Direct Application → Minor Adaptation → Major Modification → Inspired-By
      │                    │                   │                  │
   Same theory         Adjust for          Rewrite theory      New method,
   applies            new setting          for new setting     similar spirit

# Template for source method extraction
extract_source_method <- function(method_name, reference) {
  list(
    name = method_name,
    estimand = "formal expression of what is estimated",
    estimator = "formula for the estimator",
    assumptions = c("A1: condition", "A2: condition"),
    properties = c("consistency", "asymptotic normality"),
    algorithm = c("Step 1: ...", "Step 2: ..."),
    complexity = "O(n^2) or similar"
  )
}

# Example: Extract Lasso from signal processing
lasso_extraction <- list(
  name = "Lasso/Basis Pursuit",
  field = "Signal Processing / Compressed Sensing",
  estimand = "argmin ||y - Xb||_2^2 + lambda * ||b||_1",
  key_insight = "L1 penalty induces sparsity via soft thresholding",
  assumptions = c("RIP condition", "Incoherence"),
  properties = c("Sparse solution", "Variable selection consistency")
)

# Abstract structure identification
identify_abstraction <- function(source_method) {
  list(
    mathematical_structure = "e.g., M-estimation, U-statistics, kernels",
    core_operation = "e.g., reweighting, regularization, projection",
    information_used = "e.g., first moments, covariance, distributional",
    key_invariance = "what property makes it work",
    generalization_path = "how to extend beyond original setting"
  )
}

# Example: Abstraction of propensity score methods
propensity_abstraction <- list(
  mathematical_structure = "Reweighting to balance distributions",
  core_operation = "Inverse probability weighting",
  invariance = "Balances covariate distribution across groups",
  generalization = "Any selection mechanism with known probabilities"
)

## Source Method Profile

### Basic Information
- Name: [Method name]
- Source field: [Domain/area]
- Key reference: [Citation]
- What it does: [One sentence]

### Problem Solved
- Input: [What data/information goes in]
- Output: [What estimate/inference comes out]
- Setting: [When it applies]

### Mathematical Structure
- Estimand: [What it estimates, formally]
- Estimator: [How it estimates, formula]
- Loss/objective: [What it optimizes]

### Assumptions Required
1. [Assumption 1]: [Mathematical statement]
   - Why needed: [Role in proof/method]
   - When violated: [Failure mode]

2. [Assumption 2]: ...

### Theoretical Properties
- Consistency: [When/how proved]
- Rate: [Convergence rate]
- Asymptotic distribution: [If known]
- Efficiency: [Relative to what]
- Robustness: [To what violations]

### Computational Aspects
- Algorithm: [How implemented]
- Complexity: [Time/space]
- Software: [Available implementations]

## Target Problem Profile

### Basic Information
- Problem name: [Description]
- Target field: [Domain/area]
- Motivation: [Why solve this]

### Problem Structure
- Data available: [What's observed]
- Estimand: [What you want to estimate]
- Challenges: [Why existing methods inadequate]

### Current Approaches
- Method 1: [Name, limitations]
- Method 2: [Name, limitations]
- Gap: [What's missing]

### Constraints
- Assumptions willing to make: [List]
- Assumptions NOT willing to make: [List]
- Computational constraints: [If any]

# Target mapping framework
create_target_mapping <- function(source, target) {
  mapping <- list(
    objects = data.frame(
      source = c("treatment", "outcome", "confounder"),
      target = c("mediator", "effect", "moderator"),
      relationship = c("direct", "indirect", "modifies")
    ),
    assumptions = data.frame(
      source_assumption = c("SUTVA", "Ignorability"),
      target_version = c("Consistency", "Sequential ignorability"),
      status = c("transfers", "needs modification")
    )
  )

  mapping
}

# Example: IV to Mendelian randomization mapping
iv_to_mr <- list(
  price_instrument = "genetic_variant",
  demand = "biomarker_exposure",
  endogeneity = "unmeasured_confounding",
  exclusion = "pleiotropic_effects",
  key_difference = "biological vs economic mechanisms"
)

## Structure Map

### Object Correspondence

| Source | Target | Notes |
|--------|--------|-------|
| [Source object 1] | [Target object 1] | [How they relate] |
| [Source object 2] | [Target object 2] | [How they relate] |
| ... | ... | ... |

### Assumption Correspondence

| Source Assumption | Target Version | Status |
|-------------------|----------------|--------|
| [Source A1] | [Target A1'] | ✓ Transfers / ✗ Fails / ? Modify |
| [Source A2] | [Target A2'] | ... |
| ... | ... | ... |

### What Transfers Directly
- [Property 1]: Because [reason]
- [Property 2]: Because [reason]

### What Needs Modification
- [Element 1]: From [source version] to [target version]
  - Why: [Reason for change]
  - How: [Specific modification]

### What Doesn't Transfer
- [Element 1]: Because [reason]
  - Impact: [What we lose]
  - Alternative: [How to address]

# Gap analysis framework
analyze_transfer_gaps <- function(source, target, mapping) {
  gaps <- list(
    assumption_gaps = list(
      violated = c("iid assumption in clustered data"),
      modified = c("independence -> conditional independence"),
      new_required = c("mediator positivity")
    ),

    property_gaps = list(
      lost = c("efficiency under misspecification"),
      weakened = c("convergence rate n^{-1/2} -> n^{-1/4}"),
      preserved = c("consistency", "asymptotic normality")
    ),

    computational_gaps = list(
      new_challenges = c("non-convex optimization"),
      workarounds = c("ADMM algorithm", "approximate methods")
    ),

    bridging_strategies = c(
      "Add regularization for new setting",
      "Derive modified variance estimator",
      "Implement robustness check"
    )
  )

  gaps
}

## Adapted Method Design

### Overview
[One paragraph describing the adapted method]

### Formal Definition

**Estimand**:
$$\psi = [target estimand formula]$$

**Estimator**:
$$\hat{\psi}_n = [adapted estimator formula]$$

**Algorithm**:
1. [Step 1]
2. [Step 2]
3. ...

### Modified Assumptions
1. [Assumption A1']: [New statement for target setting]
   - Analogous to: [Source assumption]
   - Modified because: [Reason]

### Expected Properties
- Consistency: [Conjecture/claim]
- Rate: [Expected]
- Efficiency: [Expected]

### Key Differences from Source
1. [Difference 1]: [Explanation]
2. [Difference 2]: [Explanation]

# Comprehensive validation framework for method transfer
validate_transfer <- function(adapted_method, n_sims = 1000) {
  results <- list()

  # 1. Bias check: Is estimator unbiased at truth?
  results$bias <- run_bias_simulation(adapted_method, n_sims)

  # 2. Coverage check: Do CIs achieve nominal coverage?
  results$coverage <- run_coverage_simulation(adapted_method, n_sims)

  # 3. Efficiency check: Compare to alternatives
  results$efficiency <- compare_to_alternatives(adapted_method)

  # 4. Robustness check: Behavior under violations
  results$robustness <- test_assumption_violations(adapted_method)

  # 5. Edge cases: Extreme scenarios
  results$edge_cases <- test_edge_cases(adapted_method)

  # Validation report
  list(
    passed = all(sapply(results, function(x) x$passed)),
    details = results,
    recommendations = generate_recommendations(results)
  )
}

# Simulation template for validation
run_transfer_validation <- function(n = 500, n_sims = 1000) {
  estimates <- replicate(n_sims, {
    # Generate data under true model
    data <- generate_dgp(n)

    # Apply transferred method
    est <- adapted_method(data)

    c(estimate = est$point, se = est$se)
  })

  list(
    bias = mean(estimates["estimate", ]) - true_value,
    rmse = sqrt(mean((estimates["estimate", ] - true_value)^2)),
    coverage = mean(abs(estimates["estimate", ] - true_value) <
                   1.96 * estimates["se", ])
  )
}

## Verification Plan

### Theoretical Verification
- [ ] Consistency proof
  - Approach: [Proof strategy]
  - Key lemma: [What needs to be shown]

- [ ] Asymptotic normality
  - Approach: [Proof strategy]
  - Influence function: [If applicable]

- [ ] Efficiency (if claiming)
  - Approach: [Efficiency bound derivation]

### Simulation Verification
- [ ] Scenario 1: [Description]
  - DGP: [Data generating process]
  - Expected result: [What should happen]

- [ ] Scenario 2: Comparison to oracle
  - Purpose: [Verify optimality]

- [ ] Scenario 3: Stress test
  - Purpose: [Find failure modes]

### Empirical Verification
- [ ] Benchmark dataset: [If available]
- [ ] Real application: [Domain]

## Transfer Documentation

### Contribution Statement
"We adapt [source method] from [source field] to [target setting] by
[key modification]. Our adapted method [key property]. Unlike [alternative],
our approach [advantage]."

### Theoretical Contribution
- New result 1: [Theorem statement]
- New result 2: [If applicable]

### Methodological Contribution
- Adaptation insight: [What's novel about the transfer]
- Practical guidance: [When to use]

### What We Learned
- About source method: [New understanding]
- About target problem: [New understanding]
- General principle: [Broader insight]

Source: Horvitz-Thompson estimator
        E[Y] ≈ Σᵢ Yᵢ/πᵢ where πᵢ = P(selected)

Target: IPW for ATE
        E[Y(1)] ≈ Σᵢ Yᵢ·Aᵢ/e(Xᵢ) where e(x) = P(A=1|X=x)

Mapping:
- Selection indicator → Treatment indicator
- Selection probability → Propensity score
- Survey weights → Inverse propensity weights

Key insight: Both correct for selection bias via reweighting

Source: Augmented IPW for missing data
        DR = IPW + Imputation - (IPW × Imputation)

Target: AIPW for causal effects
        Same structure but for counterfactual outcomes

Mapping:
- Missing indicator → Treatment indicator
- Missingness model → Propensity model
- Imputation model → Outcome model

Key insight: Product-form bias enables robustness to one misspecification

Source: IF for smooth functional of CDF
        √n(T(Fₙ) - T(F)) → N(0, E[φ²])

Target: IF for mediation effect functional
        Requires: mediation-specific tangent space

Mapping:
- General functional → Mediation estimand
- CDF → Joint distribution (Y,M,A,X)
- Generic IF → Mediation-specific IF

Key insight: EIF theory applies to any pathwise differentiable functional

Source: Instrumental variables for demand estimation
        Z → A → Y, Z ⫫ U

Target: MR for causal effects of exposures
        Gene → Biomarker → Outcome

Mapping:
- Price instrument → Genetic variant
- Demand → Exposure level
- Endogeneity → Confounding

Key insight: Exogenous variation strategy is general

Source: Stochastic gradient descent for ERM
        θₜ₊₁ = θₜ - ηₜ∇L(θₜ; Xₜ)

Target: Online updating for streaming causal data
        Sequential estimation as data arrives

Mapping:
- Loss function → Estimating equation
- Gradient → Score contribution
- Learning rate → Weighting scheme

Key insight: Streaming updates possible for M-estimators