Cross Disciplinary Ideation

The Innovation Cycle

Problem in Statistics → Abstract Structure → Search Other Fields
         ↑                                           ↓
    Validate/Adapt ←── Identify Analogues ←── Find Connections

Machine Learning Connections

Deep Learning for Causal Mediation

# Double ML for mediation effect estimation
library(DoubleML)

# Estimate nuisance parameters with ML
estimate_dml_mediation <- function(Y, A, M, X) {
  # First stage: E[M|A,X]
  mediator_model <- cv.glmnet(cbind(A, X), M)
  M_hat <- predict(mediator_model, cbind(A, X))

  # Second stage: E[Y|A,M,X]
  outcome_model <- cv.glmnet(cbind(A, M, X), Y)

  # Debiased estimation
  residuals_M <- M - M_hat

  list(
    direct = coef(outcome_model)["A"],
    indirect_component = residuals_M
  )
}

Physics Analogies

Energy-Based Statistical Models

Productive Questions:

• "What is the energy landscape of this estimation problem?"
• "What physical system has this equilibrium?"
• "How would a physicist think about this constraint?"

Computer Science Algorithms

Algorithmic Approaches to Statistical Problems

# Dynamic programming for sequential mediation paths
compute_path_effects <- function(effect_matrix, n_mediators) {
  # effect_matrix[i,j] = effect from node i to node j
  n <- nrow(effect_matrix)

  # Initialize path effects (like shortest path, but products)
  path_effects <- matrix(0, n, n)
  diag(path_effects) <- 1

  # DP recurrence: path[i,j] = sum over k of path[i,k] * edge[k,j]
  for (len in 1:n_mediators) {
    for (i in 1:n) {
      for (j in 1:n) {
        for (k in 1:n) {
          if (effect_matrix[k, j] != 0) {
            path_effects[i, j] <- path_effects[i, j] +
              path_effects[i, k] * effect_matrix[k, j]
          }
        }
      }
    }
  }

  path_effects
}

Statistics ↔ Computer Science

Productive Questions:

• "What's the computational complexity of this estimator?"
• "Is there an online version of this method?"
• "What optimization algorithm solves this?"

Statistics ↔ Economics

Productive Questions:

• "What are the incentives in this data collection?"
• "Is there a game-theoretic interpretation?"
• "What market mechanism generates this data?"

Biology Applications

Evolutionary and Systems Biology Connections

# Mendelian randomization for mediation
# Using genetic variants as instruments
mr_mediation <- function(snp, exposure, mediator, outcome) {
  # Stage 1: SNP -> Exposure
  gamma_A <- coef(lm(exposure ~ snp))["snp"]

  # Stage 2: SNP -> Mediator (genetic effect on M)
  gamma_M <- coef(lm(mediator ~ snp + exposure))["snp"]

  # Stage 3: Instrument-based mediation
  # Indirect via genetic pathway
  iv_model <- ivreg(outcome ~ mediator + exposure | snp + exposure)

  list(
    genetic_effect_exposure = gamma_A,
    genetic_effect_mediator = gamma_M,
    iv_mediation_estimate = coef(iv_model)["mediator"] * gamma_M
  )
}

Statistics ↔ Biology

Productive Questions:

• "What evolutionary pressure shapes this distribution?"
• "Is there a biological network analog?"
• "How does selection affect what we observe?"

Statistics ↔ Mathematics

Productive Questions:

• "What's the geometric structure of this problem?"
• "Is there a measure-theoretic generalization?"
• "What invariance does this exploit?"

Structured Ideation Process

Step 1: Problem Decomposition

Break the statistical problem into abstract components:

Problem: "Estimate mediation effects with measurement error"

Components:
1. Causal structure (DAG with mediator)
2. Latent variable (true M vs observed M*)
3. Identification (what assumptions needed?)
4. Estimation (how to account for error?)
5. Inference (variance under misspecification?)

Step 2: Abstract Pattern Recognition

Identify the mathematical essence:

Abstract patterns in measurement error mediation:
- Signal + noise model
- Latent variable with proxy
- Product of uncertain quantities
- Attenuation toward null

Step 3: Cross-Field Search

For each abstract pattern, search analogues:

Step 4: Deep Dive on Promising Connections

For each promising analogue:

1. Understand the source method deeply

   • What problem does it solve?
   • What assumptions does it make?
   • What are its limitations?

2. Map to target domain

   • What corresponds to what?
   • What assumptions translate?
   • What doesn't transfer?

3. Identify the gap

   • What modification is needed?
   • Is the gap a feature or bug?
   • Can we fill it?

Step 5: Synthesis and Evaluation

Evaluation Criteria:
□ Does it solve a real problem?
□ Is it novel (not already done)?
□ Are assumptions reasonable?
□ Is it computationally feasible?
□ Can it be proven to work (theory)?
□ Does it work in practice (simulation)?

Ideation Prompts by Problem Type

When Stuck on Identification

• "How do economists identify effects in similar settings?"
• "What instrumental variable approach might work here?"
• "Is there a regression discontinuity analog?"
• "What if this were a designed experiment?"

When Stuck on Estimation

• "How would a machine learner approach this?"
• "Is there an EM algorithm formulation?"
• "What loss function captures my goal?"
• "Can I frame this as optimization?"

When Stuck on Computation

• "What physics simulation technique applies?"
• "Is there an approximate algorithm from CS?"
• "Can I use stochastic approximation?"
• "What variational approach might work?"

When Stuck on Theory

• "What's the information-theoretic limit?"
• "Is there a minimax lower bound?"
• "What geometry characterizes this problem?"
• "Can I use empirical process theory?"

When Stuck on Robustness

• "What's the worst-case distribution?"
• "How would a game theorist think about this?"
• "What's the sensitivity to assumptions?"
• "Can I bound instead of point estimate?"

Successful Transfer Examples

Example 1: Propensity Scores from Survey Sampling

Source: Survey sampling (Horvitz-Thompson estimator) Target: Causal inference (propensity score weighting)

Transfer insight:

• Selection into treatment ≈ selection into sample
• Inverse probability weighting corrects both
• Same variance inflation issues

Innovation: Rosenbaum & Rubin (1983) - propensity score methods

Example 2: Lasso from Signal Processing

Source: Basis pursuit in signal processing Target: Variable selection in regression

Transfer insight:

• Sparse signals ≈ sparse coefficients
• L1 penalty induces sparsity
• Convex relaxation of L0

Innovation: Tibshirani (1996) - Lasso regression

Example 3: Double Robustness from Missing Data

Source: Missing data augmented IPW Target: Causal inference estimators

Transfer insight:

• Missing outcomes ≈ counterfactual outcomes
• Augmentation improves efficiency
• Protection against model misspecification

Innovation: Robins et al. - AIPW estimators

Example 4: Influence Functions from Robustness

Source: Robust statistics (Hampel) Target: Semiparametric efficiency

Transfer insight:

• Influence function measures sensitivity
• Also characterizes asymptotic variance
• Efficient influence function = optimal

Innovation: Bickel et al. - semiparametric theory

Domain-Specific Prompts for Mediation Research

From Causal Inference Literature

• "How do IV methods handle unmeasured confounding? Can this apply to A-M confounding?"
• "What do DID approaches suggest for mediation in panel data?"
• "How does synthetic control relate to mediation counterfactuals?"

From Machine Learning

• "Can representation learning separate direct/indirect pathways?"
• "How would a VAE model the mediation structure?"
• "What does causal forest suggest for heterogeneous mediation?"

From Econometrics

• "How do structural equation models in econ differ from psychology?"
• "What do control functions offer for endogeneity in mediators?"
• "How does Heckman selection relate to mediator measurement?"

From Biostatistics

• "How does survival analysis handle time-varying mediators?"
• "What do competing risks suggest for multiple mediators?"
• "How does Mendelian randomization inform mediator instruments?"

From Physics/Information Theory

• "What does information decomposition say about mediation?"
• "How do Markov blankets relate to mediation assumptions?"
• "What does the data processing inequality imply?"

Innovation Documentation Template

When you discover a promising connection:

## Connection: [Source Method] → [Target Application]

### Source Domain
- **Method**: [Name and citation]
- **Problem it solves**: [Description]
- **Key insight**: [Core idea]
- **Assumptions**: [What it requires]

### Target Domain
- **Problem**: [Statistical problem to solve]
- **Current approaches**: [Existing methods and limitations]
- **Gap**: [What's missing]

### Transfer Analysis
- **Structural correspondence**:
  - [Source concept] ↔ [Target concept]
  - [Source assumption] ↔ [Target assumption]

- **What transfers directly**: [List]
- **What needs modification**: [List]
- **What doesn't transfer**: [List]

### Proposed Innovation
- **Core idea**: [How to adapt]
- **Novel contribution**: [What's new]
- **Theoretical questions**: [What to prove]
- **Empirical questions**: [What to simulate]

### Feasibility Assessment
- [ ] Theoretically sound
- [ ] Computationally tractable
- [ ] Practically relevant
- [ ] Sufficiently novel
- [ ] Publishable venue: [Journal]

### Next Steps
1. [Immediate action]
2. [Follow-up]
3. [Validation approach]

Transfer Opportunities

High-Priority Cross-Disciplinary Transfers for Statistical Research

Immediate Research Directions

# Transfer: Control functions from economics to mediation
# Relaxes sequential ignorability assumption
control_function_mediation <- function(Y, A, M, X, Z) {
  # Z is instrument for A

  # First stage: A on Z and X
  stage1 <- lm(A ~ Z + X)
  A_residual <- residuals(stage1)

  # Second stage with control function
  # Includes residual to correct for endogeneity
  stage2 <- lm(M ~ A + X + A_residual)

  # Third stage: outcome with control
  stage3 <- lm(Y ~ A + M + X + A_residual)

  list(
    a_to_m = coef(stage2)["A"],
    m_to_y = coef(stage3)["M"],
    indirect = coef(stage2)["A"] * coef(stage3)["M"],
    control_function_coef = coef(stage2)["A_residual"]
  )
}

Transfer Success Criteria

For any cross-disciplinary transfer, evaluate:

1. Structural Match: Does the source problem structure map to target?
2. Assumption Compatibility: Do source assumptions make sense in target?
3. Computational Feasibility: Is the transferred method tractable?
4. Novel Contribution: Is this genuinely new in the target field?
5. Practical Value: Does it solve a real problem researchers face?

Integration with Other Skills

This skill works with:

• literature-gap-finder - Identify where innovation is needed
• method-transfer-engine - Formalize the transfer
• proof-architect - Prove the transferred method works
• identification-theory - Check identification in new setting
• methods-paper-writer - Write up the innovation

Key References

Cross-Disciplinary Statistics

• Efron, B. & Hastie, T. (2016). Computer Age Statistical Inference
• Hastie, T., Tibshirani, R., & Friedman, J. (2009). Elements of Statistical Learning
• Cover, T.M. & Thomas, J.A. (2006). Elements of Information Theory

Physics-Statistics Connection

• MacKay, D.J.C. (2003). Information Theory, Inference, and Learning Algorithms
• Jaynes, E.T. (2003). Probability Theory: The Logic of Science

CS-Statistics Connection

• Shalev-Shwartz, S. & Ben-David, S. (2014). Understanding Machine Learning
• Vershynin, R. (2018). High-Dimensional Probability

Version: 1.0 Created: 2025-12-08 Domain: Research Innovation, Method Development