35  Concept Reference: Three Worlds and Three Levels of Reason

35.1 Concept Reference Tables

This appendix provides comprehensive reference tables mapping key concepts covered in this book to the three modelling layers (Structural, Dynamical, Observable) and the three levels of Reason (Association, Intervention, Counterfactual). These tables serve as a navigation aid and help clarify how concepts relate across the book’s framework.

A condensed version of these tables appears in the Introduction for quick reference.

35.1.1 Classification Principles

Layering: Many concepts are introduced first at one layer and then reused at other layers with different semantics. The Primary World listed in the tables is where a concept is introduced first.

Examples: - Graph Theory is introduced at Structural (structure/invariances), and appears in Dynamical (dynamic semantics) and Observable (data/estimation) - State-Space Models are introduced at Structural (latent structure and mechanisms), and used in Dynamical and Observable - ODEs come into being at Dynamical (time-dependent processes), and exist in Observable

Bridge Concepts: Some concepts inherently span multiple worlds because they address the relationship between worlds. These are marked with both worlds and a directional arrow to show the flow:

  • Centrifugal (β†’): Concepts that flow from inner to outer worlds (e.g., Structural β†’ Observable). These are structural rules/principles applied to observable data, like layers of an onion where inner layers inform outer layers.
  • Centripetal (←): Concepts that flow from outer to inner worlds (e.g., Observable β†’ Structural). These are methods that use observable data to infer/reason about inner world possibilities, like peeling back layers of an onion to reveal inner structure.

The Onion Metaphor: The three layers are like layers of an onion. Bridge concepts show how we move between layersβ€”either applying structural assumptions to data (centrifugal) or using data to infer structural quantities (centripetal).

Methods vs Concepts: Some entries distinguish between: - Concepts/theories (what exists at each world): The ontological structure - Methods/computations (what we do): Practical operations that may span worlds

This classification is purely organisational: it keeps separate what we assume structurally, what we model dynamically, and what we observe/estimate.

35.2 Table 1: Concepts by World

Concept Primary World Description
Graph Theory Structural Causal structure assumptions, DAGs, directed dependencies (includes network structure and properties)
d-Separation Structural Graph-theoretic conditional independence
Markov Boundary Structural Minimal sufficient set for causal reasoning
Identification Structural β†’ Observable Centrifugal bridge: Structural question about what can be learned from observable data
Do-Calculus Structural β†’ Observable Centrifugal bridge: Structural rules applied to compute interventional distributions from observable data
Counterfactuals (concept) Observable β†’ Structural Centripetal bridge: Use observable data to reason about structural alternative possibilities
Transportability Structural β†’ Observable Centrifugal bridge: Structural question about whether causal claims can be generalised across observable domains
State-Space Models (concept) Structural Latent process structure and mechanisms (often treated as time-invariant within a modelling context)
Filtering Observable β†’ Structural Centripetal bridge: Method using observable data to infer current structural state
Smoothing Observable β†’ Structural Centripetal bridge: Method using observable data to infer past structural states
Identifiability Structural β†’ Observable Centrifugal bridge: Structural question about whether mechanisms are learnable from observable data
Model Criticism Observable β†’ Structural Centripetal bridge: Method using observable data to test structural model validity
Free Energy Principle Structural β†’ Dynamical Centrifugal bridge: Structural principle applied to dynamical systems
Markov Blanket Structural/Dynamical Internal/external state separation
ODEs Dynamical Deterministic dynamics, flows, equilibria
SDEs Dynamical Stochastic dynamics, process noise
Regime Switching Dynamical Tipping points, attractor transitions
Resilience Dynamical Recovery from perturbations
Robustness Dynamical Maintaining function under variation
CDMs Observable Unified causal-dynamical framework linking structure, dynamics, and data
Correlation Analysis Observable Method: Computing correlations from observed data
Conditional Forecasting Observable Method: Forecasting future observations given past data
Interventional Forecasting Observable β†’ Structural Centripetal bridge: Method using observable data to reason about structural interventions
Counterfactual Simulation Observable Method: Computing unit-level alternative outcomes
G-Methods Observable Methods for time-varying confounding
TMLE Observable Method: Targeted maximum likelihood estimation
Policy Evaluation Observable Method: Evaluating dynamic treatment strategies
Experimental Design Observable Method: Optimal measurement strategies

35.3 Table 2: Concepts by Level of Reason

Concept Level 1 (Association) Level 2 (Intervention) Level 3 (Counterfactual)
Graph Theory Conditional independence (structural property) Intervention structure Counterfactual structure
d-Separation βœ“ (tested on observable data) βœ“ βœ“
Markov Boundary βœ“ (applied to observable models) βœ“ βœ“
Identification β€” βœ“ βœ“
Do-Calculus β€” βœ“ βœ“
Counterfactuals β€” β€” βœ“
State-Space Inference βœ“ (infer structural from observable) βœ“ βœ“
Filtering βœ“ (infer current state) βœ“ (infer under intervention) βœ“ (infer for counterfactual)
Smoothing βœ“ (infer past states) βœ“ βœ“
Conditional Forecasting βœ“ β€” β€”
Interventional Forecasting β€” βœ“ β€”
Counterfactual Simulation β€” β€” βœ“
G-Methods β€” βœ“ βœ“
TMLE β€” βœ“ βœ“
Policy Evaluation β€” βœ“ βœ“
ODEs/SDEs (as mechanisms) βœ“ (simulate dynamics) βœ“ (simulate under intervention) βœ“ (simulate counterfactual)
Resilience Analysis βœ“ (measure from data) βœ“ (test under intervention) βœ“ (compare counterfactuals)
Experimental Design βœ“ (design observational studies) βœ“ (design interventions) βœ“ (design for counterfactuals)
Correlation Analysis βœ“ β€” β€”

Legend: βœ“ = Concept applies at this level; β€” = Concept does not apply at this level

35.4 Table 3: Mathematical Methods by World and Level

Method World L1 (Association) L2 (Intervention) L3 (Counterfactual)
Graph algorithms Structural d-separation Backdoor criterion Counterfactual structure
Do-calculus Structural β€” βœ“ βœ“
Identification theory Structural β€” βœ“ βœ“
Kalman filter Observable βœ“ βœ“ βœ“
Particle filter Observable βœ“ βœ“ βœ“
Kalman smoother Observable βœ“ βœ“ βœ“
Posterior predictive checks Observable βœ“ βœ“ βœ“
G-computation Observable β€” βœ“ βœ“
IPTW Observable β€” βœ“ βœ“
TMLE Observable β€” βœ“ βœ“
Off-policy evaluation Observable β€” βœ“ βœ“
ODE integration Observable βœ“ βœ“ βœ“
SDE simulation Observable βœ“ βœ“ βœ“
Correlation analysis Observable βœ“ β€” β€”

35.5 Table 4: Philosophical Concepts by World

Concept World Description
Actual Occasions All Discrete units of experience (Whitehead)
Prehensive Relations Structural How occasions β€œgrasp” predecessors
Concrescence All Process of becoming, state transitions
Gradual Embodiment All Structural β†’ Dynamical β†’ Observable
Perfect Attractors Structural Timeless, spaceless ideal forms
Invariant Attractors Structural Stable, environment-independent
Dynamic Attractors Dynamical Time-dependent, environment-dependent
Actualised States Observable Fully material, observed
Creative Advance All Exogenous noise, novelty in process
Causal Efficacy Structural How past influences present
Presentational Immediacy Observable What we observe directly
Alternative Concrescences Structural Counterfactual possibilities
Markov Blanket Structural/Dynamical System boundaries (FEP)

35.6 Table 5: Methods for Each Level of Reason

35.6.1 Level 1: Association (Seeing)

Method World Purpose
Conditional forecasting Observable Predict future given past observations
Filtering Observable Infer current structural state from observations
Smoothing Observable Infer past structural states from all observations
Correlation analysis Observable Find associations in observed data
d-separation Structural Graph-theoretic conditional independence
Model criticism Observable Validate structural model fit using observable data

35.6.2 Level 2: Intervention (Doing)

Method World Purpose
Do-calculus Structural Rules for computing interventional distributions
Interventional forecasting Observable Forecast under interventions
G-methods Observable Handle time-varying confounding
TMLE Observable Robust causal effect estimation
Policy evaluation Observable Evaluate treatment strategies
Structural interventions Structural Concept of modifying mechanisms

35.6.3 Level 3: Counterfactual (Imagining)

Method World Purpose
Counterfactual simulation Observable Unit-level alternative outcomes
Shared exogenous noise Structural Concept of unit identity
Alternative concrescences Structural Concept of possibilities
Bounds Observable Partial identification
Sensitivity analysis Observable Test counterfactual assumptions

35.7 Table 6: Cross-World Concepts

Some concepts span multiple worlds, showing how the framework unifies:

Concept Worlds Description
Edges (Prehensive Relations) All Fundamental unit connecting all worlds
Attractors All Perfect β†’ Invariant β†’ Dynamic β†’ Actualised
State Variables Structural, Dynamical, Observable \(X_t\) exists across these worlds
Observations Observable \(Y_t\) manifests from inner worlds
Interventions Structural, Observable \(do(\cdot)\) applies across worlds
Exogenous Noise All Creative advance in all worlds
Graph Structure Structural, Dynamical \(G\) constrains all worlds
CDMs All Unified framework across all worlds

35.8 Table 7: Practical Workflows by World and Level

35.8.1 Structural World

Workflow Level Steps
Causal Discovery L1 1. Test conditional independences
2. Apply d-separation
3. Infer graph structure
Identification L2 1. Specify estimand
2. Check identifiability
3. Apply do-calculus
Counterfactual Reasoning L3 1. Identify shared exogenous noise
2. Compute alternative concrescences
3. Compare outcomes

35.8.2 Structural World (continued)

Workflow Level Steps
State Inference L1 1. Filter
2. Smooth
3. Validate with PPCs
Interventional Inference L2 1. Infer state under intervention
2. Propagate through mechanism
3. Compare to baseline
Counterfactual Inference L3 1. Infer unit-specific noise
2. Simulate alternative mechanism
3. Compare trajectories

35.8.3 Dynamical World

Workflow Level Steps
Dynamics Simulation L1 1. Specify ODE/SDE
2. Integrate forward
3. Analyse trajectories
Intervention Simulation L2 1. Modify mechanism
2. Integrate under intervention
3. Compare attractors
Counterfactual Dynamics L3 1. Fix exogenous noise
2. Simulate alternative dynamics
3. Compare system evolution

35.8.4 Observable World

Workflow Level Steps
Forecasting L1 1. Fit CDM
2. Condition on history
3. Predict future
Policy Evaluation L2 1. Fit CDM
2. Specify policy
3. Simulate under \(do(\pi)\)
4. Compute expected outcome
Counterfactual Analysis L3 1. Infer unit-specific noise
2. Simulate counterfactual
3. Compare to observed

35.9 How to Use These Tables

  1. Finding concepts by world: Use Table 1 to see which concepts belong to which world(s)
  2. Finding concepts by level: Use Table 2 to see which concepts apply at each level of Reason
  3. Finding methods: Use Table 3 for mathematical methods, Table 5 for methods by level
  4. Understanding philosophy: Use Table 4 for Whiteheadian/process philosophy concepts
  5. Cross-world connections: Use Table 6 to see how concepts span multiple worlds
  6. Practical workflows: Use Table 7 to see step-by-step procedures

These tables complement the book’s structure by providing a cross-cutting view of how concepts relate to the three-world ontology and three levels of Reason.