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Towards Realistic Models of Change: A Causal Perspective on Decision-Making and Dynamical Systems

Isacco Beretta

arXiv:2610.28803v1cs.LGcs.AI

TL;DR

Data-driven methods can produce non-generalizable results, and time-unaware causal recourse may ignore how long actions take. This thesis develops user-centered and time-aware recourse approaches alongside causal interventions for dynamical systems, linking stable VAR interventions to SCMs. Its findings indicate that ignoring response times can yield impractical recommendations, while the proposed temporal extension and waiting-time choices remain preliminary.

  • Problem

    Data-driven methods risk non-generalizable results, and time-unaware recourse does not account for temporal dependencies between features and interventions.

    Source: §1.1 · p. 8, §5.2.2 · p. 67

  • Method

    The thesis develops an interactive feature-space tool, incorporates response time into recourse evaluation, and defines interventions for dynamical systems linked to SCMs.

    Source: §4.6 · p. 60, §5.7 · p. 79, §7.8 · p. 128

  • Results

    Time-unaware causal models could recommend actions that are optimal under an SCM but too slow to be practically viable.

    Source: §5.7 · p. 79

  • Takeaways & Limitations

    Accounting for time in recourse can reveal differing time requirements for equal-cost actions and support time-constrained actionability evaluation.

    Source: §5.7 · p. 79

    The temporal extension remains preliminary, and waiting times were assigned manually without rigorous theoretical or empirical justification.

    Source: §9.1.2 · p. 156

Abstract

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While statistical learning has made remarkable progress in recent decades, driven by large datasets and data-driven protocols that have transformed scientific practice, it still faces a key limitation: it struggles to generalize beyond the data it has seen, often failing under distribution shift. Causality offers a promising solution: causal models focus on the underlying mechanisms of change rather than surface-level associations, enabling robust reasoning under external interventions. This allows systems not only to learn from data, but also to reason about hypothetical scenarios, adapt to new environments, and inform better decisions. This thesis explores how these approaches can inform each other, in two parts. The first part examines algorithmic recourse, that is, identifying which changes an individual can make to overturn an unfavorable decision. We propose an alternative perspective to the established framework, arguing for a more realistic cost function, a reformulation of the problem in causal terms, and the inclusion of a temporal dimension in policy evaluation. The second part focuses on dynamical systems, extending the notions of causal interventions and effects to discrete and continuous stochastic processes. We show that the equilibria of these processes, and how they change following an intervention, can be linked to Structural Causal Models (SCMs), viewed as compressed representations of the underlying temporal dynamics. We argue for the relevance of such temporal causal models, since in real-world scenarios actions take time to produce their effects. The common goal of both parts is practical: the utility, robustness, and trustworthiness of these models depend critically on their ability to realistically represent the underlying phenomena.

Acronyms

The acronym list provides abbreviations for causal inference, machine learning, recourse, and stochastic-process terminology.

Acronyms

  • AI means Artificial Intelligence, AR Algorithmic Recourse, XAI eXplainable Artificial Intelligence, and CEx Contrastive Explanation.Source: Acronyms · p. 7
  • CPDAG, CRe, DAG, dE, OdE, and ODE abbreviate graph and equation terms, including Completed Partially Directed Acyclic Graph and Ordinary Differential Equation.Source: Acronyms · p. 7
  • DSP, GF, MEC, ML, PDAG, RDE, RdE, SCM, SDE, SdE, SP, VAR, BM, BN, and CD cover processes, models, graphs, and causal methods.Source: Acronyms · p. 7

Introduction

The introduction frames model-driven and data-driven inquiry as complementary approaches with different strengths and limitations. It motivates causal and temporal extensions, then outlines the thesis’s work on recourse and dynamical systems.

1.1 Motivation

  • Data-driven methods handle poorly understood systems but may lack interpretability and produce non-generalizable results, while model-driven approaches can struggle with complex systems.Source: §1.1 · p. 8

1.1.1 The Role of Causality

  • Causal reasoning addresses questions about interventions and counterfactuals that associative methods focused on observational dependencies do not answer directly.Source: §1.1.1 · p. 9

1.2 Research Questions

  • The research questions ask how causality can improve realistic decision-making and how classical causal frameworks can incorporate temporal structure.Source: §1.2 · p. 10

1.3 Thesis Contributions and Outline

  • The thesis develops user-centered recourse, time-aware causal recourse, and causal inference frameworks for discrete- and continuous-time stochastic systems.Chapter 4 introduces Feature Space Navigator; Chapter 5 incorporates time into recourse costs; Chapters 7 and 8 develop temporal causal methods.Source: §1.3 · pp. 11–12

1.4 Publications

  • The thesis lists publications forming its basis and separately identifies a publication that does not contribute to its contents.Source: §1.4 · pp. 12–13

Mathematical Framework

This section sets out a mathematical framework for causal reasoning, moving from observational distributions and DAG structure through interventions to counterfactual queries while making explicit the assumptions required for causal interpretation. It also describes how limits such as sparse data, ambiguous causal structure, and failures of mechanism, modularity, or invariance assumptions constrain reliable inference and out-of-distribution generalization.

Mathematical Framework

  • The causal ladder moves from probability-based observation to graph-based interventions and mechanism-based counterfactuals, while stronger assumptions narrow applicability (Barocas, Selbst, and Raghavan, 2020).Observational data describe patterns; interventions ask how outcomes change when a variable is set, and counterfactuals ask how a specific outcome would have differed.Source: Mathematical Framework · pp. 15–16

2.1 The Ladder of Causation

  • Under distribution shift, interventions and counterfactuals can alter the data-generating process, so models relying on surface associations may fail in changed contexts.The ladder structures reasoning from observational data toward interventions and counterfactuals, where causal assumptions become increasingly important.Source: §2.1 · p. 16
  • With observational p(X, Y) alone, causal direction remains unidentified, and confounding or shared temporal dynamics can make associations spurious.The section proposes a modular causal pipeline to clarify assumptions, methodological choices, bottlenecks, and potential failure modes across analysis stages.Source: §2.1 · pp. 18–20

2.2 Observational Level

  • The observational level estimates P(X) from i.i.d. samples; independence structures can compress its representation through an appropriate factorization.The i.i.d. assumption makes the inferential target well-defined and keeps its complexity fixed as sample size grows.Source: §2.2 · pp. 20–22

2.2.1 Finite-data problem

  • Finite data make conditional-independence testing increasingly impractical as the conditioning set grows, while continuous-data binning and finite-sample algorithm weaknesses add limitations.The passage notes that PC becomes order-dependent and GES no longer guarantees a global score optimum in this regime.Source: §2.2.1 · pp. 22–23

2.3 Interventional Level

  • A causal graph compactly encodes dependencies and represents external interventions as modifications to underlying factors; observational data alone generally cannot identify causal structure.Answering interventional questions from observational data therefore requires causal assumptions, including a graph that correctly captures cause-effect relationships.Source: §2.1 · p. 17, §2.3 · p. 23
  • A DAG’s Markov property links its structure to distributional conditional independencies and factorization, but faithfulness can fail when causal paths cancel, and a P-compatible graph need not be the true causal structure.Finite-sample tests with margins of error can make faithfulness violations more problematic than traditionally assumed.Source: §2.3 · pp. 24, 26, 27

2.3.1 Markov Equivalence Class

  • DAGs in the same Markov equivalence class share a skeleton and v-structures, and their CPDAG represents alternative causal structures with shared directed cause-effect relations.The class captures structures capable of generating the same probability distribution.Source: §2.3.1 · pp. 27–28

2.3.2 Causal Sufficiency

  • Causal sufficiency assumes all direct common causes of modeled variables are included; excluding latent variables can yield unrealistic models, while unrestricted hidden-variable models make causal structure unfeasible to determine.Choosing a simple model is a heuristic and does not guarantee that it captures the underlying mechanism.Source: §2.3.2 · pp. 28–29

2.3.3 Interventions

  • Causal interventions replace a variable’s conditional mechanism; hard do-interventions set its value and remove incoming arrows, while modular changes can support generalization from one problem to another.The framework defines causal effect as the difference in expected variables before and after intervention.Source: §2.3.3 · pp. 29–31

2.4 Counterfactual Level

  • The framework moves from a graph specifying which variables directly determine others and interventional distributions to the counterfactual level.The passage describes this as the perspective adopted so far before proceeding to the counterfactual level.Source: §2.4 · p. 31
  • SCMs represent each variable through a structural assignment using its parents and exogenous noise, enabling counterfactual queries under a fixed noise realization.The assignment mechanisms induce a joint distribution; after a realization occurs, the system is deterministic.Source: §2.4 · pp. 32–33

2.4.1 Interventions and Counterfactuals

  • An SCM intervention replaces one or more mechanisms; hard interventions set a mechanism to a constant, while soft interventions retain at least one argument.The average causal effect can differ from a particular person's counterfactual effect: the drug example has an average effect of 0.5 but is harmful for a patient with E_S = 0.Source: §2.4.1 · pp. 33–34
  • Counterfactual queries require an observed realization, and their alternative outcomes cannot be directly verified against ground truth.In the drug example, administering treatment was the best decision given the information available then, despite a harmful counterfactual outcome for one patient.Source: §2.4.1 · p. 34

2.4.2 Modularity and Invariance

  • SCMs assume each mechanism depends only on its specific inputs, but this fails when outcomes depend on group-relative properties, as in selecting the top 10 competitors.Increasing every participant's score by five points would leave their relative ranking and selection unchanged.Source: §2.4.2 · p. 35
  • Modularity and interventional invariance can fail when mechanisms adapt: immune responses may change with past exposures, or a virus may develop immunity after vaccination.Interventional invariance assumes mechanisms not directly targeted by an intervention remain unchanged.Source: §2.4.2 · pp. 35–36

Algorithmic Recourse

This section examines algorithmic recourse as a way to identify actions that can improve an individual’s outcome, emphasizing causal mechanisms and action-focused recommendations when features depend on one another. It also considers the computational complexity of causal recourse and conditions under which it can match counterfactual recourse’s complexity or simplify the search.

Algorithmic Recourse

  • Contrastive Explanations show how an outcome could have changed, whereas Consequential Recommendations give tailored actions intended to produce a favorable outcome.The exam example contrasts hypothetical changed answers with advice to improve understanding of derivatives and integrals, which remains useful for a retake.Source: Algorithmic Recourse · pp. 38–39

3.1 Preliminaries

  • A counterfactual instance is one whose binary model prediction differs from the factual instance’s prediction; recourse seeks an optimal such instance under a cost function.The chapter’s setup treats outcome 0 as unfavorable and 1 as favorable.Source: §3.1 · pp. 39–40

3.2 First Challenge: Choosing Cost

  • The cost function is often left abstract, and the chapter frames choosing a realistic cost as a challenge that may require moving beyond a purely algorithmic approach.The following discussion examines common cost choices and their limitations.Source: §3.2 · pp. 40–41

3.2.1 Common choices

  • Common recourse costs use weighted ℓp distances, whose choices encode preferences such as sparsity but measure feature-space changes rather than realistic effort.The ℓ0 norm counts changed features, while weights can address differing feature scales.Source: §3.2.1 · p. 41

3.2.2 Constraints and Limitations

  • Actionability, plausibility, sparsity, diversity, and robustness constraints improve realism, but penalty terms cannot resolve all limitations of distance-based costs.The text identifies translational invariance and symmetry as persistent problems, alongside later challenges involving causality, user dependence, and time awareness.Source: §3.2.2 · pp. 41–42

3.2.3 Limitations

  • Distance metrics can misrepresent effort because they are translation-invariant and symmetric, treating equal-sized changes alike despite different starting points or direction.For example, the same $700 salary increase imposes different relative effort at $8,000 versus $1,200 monthly income; ℓp and percentile shifts are symmetric even when improvement should cost more than deterioration.Source: §3.2.3 · pp. 42–43

3.3 Second Challenge: Dependent Variables

  • Because real-world variables interact, changing one can help one outcome while harming another, making independent-feature recourse inadequate.The winemaker’s trade-off between grape quality and yield illustrates how actions affect profit through multiple dependent variables.Source: §3.3 · p. 43

3.3.1 Causality in Algorithmic Recourse

  • Causal Consequential Recommendations optimize an action over possible interventions, accounting for how system structure affects the outcome.Unlike distance-based recommendations that identify a destination, action-based recommendations specify how to reach it; the resulting counterfactual need not match a Contrastive Explanation.Source: §3.3.1 · pp. 45–46
  • Causal recourse offers a more meaningful and realistic formulation but adds complexity; counterfactual recourse is usually computationally manageable with gradient-based or approximate methods.Causal recourse is at least as computationally complex as counterfactual recourse.Source: §3.3.1 · pp. 46–47
  • When variables are completely independent, causal and counterfactual recourse become entirely equivalent.This is the best-case scenario for simplifying the action space.Source: §3.3.1 · p. 47
  • In linear SCMs with additive interventions, superposition lets the search proceed one variable at a time, reducing the search-space size to the system’s dimension.The total effect of interventions on different variables is the sum of their individual effects.Source: §3.3.1 · pp. 47–48

Cost is Subjective

This section reframes counterfactual recourse as interactive, user-directed exploration shaped by what users judge feasible, desirable, and useful, rather than fixed universal costs that can yield unrealistic changes. It also highlights the need to evaluate such recourse for usability, plausibility, and responsibility.

4.1 Contributions

  • The chapter proposes a subjective, user-driven notion of recourse cost and interactive feature modification instead of assuming one fixed cost applies to everyone.It also presents Feature Space Navigator as a lightweight, local, model-agnostic prototype for experts and lay users.Source: §4.1 · p. 51

4.2.1 Counterfactual generation techniques

  • Existing counterfactual methods assess multiple qualities, but their varied metrics complicate comparison, and most do not support user interaction.These methods include optimization, heuristic search, instance-based approaches, and decision trees.Source: §4.2.1 · p. 52

4.2.2 Human-Centered Approach

  • A human-centered explanation should be comprehensible and useful to its recipient, and interactive dialogue can give users control over examining model behavior.The chapter presents counterfactuals as an intuitive, contrastive form of explanation for both domain experts and lay users.Source: §4.2.2 · pp. 52–53

4.2.3 Interactive explanations and visualizations

  • Feature Space Navigator prioritizes user control through intuitive sliders, works with any model without dataset access, and focuses locally on the instance at hand.Unlike methods that automate generation using predefined metrics or provide control through constraints, users adjust features directly and indirectly define the optimization metric.Source: §4.2.3 · pp. 53–54

4.3 User-Centered Limitations

  • Fixed cost functions cannot adapt to feedback, while universal costs ignore personal contexts that shape action feasibility, making recommendations potentially unrealistic.For example, users may find changing jobs more attainable than pursuing education, and job-change feasibility can depend on local opportunities, transportation, or caregiving responsibilities.Source: §4.3 · p. 55

4.4 Feature Space Navigator

  • Feature Space Navigator lets users directly explore a model’s decision space and craft counterfactuals according to their own knowledge, preferences, and circumstances.Users act as the domain experts of their real-world scenarios, without an explanation algorithm mediating their interaction with the decision model.Source: §4.4 · p. 56
  • Users adjust features with sliders and see model-score gradients update after each change, enabling iterative exploration of prototypes and counterfactuals.The interface permits movement along individual feature axes rather than diagonal moves; cumulative adjustments can reach any point, and updates require predictions scaling linearly with feature count.Source: §4.4 · pp. 56–57

4.5 Scenarios

  • The prototype is implemented on two UCI datasets; the German Credit dataset contains 1,000 instances and 20 categorical or integer features.Its categorical features are handled by preserving the original category alongside the LabelEncoder-generated value.Source: §4.5 · pp. 57–58

4.5.1 Understanding the Effects of Interventions

  • In the Statlog loan scenario, reducing the requested credit amount improves the score but does not change the denial; adjusting loan duration changes the prediction to good credit risk.The example illustrates exploration as a negotiation among changes users consider desirable, feasible, and effective, rather than reliance on a fixed optimization routine.Source: §4.5.1 · p. 59
  • Changing only flavanoids moves a Wine sample across all three cultivar classes, demonstrating nonlinear shifts in the model’s decision boundary.The Wine dataset contains 178 instances, 13 chemical features, and three classes; increasing flavanoids across a narrow range changes the prediction from class 2 to class 1 and then class 0.Source: §4.5 · p. 58, §4.5.1 · pp. 59–60

4.6 Conclusions

  • Feature Space Navigator replaces fixed counterfactual optimization with an interactive view of a black-box decision boundary, giving users greater transparency and control.The authors argue that evaluation should include a user study; without a predefined cost function, optimality comparisons with algorithmic methods are not useful, and effectiveness cannot be assessed solely with data-driven metrics.Source: §4.6 · p. 60
  • Users may judge a formally valid counterfactual implausible or unachievable, so the interface needs plausibility guidance that preserves their freedom to explore.The authors propose soft constraints, visual cues, or lightweight warnings as possible directions, while acknowledging that this issue is not fully addressed.Source: §4.6 · pp. 60–62
  • Feature combinations can be technically valid yet inconsistent in practice, and adding plausibility checks without sacrificing simplicity and transparency remains an open challenge.Without statistical constraints or generative models, the system risks showing combinations that do not make sense in real-world scenarios.Source: §4.6 · p. 62
  • Manual exploration assumes users make fewer, meaningful changes, but many editable features can increase cognitive burden and make complex boundaries tiring or confusing to navigate.Highlighting influential features or prioritizing promising directions could help users without technical or domain-specific expertise.Source: §4.6 · p. 62
  • Counterfactual explanations can enable gaming or recommend formally valid actions that are ethically questionable or socially undesirable, concerns the authors say responsible recourse systems must take seriously.The cited work includes Barocas, Selbst, and Raghavan (2020), Kasirzadeh and Smart (2021), and Slack et al. (2021).Source: §4.6 · p. 62

Causal Algorithmic Recourse and Time

The chapter argues that causal recourse must account for the time required for effects to propagate through a system: otherwise, recommendations can be impractical and may favor interventions on root nodes. It introduces graph-based temporal costs and considers how time affects fairness and actionability.

Causal Algorithmic Recourse and Time

  • Existing causal recourse methods omit temporal dependencies, which can lead to ineffective or suboptimal recommendations and undermine their realism and feasibility.The chapter proposes incorporating causal-graph topology into cost evaluation and treating the causal model as a representation of a process evolving over time.Source: §5.1 · p. 65, Causal Algorithmic Recourse and Time · p. 65

5.1 Contributions

  • The chapter introduces a method that incorporates time into recourse-action costs by using the topology of the causal graph to model action effects.It also examines how temporal factors may affect fairness, particularly for marginalized groups.Source: §5.1 · pp. 65–66

5.2.1 Preliminaries

  • The chapter identifies a more realistic intervention cost and a causal reformulation of recourse as broader aims, while noting that common ℓp costs are widely used partly because effective alternatives require supplementary information.The cost function is expected to increase monotonically with the magnitude of the feature change.Source: §5.2.1 · p. 66

5.2.2 The Role of Time

  • The usual assumption that changing a causal-graph variable immediately changes its descendants omits potentially long delays in real processes.The education–skills–salary example illustrates that gaining skills through college may take years, while a training course may be a faster route to the same target.Source: §5.2.2 · p. 67
  • When the education-to-skills coefficient is sufficiently large, a time-unaware model may favor intervening on education over skills, suggesting that recourse can be biased toward root nodes.The chapter notes that nodes with many children or a long downstream chain are likely to enter the intervention set.Source: §5.2.2 · p. 67

5.3 Time Weighted Causal Graphs

The proposed time-weighted causal graph uses edge response times and causal-graph structure to account for when an intervention’s effects reach a target. The chapter applies this perspective to fairness and reframes actionability to include a feasible timeframe.

5.3 Time Weighted Causal Graphs

  • Edge response-time weights can represent how long an intervention on a parent variable takes to affect its child, allowing analysis of causal-effect propagation over time.The chapter proposes assigning these weights to edges in the causal graph.Source: §5.3 · p. 68
  • The time component of the recourse cost can be the longest path distance from intervention variables to the target, reflecting when all causal paths have finished.The chapter cautions that this maximum can be overly rigid: a single high-weight edge may dominate regardless of its causal effect.Source: §5.3 · pp. 68–69
  • A weighted average of causal-path propagation times, weighted by each path’s causal impact, is proposed as a more flexible time cost than using only the longest path.The cost represents when a significant portion of the causal effect becomes observable, not necessarily when the entire effect has arrived.Source: §5.3 · p. 69

5.4 Time Awareness for Fairer Recommendations

  • Similar feature-space costs for two people may correspond to substantially different temporal costs, and the time-cost weight λ may vary with individual circumstances.The chapter gives older loan applicants as an example of a group that might have a higher λ because of a shorter time to recover from financial setbacks.Source: §5.4 · p. 71
  • Actionability is expanded to require that a recommendation be achievable within a reasonable timeframe, with a maximum-time constraint that can be set in advance or adapted to an applicant’s needs.The proposed fairness interpretation requires recommendations to remain fair under any fixed time constraint.Source: §5.4 · pp. 71–72

5.5 Empirical Evaluation

The German Credit semi-synthetic evaluation examines how action costs influence selected variables and whether time-unaware causal recourse favors graph roots. Its results show that cost normalization can distort estimated intervention effects, while equal-cost interventions on education and income have different impacts on the target.

5.5.1 Research Questions

  • The experiment is organized around whether action cost changes the selected actionable variables and whether time-unaware causal recourse favors interventions on graph roots.The study evaluates these questions using the German Credit-based setup.Source: §5.5.1 · p. 72, §5.5.2 · p. 73

5.5.2 Experimental Settings

  • The evaluation uses a semi-synthetic SCM based on German Credit data, with Age, Gender, and Loan Duration modeled as non-actionable and Education and Income selected for analysis.Loan Duration can change as a consequence of an intervention even though it cannot be manipulated directly.Source: §5.5.2 · p. 73, §5.5.1 · p. 73

5.5.3 Results

  • Using total feature variance to normalize intervention costs can make interventions cheaper farther down the graph, regardless of variables’ internal properties.The authors instead normalize using the proper variances of the manipulated variables.Source: §5.5.3 · p. 75
  • In the earlier SCM, nearly every estimated Causal Effects Derivative is close to zero; in the proposed SCM, variables nearer Y have the highest CED values.The four highest CED variables are at distance 1 from Y, while variables at distance 2 or more have lower CED values.Source: §5.5.3 · p. 76, Table 5.1 · p. 76
  • For the same cost, intervention on Education has a stronger causal effect on the target Y than intervention on Income, leading the analysis to prioritize Education and confirming the hypothesized root-node bias.The pair plot compares observational and interventional distributions for Education and Income, and the text reports that Education has the stronger effect on Y.Source: §5.5.3 · p. 77, Figure 5.8 · p. 77

5.6 Broader Impact

The chapter argues that incorporating time into causal recourse can make recommendations more practically timely and psychologically congruent with users’ expectations. Its conclusion reports that time-unaware models may recommend actions that are optimal under an SCM but too slow to be viable in many applications.

5.6 Broader Impact

  • Time-aware causal reasoning may make recourse explanations more psychologically congruent with users’ expectations, fostering trust and engagement with automated systems.The conclusion also connects time-aware actionability with practical, timely recommendations.Source: §5.6 · p. 78
  • The chapter distinguishes predicted outcomes from actual outcomes, noting that interventions on effects such as symptoms do not change their causes, such as disease status.The work defines a causal-aware predictive model so that its prediction aligns more closely with the actual outcome.Source: §5.6 · p. 78

5.7 Conclusions

  • In the German Credit-inspired experiment, models unaware of response times could recommend actions that are optimal under the SCM but too slow to be practically viable in many applications.The conclusion connects time-aware models with fairness and alignment with human psychological expectations.Source: §5.7 · p. 79

Autonomous Dynamical Systems

The section develops a causal perspective on dynamical systems, connecting long-run equilibrium changes to interventions in static SCMs while recognizing that some questions require temporal models. It also identifies autonomy and the relevance of transient behavior as conditions shaping what these models can explain.

Autonomous Dynamical Systems

  • Temporal causal analysis addresses limits of observational time-series methods, which estimate dependencies but lack formal tools for reasoning about interventions and effects unfolding across time.The section proposes incorporating time into SCM equations as an elementary approach.Source: Autonomous Dynamical Systems · p. 81

6.1 A Dynamical System Perspective

  • SCMs can represent a dynamical system’s equilibrium as a compressed account of its long-run structure, with interventions shifting the equilibrium distribution and its causal interpretation.The proposed bridge treats invariant distributions as the key objects for studying long-term causal behavior.Source: §6.1 · pp. 82, 84

6.2 Autonomy

  • Autonomy assumes each mechanism remains invariant over time and interventions, supporting stable causal analysis and time-invariant asymptotic behavior.The assumption treats mechanisms as modular components that determine variables from their direct causes.Source: §6.2 · pp. 84–85

6.2.1 Is Climate Change Caused by Humans? Causal Ambiguity in Non-Stationary Processes

  • In climate systems, changing natural influences can destabilize relationships between emissions and temperature, while omitted variables can make mechanisms appear to change and weaken causal claims.Measuring and adding hidden influences could restore stable mechanisms, but complex natural factors and long-term trends make this difficult in practice.Source: §6.2.1 · p. 85

6.3 Not Only Long-Term Effects

  • Equilibrium compression suits systems where fast dynamics are irrelevant or too complex, but it loses useful information when researchers need effects during transitions.Policy evaluation may focus on short- to medium-term population changes, and systems may never reach equilibrium because policies or social conditions change.Source: §6.3 · p. 86

6.4 Objective of the Next Chapters

  • The next chapters aim to map dynamical models to equilibrium SCMs and extend causal inference to stochastic processes with interventions and effects estimated over time.The proposed temporal models preserve continuity with SCMs by treating mechanisms as functional relationships described by differential or difference equations.Source: §6.4 · pp. 87–88

Causal Inference over Time

The section develops causal inference over time by connecting dynamical systems and VAR dynamics with structural causal models, and by formulating temporal interventions, counterfactual queries, and intervention effects. It examines equilibrium-based relationships between these models and how intervention effects unfold over time.

Causal Inference over Time

  • Dynamical systems involve delayed responses and feedback across time scales, making temporal causal inference important for assessing effects that may emerge only in the medium or long term.The chapter motivates the problem with monetary policy effects on inflation and growth and delayed consequences of human actions on climate change.Source: Causal Inference over Time · p. 89

causality (Granger, 1969).

7.1 Contributions

  • The framework extends SCMs to discrete-time stochastic processes, defines additive and forcing interventions, maps VARs to equilibrium SCMs, and proves causal semantics are preserved under stability conditions.The mapping preserves observational and interventional distributions between time-based and equilibrium representations.Source: §7.1 · p. 90, • Mapping VAR to SCM · p. 91

7.2.1 Structural Causal Models

  • SCMs represent variable-generating mechanisms and support observational, interventional, and counterfactual queries by changing one or more mechanisms.Interventions may be hard, replacing a mechanism with a constant, or soft, retaining at least one of its arguments.Source: §7.2.1 · p. 91

7.2.2 Discrete-time Stochastic Processes

  • Difference equations describe discrete-time processes through past values and can approximate differential equations; ongoing stochastic noise prevents stochastic difference equation trajectories from converging to equilibrium.Source: §7.2.2 · pp. 92, 94

7.2.3 Vector Autoregressive models

  • VARs lack causal interpretability when their noise components are cross-correlated and act as hidden confounders; the proposed framework complements SVAR causal discovery.VAR and SVAR are equivalent representations, and choosing between them does not affect causal interpretation when the contemporaneous transformation is known.Source: §7.2.3 · pp. 95–96

7.3 Causal perspective on Discrete-time Stochastic Processes

  • The theory extends results for ordinary and random difference equations to stochastic difference equations, formalizes causal interventions, and uses SCMs to represent dynamical systems’ asymptotic behavior.Source: §7.3 · p. 96

7.3.1 Stochastic Processes Equilibration - dEs

  • Intervention and equilibration commute for random difference equations, but not for time-localized stochastic difference equations, which cannot strongly equilibrate unless their stochastic term vanishes asymptotically.Source: §7.3.1 · p. 97

Equilibration as a map from RdEs to SCMs

7.3.2 Causal Interventions on SdEs

  • As time tends to infinity, a stochastic dynamical equation’s causal effect converges to the corresponding SCM intervention effect.The temporal effect is defined at each time step for an intervention on the dynamical process.Source: §7.3.2 · p. 98

Equilibration as a map from SdEs to SCMs

7.3.3 Mapping SdEs to SCMs - The linear case

  • Cross-sectional distributions can miss temporal causal dependencies, but a long-run normalized-mean transformation yields an SCM that models intervention-induced distribution shifts in the linear case.The chapter discusses the SdE-to-SCM mapping only for linear systems; explicitly characterizing it for nonlinear systems is deferred to the next chapter.Source: Equilibration as a map from SdEs to SCMs · p. 98, §7.3.3 · p. 101

7.4.1 Implementation of Causal Interventions

  • The VAR intervention implementations cover additive shifts and target-forcing approaches, whose stability conditions differ.The chapter presents these forms as options for different application scenarios.Source: §7.4.1 · p. 102

Additive Interventions

  • Additive VAR interventions apply a persistent force from a specified time without changing process dynamics, producing deterministic, trajectory-independent effects while preserving stability.Source: Additive Interventions · pp. 102–103

Forcing Interventions

  • Forcing VAR interventions attract a variable toward a target and can implement do(Xi = target) as force grows, but may destabilize the system.For triangular VAR coefficient matrices, including matrices that allow self-loops, stability is preserved under forcing interventions.Source: Forcing Interventions · pp. 103–104

7.4.2 Mapping VARs to SCMs

  • A stable VAR(p) can be represented by a linear SCM sharing its causal graph and matching both equilibrium observational and interventional distributions.The chapter introduces two VAR intervention implementations and shows how linear SCMs model stable VARs’ long-term effects. The mapping theorem states the distributional equivalences for the transformed equilibrium variables.Source: §7.4 · p. 102, §7.4.2 · p. 105
  • At equilibrium, VARs map to linear SCMs that may contain causal cycles and correlated exogenous variables, and additive and forcing interventions commute with equilibration.The equilibrium SCMs retain the full covariance matrix of the VAR innovations.Source: §7.4.2 · pp. 107–108

7.5 Practical implications

  • VAR causal interventions extend beyond Granger causality to forecast effects over time and answer retrospective questions about past interventions.The retrospective procedure follows SCM-style abduction, action, and prediction: infer residuals from the observed trajectory, modify the model, and simulate using those residuals; without intervention, it recovers the factual trajectory perfectly.Source: §7.5 · p. 109, §7.5.2 · pp. 113–114
  • VARs can approximate any purely nondeterministic covariance-stationary process, but the guarantee gives no approximation rates and linear Gaussian representations can miss localized nonlinearities or non-Gaussian steady states.The reflected random walk example behaves mostly like an AR(1), but boundary reflections introduce a localized nonlinearity that its intractable MA(∞) representation fails to capture; feedback loops are natural in time-series models but less developed in practical SCM approaches.Source: §7.5 · pp. 109–110

Forecasting

  • Under i.i.d. white-noise innovations, optimal VAR forecasts recursively apply the model without future noise, while forecast-error covariance grows monotonically and long-range forecasts converge to the process mean.As the horizon grows, the past contains no information about the process’s distant future.Source: Forecasting · p. 111

Interventional Forecasting

  • Additive interventions shift the forecast while leaving its MSE unchanged; forcing interventions modify the VAR matrices and require comparing interventional and observational forecasts.For additive interventions, the causal effect can be estimated directly and added to the observational forecast; forcing effects are estimated through the difference between forecasts.Source: Interventional Forecasting · pp. 112–113

7.5.2 Retrospective Counterfactuals

  • Retrospective counterfactuals simulate the trajectory from the intervention time to the present using inferred residuals and the intervened model.Forcing interventions require left-multiplying the simulation’s right-hand side by [I + Fdiag]^-1.Source: §7.5.2 · p. 113

7.6 Empirical evaluation

  • The empirical evaluation compares VAR forecasting performance with state-of-the-art models, examines how additive and forcing interventions affect system dynamics, and evaluates causal VAR estimates of intervention effects over time.It uses German9, Pendulum, and Census; forecasting error is MAE on target variables on the test set, averaged over ten runs.Source: §7.6.1 · p. 114, Experimental Settings · pp. 114–115

7.6.2 Observational Forecasting

  • VAR consistently matched or closely approached the Oracle across datasets; on Census, TiDE slightly outperformed VAR at the 5-step horizon.DLinear was usually close to VAR for 1-step forecasts, while TiDE and TSMixer underperformed VAR and DLinear on German and Pendulum; all models performed worse on Pendulum than German.Source: §7.6.2 · p. 115, Table 7.1 · p. 116

Interventional Forecasting

  • Additive Expertise interventions produce delayed Credit Score effects without changing forecast covariance, while forcing interventions align forecasts to targets and can resemble do-interventions even at low force.The chapter also demonstrates that Responsibility affects Income but not Loan Duration, Loan Amount affects Loan Duration but not Income, and simultaneous forcing interventions can jointly affect Credit Score.Source: §7.6.3 · p. 116, Interventional Forecasting · pp. 122–123

7.6.4 Experimental details and extra results

  • The experiments simulate a stable SVAR(4) for a 7-dimensional stationary series, a stable SVAR(1) for a 2-dimensional Pendulum series, and use Census data for 50 high-income countries from 1991 to 2022.The Census panel contains 1,600 instances; forecast targets are Credit Score for German, X1 for Pendulum, and age groups for Census.Source: §7.6.4 · pp. 118–119

Observational Forecasting

  • VAR leads observational forecasting on German and Pendulum, often near the Oracle; DLinear is competitive at shorter horizons, while Census results favor TiDE at five steps.TSMixer and TiDE generally trail VAR and DLinear on German and Pendulum; on Census, VAR and TiDE are similar at one step, and TiDE slightly outperforms VAR at five steps in both metrics.Source: Observational Forecasting · pp. 120–122, Table 7.4 · p. 121, Table 7.5 · p. 121

7.6.3 Interventional Forecasting

  • Interventional forecasts are highly accurate at 10 steps; Pendulum effects appear immediately, whereas German variables may require several steps to change measurably.The German Credit Score requires at least three time steps for an intervention to take effect; the reported intervention scores remain accurate over longer horizons.Source: §7.6.3 · p. 118, Table 7.6 · p. 124, Table 7.7 · p. 124

Retrospective Counterfactuals

  • Retrospective counterfactuals show that a 2004 Expertise increase would raise Responsibility but leave Loan Amount unchanged, while a 2011 Births increase lowers average age more in Germany and Spain than Singapore.The Loan Amount result follows the causal graph because Expertise is not its ancestor; Singapore’s more moderate age response is consistent with its younger demographic profile at intervention time.Source: Retrospective Counterfactuals · pp. 125–126

7.7 Use cases

  • Use cases show that similar applicant scores can conceal different histories and intervention outcomes, while Census interventions illustrate changes in population age and births over time.Increasing Expertise can help quantify threshold-crossing times, but an intervention may be inadequate for a particular applicant; increasing Births lowers average age more evidently in Japan than Chile, and increased Migration raises births.Source: §7.7 · pp. 126–128

7.8 Conclusions

  • The framework maps VARs to linear SCMs and, under specified stability conditions, makes dynamical-system and SCM interventions equivalent; it requires a known causal graph and may be less effective for strongly nonlinear systems.The chapter introduces additive and forcing VAR interventions to support causal inference over time.Source: §7.8 · p. 128
  • The authors plan to investigate nonlinear DSPs for potentially better convergence and causal interpretation of a broader family of dynamical models, including high-dimensional applications such as climate science.The work also identifies further research directions, with concrete examples referenced in Section 7.5.Source: §7.8 · p. 128

Continuous Time and Non-linear Systems

This chapter extends the earlier discrete-time, linear framework for causal inference over time to continuous-time and potentially nonlinear systems. It uses SDEs and Gradient Flow to define continuous-time interventions and effects, analyze asymptotic behavior, and connect discrete- and continuous-time frameworks.

Continuous Time and Non-linear Systems

8.1 Differential Equations

This section develops continuous-time stochastic modeling, moving from deterministic differential equations to Brownian motion, Itô processes, stochastic integration, and density-level dynamics. It derives the Fokker–Planck equation from expectations under an SDE, extends stochastic calculus to multidimensional processes, and considers simplifications under constant isotropic diffusion and limiting cases.

8.1 Differential Equations

  • ODE vector fields determine trajectories from initial conditions, while random initial states or parameters yield RDEs whose long-run behavior SCMs can compress.With a random limit value k, logistic dynamics converge to a limiting distribution X∞.Source: §8.1 · pp. 130–132

8.1.1 Brownian Motion

  • Brownian motion models continuous random disturbances, but its nondifferentiable paths and history-independent increments make it inexpressible as an RDE with finite random parameters.Its increments are independent and normally distributed, and its trajectories are continuous almost surely.Source: §8.1.1 · pp. 133, 135

8.1.2 Itô Processes and Stochastic Differential Equations

  • Itô processes combine drift and Brownian diffusion in an SDE: μ controls average infinitesimal displacement, while σ controls the infinitesimal diffusion rate.As σ approaches zero, the process reduces to the equation without Brownian perturbations.Source: §8.1.2 · pp. 135–136

8.1.3 Stochastic Integrals

  • Because Brownian motion is non-rectifiable, stochastic integration requires a constructed integral; Itô’s formula adds a second-order correction and enables mean and variance analysis.The formula extends to Itô processes and can be used to estimate observables over time.Source: §8.1.3 · pp. 136–138

8.1.4 Fokker Plack Equation

  • The Fokker–Planck equation translates an SDE into a partial differential equation for the probability density p(x, t), a coarser description useful for tracking mean and variance.Source: §8.1.4 · p. 139
  • The Fokker–Planck equation is derived by applying Itô’s formula to arbitrary f(X_t), taking expectations, integrating drift and diffusion terms by parts, and using the arbitrariness of f.Source: §8.1.4 · pp. 139–140

8.1.5 Multidimensional Stochastic Calculus

  • For n-dimensional SDEs driven by d-dimensional Brownian motion, the text extends differential-product rules, Itô’s formula, and the Fokker–Planck equation, assuming positive-definite Σ.Source: §8.1.5 · pp. 140–141
  • With constant isotropic diffusion, the divergence term simplifies to σ²∇p; setting µ = 0 gives the heat equation, while σ = 0 gives the continuity equation.Source: §8.1.5 · p. 141

8.2 Gradient Flow

This section seeks to extend the established mapping between discrete-time dynamical systems and SCMs beyond linear systems, using invariant distributions as a bridge. It develops stochastic gradient flow as a tractable setting where a potential determines the equilibrium distribution and interventions shift that distribution.

8.2 Gradient Flow

  • The earlier explicit mapping between discrete-time dynamical systems and SCMs was established only for linear systems, motivating extension to nonlinear and continuous-time settings.Source: §8.2 · p. 141
  • Invariant distributions in observational and interventional systems serve as the bridge between the temporal framework and static SCMs.Source: §8.2 · p. 142
  • Finding conditions for an SDE to admit an invariant distribution is challenging in general, but stochastic gradient flow offers a tractable special case.It is also known as the Overdamped Langevin Equation.Source: §8.2 · p. 142

8.2.1 Stochastic Gradient Flow

  • For stochastic gradient flow, the drift is the negative gradient of a potential function, and the diffusion matrix is constant and isotropic.Source: §8.2.1 · p. 142
  • When the potential grows faster than logarithmically as |x| →∞, the SDE has a well-defined invariant Gibbs distribution.Source: §8.2.1 · p. 142
  • An intervention modifies the potential, shifting the equilibrium distribution to a new Gibbs distribution.The potential determines the system’s equilibrium distribution.Source: §8.2.1 · p. 143

8.3 A Causal Perspective on SDEs

This section adopts an intervention-based definition of causality and develops a causal perspective for SDEs. It treats interventions as changes to drift mechanisms over time, while noting that the meaning of causal influence can become less clear in high-dimensional systems.

8.3 A Causal Perspective on SDEs

  • The section defines causality through whether an external agent’s deliberate intervention on one variable changes another, framing causal identification as experimental.The authors describe this as an intermediate stance between control-theoretic and Pearl-style perspectives.Source: §8.3 · p. 143
  • The boundary between intervention-based causality and the weaker control-theoretic notion is not always clear, especially when individual influences become less meaningful in high-dimensional systems.The passage illustrates this concern with a gas of many interacting particles.Source: §8.3 · p. 144
  • Causalization is presented as a coarse-grained representation that preserves essential behavior while omitting details, with a causal graph described as the most compressed representation consistent with the system’s probability distribution.The passage connects this perspective to an agent’s ability to intervene and generalize under distribution shifts (Peters, Janzing, and Schölkopf, 2017).Source: §8.3 · p. 144

8.3.1 Causal Interventions on SDEs

  • Interventions on an SDE modify one or more drift functions from an intervention time onward, replacing the mechanism driving the affected variable.Source: §8.3.1 · p. 144
  • The section distinguishes additive interventions, which add a constant to drift, from forcing interventions, which pull a variable toward a target value.Source: §8.3.1 · p. 144

8.4 Estimating the Long-Run Causal Effect

The section estimates long-run causal effects through stationary distributions, seeking alternatives to costly simulation. It derives equilibrium distributions and considers how additive and forcing interventions change them, including conditions that affect stability.

8.4 Estimating the Long-Run Causal Effect

  • Long simulations can be computationally expensive because the convergence rate to the stationary distribution is typically unknown and may be slow.The section therefore favors direct estimation of the stationary distribution when available.Source: §8.4 · p. 145
  • Additive interventions do not always preserve stability: with V(x) = |x|, F > 1 pushes probability mass toward increasingly large x.A sufficient condition for stability preservation is that V(x) grows faster than linearly in the tails.Source: §8.4 · p. 146
  • As forcing strength F →∞, the stationary distribution approaches a Dirac delta at ˆx, making forcing intervention equivalent to do(x = ˆx).For large F, the distribution becomes increasingly similar to a Gaussian with variance σ2/F.Source: §8.4 · p. 146
  • Acyclic systems permit long-run effects to be estimated by computing each variable’s conditional stationary distribution in causal order, then reconstructing the joint distribution.The resulting factorization means an intervention changes only the conditional mechanism for the intervened variable; cyclic systems do not admit this independent-component factorization here.Source: §8.4 · pp. 147–148
  • Interventions are identifiable from the added potential term: additive interventions add a linear term, while forcing interventions add a quadratic term specific to the intervened variable.That variable-specific term uniquely identifies which intervention was applied.Source: §8.4 · pp. 148–149

8.5 Connection with Discrete-Time Models

The section connects discrete- and continuous-time causal models, deriving an SDE-to-SCM mapping through discretization and equilibrium. It also gives convergence guarantees for approximate discretization under stated conditions.

8.5 Connection with Discrete-Time Models

  • The continuous-time and discrete-time frameworks are linked through results intended to transfer between them, but nonlinear systems with feedback loops remain unhandled in either framework.The identified transfer directions include continuous-time linear systems with loops and discrete-time nonlinear acyclic systems.Source: §8.5 · p. 149, Table 8.1 · p. 149

8.5.1 From SDEs to VARs

  • A linear SDE sampled at time steps ∆t yields a VAR(1), and composing this discretization with equilibrium mapping gives a linear SCM.Theorem 8 allows arbitrary ∆t, yielding a family of equivalent SCMs associated with different time steps.Source: §8.5.1 · pp. 150–151, Figure 8.4 · p. 150
  • The SDE and its VAR representation need not match pathwise, but their distributions at each time and their equilibrium states agree.The continuous process retains infinitesimal fluctuations between discrete time steps.Source: §8.5.1 · p. 151

8.5.2 From SdEs to SDEs

  • For SDEs satisfying global Lipschitz conditions, Euler–Maruyama discretization converges strongly with order 1/2 as the timestep decreases.The associated bound estimates the long-term causal effect up to error of order 2C √∆t, but does not guarantee an explicit SCM or an explicitly identifiable invariant distribution.Source: §8.5.2 · p. 153
  • The convergence bound concerns the observational process; a tighter empirically established bound can give finer control over expected error in intervention claims.Additive interventions are expected to leave the error bound unchanged because they do not affect the drift’s Lipschitz constant.Source: §8.5.2 · pp. 153–154

Final Discussion

The final discussion reviews two strands: more realistic algorithmic recourse and causal inference for evolving systems. It summarizes their contributions while identifying assumptions, unresolved cases, and directions for further work.

Final Discussion

  • The thesis frames realistic representation of underlying phenomena as central to the utility, robustness, and trustworthiness of causal models.The chapter reviews algorithmic recourse and causal inference over time, including their limitations and future directions.Source: Final Discussion · p. 155, §9.1 · p. 155, §9.2 · p. 157

9.1 Algorithmic Recourse

  • The recourse discussion treats default distance-based costs as arbitrary and potentially mismatched to user preferences, motivating context-informed recommendations.The proposed approach uses contextual knowledge to make recommendations more realistic and user-relevant.Source: §9.1 · p. 155

9.1.1 Response to the Research Question

  • The recourse work combines user-specific context, causal structure for cost design, and temporal constraints on actionable recommendations.It distinguishes counterfactual explanation, which identifies an optimal instance, from recourse, which specifies a sequence of changes to reach it.Source: §9.1.1 · p. 155

9.1.2 Limitations

  • The Chapter 4 approach assumes users can manage optimization, but plausibility checks are absent and the cognitive burden grows with actionable features.High-dimensional or nonmonotonic settings can make navigating the decision boundary demanding.Source: §9.1.2 · p. 156
  • Chapter 5 assumes known causal structure, uses manually assigned waiting times without a rigorous basis, and models temporal change through a simplified cost term.The discussion proposes automatically estimating waiting times or allowing users to select them for their context.Source: §9.1.2 · p. 156

9.1.3 Future Work

  • Future recourse work could learn an average cost function from user interactions, while recognizing that it cannot fully capture subjective individual preferences.Users could then tailor the resulting preliminary recommendations.Source: §9.1.3 · p. 156

9.2 Causal Inference over Time

  • The dynamical-systems framework links equilibrium behavior to SCMs and extends causal reasoning to interventions over time.The discrete-time framework maps stable systems to equivalent SCMs, while the continuous-time extension uses Itô stochastic integration and examines transfer conditions.Source: §9.2 · p. 157, §9.2.1 · p. 157

9.2.2 Limitations

  • The frameworks assume known causal structure, additive stochastic noise, and stability; they do not cover time-varying interventions or nonlinear feedback systems.Equivalence with SCMs would no longer apply to systems with long-term trends.Source: §9.2.2 · p. 157

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