using S-Learner, T-Learner, TARNet, and a post-hoc ensemble, with either MLP or LSTM encoders, and fits a separate treatment model $t=1,\dots,T$22 (Moor et al., 31 Aug 2025). In synthetic evaluations using SimBank, the ensemble was reported to yield the lowest Wasserstein-1 distance and highest Kendall’s $t=1,\dots,T$23 across interventions and confounding levels, while real-world evaluation on BPIC12 and BPIC17 used KS, Epps–Singleton, Wasserstein-1/2, Friedman–Rafsky, kNN, and Energy tests, with at least one ensemble configuration in each dataset showing all $t=1,\dots,T$24-values above $t=1,\dots,T$25 (Moor et al., 31 Aug 2025).

These usages share a causal vocabulary but differ in ontology, estimands, assumptions, and output objects. The protein-dynamics ProCause operates on MD-derived PC time series and returns directed networks among collective variables (Banerjee et al., 23 Jun 2026); the semiparametric ProCause returns projected probability-of-causation estimates (Cuellar et al., 2018); the formal-methods ProCause returns minimal causal action sets in program traces (Datta et al., 2015); and the process-mining ProCause returns counterfactual outcome generators for policy evaluation (Moor et al., 31 Aug 2025).

5. Relation to the broader probabilities-of-causation literature

Several papers in the supplied literature do not name their methods ProCause but are directly relevant to the probability-of-causation branch associated with the term. They collectively show that the “probability of causation” family has expanded substantially beyond the original binary treatment–binary outcome setting.

One direction generalizes probabilities of causation to continuous and vector-valued treatments and outcomes. Definitions of $t=1,\dots,T$26, $t=1,\dots,T$27, and $t=1,\dots,T$28 are given in terms of threshold events for scalar outcomes and total orders for vector outcomes, with nonparametric identification via conditional CDFs $t=1,\dots,T$29 under exogeneity, monotonicity, and common-support assumptions (Kawakami et al., 2024). The same work extends the theory to sub-population evidence and multi-hypothetical threshold queries and illustrates the framework on Portuguese student-grade data (Kawakami et al., 2024).

A second direction derives sharp bounds for ordinal outcomes. For $t=1,\dots,T$30 and ordinal $t=1,\dots,T$31, the probability of causation for a treated individual observed above threshold $t=1,\dots,T$32 is

$t=1,\dots,T$33

and the paper gives sharp nonparametric lower and upper bounds using Fréchet–Hoeffding inequalities. It further shows that when $t=1,\dots,T$34, the formulas reduce to the classical Tian–Pearl bounds, and that introducing a binary mediator under complete mediation sharpens the upper bound (Sun et al., 2024).

A third direction studies mediation-aware probability-of-causation functionals. Controlled-direct, natural-direct, and natural-indirect variants of probability of necessity and sufficiency are defined under a mediator graph $t=1,\dots,T$35, and identification theorems reduce them to conditional CDFs and a cross-world quantity

$t=1,\dots,T$36

under consistency, positivity, sequential ignorability, and monotonicity (Kawakami et al., 2024).

A fourth direction considers quasi-Markovian structural causal models with latent confounding. There, probabilities of necessity, sufficiency, and necessity-and-sufficiency are treated as partially identifiable quantities bounded via optimization, and the paper introduces counterfactual-graph reduction techniques that can substantially reduce optimization degree before using these scores for root-cause analysis over causal paths (Laurentino et al., 2 Sep 2025).

This broader literature indicates that the probability-of-causation sense of ProCause is best understood as part of a larger methodological family concerned with necessity, sufficiency, counterfactual attribution, and identification under structural assumptions, rather than as a single canonical estimator (Cuellar et al., 2018, Kawakami et al., 2024, Sun et al., 2024, Kawakami et al., 2024, Laurentino et al., 2 Sep 2025).

6. Methodological themes, limitations, and interpretive cautions

Across its different meanings, ProCause recurrently emphasizes counterfactual or asymmetric notions of causation, but the relevant asymmetry differs by domain. In protein dynamics, asymmetry means improved lagged prediction of one PC by another via imbalance gain (Banerjee et al., 23 Jun 2026). In semiparametric epidemiologic-style inference, it means the probability that an observed outcome would not have occurred without exposure (Cuellar et al., 2018). In concurrent programs, it means minimal action sets whose presence is sufficient for violation across counterfactual executions (Datta et al., 2015). In process mining, it means generating counterfactual outcomes under alternative interventions for evaluating policies (Moor et al., 31 Aug 2025).

The limitations are likewise domain-specific. In the protein-dynamics formulation, $t=1,\dots,T$37 is only a necessary, not sufficient, condition for a causal link; hidden confounding can remain, estimation requires many independent chunks, the method is pairwise rather than fully multivariate, and results depend on neighborhood size, $t=1,\dots,T$38 grid, lag grid, and chunk size (Banerjee et al., 23 Jun 2026). In the semiparametric probability-of-causation setting, validity depends on positivity, consistency, no unobserved confounding, and monotonicity, while misspecification of nuisance models can affect finite-sample performance even though the projection target is robust to working-model misspecification (Cuellar et al., 2018). In the formal-methods setting, worst-case search is exponential in log length, even though pruning is possible and the method is sound (Datta et al., 2015). In the process-mining setting, evaluation still relies on causal-identification assumptions and was validated on a single simulator, so realism checks and learner diagnostics remain necessary (Moor et al., 31 Aug 2025).

A common misconception would be to treat the shared label “ProCause” as evidence of a single transferable formalism. The literature supplied does not support that view. The commonality is thematic—causal explanation or causal attribution—whereas the technical objects range from rank-based neighborhood statistics to efficient influence functions, projected sublogs, and generative outcome models (Banerjee et al., 23 Jun 2026, Cuellar et al., 2018, Datta et al., 2015, Moor et al., 31 Aug 2025).

This suggests that “ProCause” functions more as a domain-local naming convention than as a cross-domain standard. A plausible implication is that any citation to ProCause should be disambiguated by field and arXiv identifier, since otherwise the intended framework, assumptions, and outputs may be unclear.

7. Prospective developments

The protein-dynamics ProCause paper identifies several extensions. One is a multivariate ProCause that would extend conditional imbalance-gain ideas to distinguish direct from mediated links among sets of PCs. Another is residue-level mapping, in which a detected PC$t=1,\dots,T$39PC$t=1,\dots,T$40 link is back-projected onto residues and combined with contact maps to hypothesize structural allosteric pathways. Further suggested applications include ligand binding, enzyme cycles, and channel gating, as well as integration with TICA or VAMP to identify “slowest causal drivers” rather than merely slow decorrelating modes (Banerjee et al., 23 Jun 2026).

In the probability-of-causation branch, ongoing work points toward richer variable types, mediation structure, and path-based explanation. Extensions to continuous and vector variables expand the query class beyond binary outcomes (Kawakami et al., 2024); ordinal-outcome theory refines partial-identification bounds and mediator-assisted sharpening (Sun et al., 2024); mediation analysis decomposes necessity-and-sufficiency across direct and indirect pathways (Kawakami et al., 2024); and quasi-Markovian models combine counterfactual bounds with root-cause ranking over causal narratives (Laurentino et al., 2 Sep 2025).

In process mining, the ProCause evaluator suggests future work on architecture selection under temporal dependence, with current evidence indicating that TARNet is not uniformly best, that ensembles provide more stable evaluation, and that LSTM encoders can help when intervention timing matters (Moor et al., 31 Aug 2025). In formal accountability, the program-actions framework establishes a basis for automated explanation and blame assignment in decentralized systems, suggesting further tooling around model checking and trace analysis (Datta et al., 2015).

Taken together, these developments show that ProCause is best treated as a polysemous technical label attached to several active research programs. In contemporary usage, the most specific and recently introduced sense is the protein-dynamics framework that augments PCA and TICA with directed, statistically tested causal asymmetries among principal components (Banerjee et al., 23 Jun 2026).