Monte Carlo Simulation Design

Help the researcher design rigorous Monte Carlo simulations to validate theoretical results. Simulations bridge the gap between theory and practice -- they demonstrate that analytical formulas work as predicted and reveal finite-sample behavior.

Step 1: Read Context

Read .papermill/state.md (Read tool) for:

• Thesis: What theoretical result needs validation.
• Experiments: Any existing simulation experiments.

If .papermill/state.md does not exist, ask the user which theoretical result needs validation. Simulation design can proceed without the state file — suggest running /papermill:init afterward.

Scan the repository for existing simulation code and results (Glob/Read tools).

Step 2: Identify What to Validate

Ask: "Which theoretical result are you validating with this simulation?"

Common simulation targets in methodology papers:

• Asymptotic distributions: Does the CLT approximation hold at finite n?
• Estimator properties: Is the MLE unbiased? What is its MSE?
• Covariance/information matrices: Does the empirical covariance match the theoretical Fisher information?
• Confidence interval coverage: Do 95% CIs actually cover 95% of the time?
• Power analysis: Can the test detect a given effect size?
• Algorithm convergence: Does the optimizer find the right answer?

Step 3: Design the Simulation

Parameters

• True parameter values: Choose several configurations spanning easy, moderate, and hard cases.
• Sample sizes: Include small (n=20-50), moderate (n=100-500), and large (n=1000+) to show convergence.
• Number of replications: Typically R=1,000-10,000. More replications reduce Monte Carlo error but cost time. The Monte Carlo standard error for a proportion p is sqrt(p(1-p)/R).

Data Generation

Specify exactly how to generate synthetic data:

1. Set random seed for reproducibility.
2. Generate from the known model with known true parameters.
3. Apply the estimation/inference procedure to each replicate.
4. Collect the quantities of interest.

Metrics

What to measure	How to summarize
Bias	Mean(estimate) - true value
Variance	Var(estimates) across replicates
MSE	Bias^2 + Variance
Coverage	Fraction of CIs containing true value
Convergence rate	Plot metric vs. n on log scale

Memory and Compute

• Batch processing: If R * n is large, process in batches to control memory. Accumulate sufficient statistics rather than storing all raw data.
• Parallelization: Independent replicates are embarrassingly parallel. Use parallel (R), multiprocessing (Python), or OpenMP (C++).
• Progress reporting: For long simulations, report progress every ~10% of replicates.

Step 4: Convergence Diagnostics

Before presenting results, verify the simulation itself is reliable:

1. Monte Carlo standard errors: Report them alongside point estimates. If MCSE is large relative to the quantity of interest, increase R.
2. Trace plots: Plot running averages to verify convergence.
3. Sensitivity to seed: Run with 2-3 different seeds. Results should be stable.

Step 5: Result Presentation

Design the output tables and figures:

Tables

• Rows: parameter configurations or sample sizes
• Columns: bias, SD, MSE, coverage, or whatever metrics are relevant
• Include Monte Carlo standard errors in parentheses

Figures

• Convergence plots: Metric vs. n (log scale) with theoretical prediction overlaid
• QQ plots: Compare empirical distribution of estimator to theoretical asymptotic distribution
• Box plots: Distribution of estimates across replicates for each configuration

Step 6: Common Pitfalls

Warn about:

• Insufficient replicates: R=100 is rarely enough. For coverage studies, you need R >= 1,000.
• Ignoring Monte Carlo error: Always report MCSE. A coverage of 0.94 with MCSE=0.007 is consistent with 0.95.
• Conditioning bias: If you filter out "non-convergent" replicates, the remaining sample is biased. Report the rejection rate.
• Single parameter configuration: Results at one parameter value do not generalize. Test across a range.
• Floating-point issues: For extreme parameter values, numerical issues can corrupt results. Include sanity checks.

Step 7: Update State File

If .papermill/state.md exists, register the simulation (Edit tool) under experiments. If it does not exist, skip registration and suggest running /papermill:init.

The entry uses the standard experiment schema with an optional config block for simulation-specific parameters:

experiments:
  - name: "simulation-name"
    type: "simulation"
    hypothesis: "Empirical covariance matches theoretical FIM as n grows"
    status: "planned"
    script: "research/simulate_covariance.R"
    last_run: null
    config:  # simulation-specific extension, not in the base experiment schema
      replications: 5000
      sample_sizes: [50, 100, 200, 500, 1000]
      parameter_configs: 3

Append a timestamped note documenting the simulation design.

Step 8: Suggest Next Steps

Based on the simulation status, suggest the most relevant next step:

• Implement: "Write the simulation script. Start with a small pilot (R=100) to debug before the full run."
• If the simulation validates a proof → "Once results confirm the theory, integrate both the proof and simulation evidence into the paper. Use /papermill:proof if the proof itself needs work."
• If simulation results are surprising or contradict theory → "The simulation suggests the theoretical result may need revisiting. Use /papermill:proof to re-examine the proof's assumptions."
• If results are written up → "Use /papermill:review to get feedback on the presentation of simulation results."