Risk Distribution Fitter

Overview

The Risk Distribution Fitter skill provides capabilities for calibrating probability distributions from historical data or expert judgment. It supports both data-driven fitting using statistical methods and expert elicitation protocols for subjective probability assessment.

Capabilities

• Maximum likelihood estimation (MLE)
• Method of moments estimation
• Bayesian parameter estimation
• Goodness-of-fit testing (KS, AD, Chi-square)
• Distribution comparison and selection
• Expert elicitation protocol support (3-point, 5-point)
• PERT distribution calculation
• Visualization of fitted distributions

Used By Processes

• Monte Carlo Simulation for Decision Support
• Predictive Analytics Implementation
• Decision Quality Assessment

Usage

Data-Driven Fitting

# Fit distributions to historical data
fitting_config = {
    "data": [/* historical observations */],
    "candidate_distributions": [
        "normal", "lognormal", "gamma", "weibull",
        "exponential", "beta", "triangular"
    ],
    "fitting_method": "mle",
    "selection_criterion": "AIC"
}

Expert Elicitation

# 3-point estimate (PERT)
expert_estimate = {
    "method": "PERT",
    "minimum": 50000,
    "most_likely": 75000,
    "maximum": 120000,
    "confidence_level": 0.90  # optional: confidence that true value is within range
}

# 5-point estimate (for more precision)
detailed_estimate = {
    "method": "5_point",
    "P10": 45000,
    "P25": 60000,
    "P50": 75000,
    "P75": 95000,
    "P90": 115000
}

Supported Distributions

Distribution	Use Case	Parameters
Normal	Symmetric, unbounded	mean, std
Lognormal	Right-skewed, positive	mu, sigma
Triangular	Bounded with mode	min, mode, max
PERT	Bounded, weighted mode	min, mode, max
Uniform	Equal probability	min, max
Beta	Bounded, flexible shape	alpha, beta
Gamma	Positive, right-skewed	shape, scale
Weibull	Reliability/time	shape, scale
Exponential	Memoryless	rate

Goodness-of-Fit Tests

• Kolmogorov-Smirnov (KS): Distribution-free, sensitive to center
• Anderson-Darling (AD): More sensitive to tails
• Chi-Square: Categorical/binned data
• Cramér-von Mises: Similar to KS, different weighting

Model Selection Criteria

• AIC (Akaike Information Criterion): Balance fit and complexity
• BIC (Bayesian Information Criterion): Stronger penalty for parameters
• Log-Likelihood: Raw fit quality

Input Schema

{
  "fitting_mode": "data_driven|expert_elicitation",
  "data_driven_config": {
    "data": ["number"],
    "candidate_distributions": ["string"],
    "fitting_method": "mle|mom|bayesian",
    "selection_criterion": "AIC|BIC|likelihood"
  },
  "expert_elicitation_config": {
    "method": "3_point|5_point|PERT|direct",
    "estimates": "object",
    "confidence_level": "number"
  },
  "options": {
    "gof_tests": ["KS", "AD", "chi_square"],
    "visualize": "boolean",
    "compare_all": "boolean"
  }
}

Output Schema

{
  "best_fit": {
    "distribution": "string",
    "parameters": "object",
    "gof_statistics": {
      "test_name": {
        "statistic": "number",
        "p_value": "number"
      }
    },
    "selection_score": "number"
  },
  "all_fits": [
    {
      "distribution": "string",
      "parameters": "object",
      "scores": "object"
    }
  ],
  "summary": {
    "mean": "number",
    "std": "number",
    "percentiles": "object"
  },
  "visualization_path": "string",
  "recommendations": ["string"]
}

Best Practices

1. Use data-driven fitting when sufficient historical data exists (n > 30)
2. Validate fitted distributions against holdout data
3. Use PERT for expert estimates when asymmetry is expected
4. Document expert credentials and elicitation process
5. Consider mixture distributions for multimodal data
6. Always visualize fitted distribution against data/estimates
7. Use multiple goodness-of-fit tests for robustness

Expert Elicitation Guidelines

1. Explain probability concepts clearly
2. Use familiar reference points
3. Ask for extreme estimates first, then middle
4. Check for overconfidence (typical: too narrow ranges)
5. Consider debiasing techniques
6. Document reasoning behind estimates

Integration Points

• Feeds into Monte Carlo Engine for simulation inputs
• Supports Calibration Trainer for expert accuracy assessment
• Connects with Bayesian Network Analyzer for CPT estimation
• Integrates with Risk Register Manager for risk quantification