risk-distribution-fitter

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Probability distribution fitting skill for calibrating uncertainty models from historical data or expert judgment

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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

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

Expert Elicitation Guidelines

Explain probability concepts clearly
Use familiar reference points
Ask for extreme estimates first, then middle
Check for overconfidence (typical: too narrow ranges)
Consider debiasing techniques
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