Bayesian Reasoning & Calibration

Bayesian Reasoning & Calibration

Table of Contents

• Purpose
• When to Use This Skill
• What is Bayesian Reasoning?
• Workflow
  • 1. Define the Question
  • 2. Establish Prior Beliefs
  • 3. Identify Evidence & Likelihoods
  • 4. Calculate Posterior
  • 5. Calibrate & Document
• Common Patterns
• Guardrails
• Quick Reference

Purpose

Apply Bayesian reasoning to systematically update probability estimates as new evidence arrives. This helps make better forecasts, avoid overconfidence, and explicitly show how beliefs should change with data.

When to Use This Skill

• Making forecasts or predictions with uncertainty
• Updating beliefs when new evidence emerges
• Calibrating confidence in estimates
• Testing hypotheses with imperfect data
• Evaluating risks with incomplete information
• Avoiding anchoring and overconfidence biases
• Making decisions under uncertainty
• Comparing multiple competing explanations
• Assessing diagnostic test results
• Forecasting project outcomes with new data

Trigger phrases: "What's the probability", "update my belief", "how confident", "forecast", "prior probability", "likelihood", "Bayes", "calibration", "base rate", "posterior probability"

What is Bayesian Reasoning?

A systematic way to update probability estimates using Bayes' Theorem:

P(H|E) = P(E|H) × P(H) / P(E)

Where:

• P(H) = Prior: Probability of hypothesis before seeing evidence
• P(E|H) = Likelihood: Probability of evidence if hypothesis is true
• P(E|¬H) = Probability of evidence if hypothesis is false
• P(H|E) = Posterior: Updated probability after seeing evidence

Quick Example:

# Should we launch Feature X?

## Prior Belief
Before beta testing: 60% chance of adoption >20%
- Base rate: Similar features get 15-25% adoption
- Our feature seems stronger than average
- Prior: 60%

## New Evidence
Beta test: 35% of users adopted (70 of 200 users)

## Likelihoods
If true adoption is >20%:
- P(seeing 35% in beta | adoption >20%) = 75% (likely to see high beta if true)

If true adoption is ≤20%:
- P(seeing 35% in beta | adoption ≤20%) = 15% (unlikely to see high beta if false)

## Bayesian Update
Posterior = (75% × 60%) / [(75% × 60%) + (15% × 40%)]
Posterior = 45% / (45% + 6%) = 88%

## Conclusion
Updated belief: 88% confident adoption will exceed 20%
Evidence strongly supports launch, but not certain.

Workflow

Copy this checklist and track your progress:

Bayesian Reasoning Progress:
- [ ] Step 1: Define the question
- [ ] Step 2: Establish prior beliefs
- [ ] Step 3: Identify evidence and likelihoods
- [ ] Step 4: Calculate posterior
- [ ] Step 5: Calibrate and document

Step 1: Define the question

Clarify hypothesis (specific, testable claim), probability to estimate, timeframe (when outcome is known), success criteria, and why this matters (what decision depends on it). Example: "Product feature will achieve >20% adoption within 3 months" - matters for launch decision.

Step 2: Establish prior beliefs

Set initial probability using base rates (general frequency), reference class (similar situations), specific differences, and explicit probability assignment with justification. Good priors are based on base rates, account for differences, honest about uncertainty, and include ranges if unsure (e.g., 40-60%). Avoid purely intuitive priors, ignoring base rates, or extreme values without justification.

Step 3: Identify evidence and likelihoods

Assess evidence (specific observation/​data), diagnostic power (does it distinguish hypotheses?), P(E|H) (probability if hypothesis TRUE), P(E|¬H) (probability if FALSE), and calculate likelihood ratio = P(E|H) / P(E|¬H). LR > 10 = very strong evidence, 3-10 = moderate, 1-3 = weak, ≈1 = not diagnostic, <1 = evidence against.

Step 4: Calculate posterior

Apply Bayes' Theorem: P(H|E) = [P(E|H) × P(H)] / P(E), or use odds form: Posterior Odds = Prior Odds × Likelihood Ratio. Calculate P(E) = P(E|H)×P(H) + P(E|¬H)×P(¬H), get posterior probability, and interpret change. For simple cases → Use resources/​template.md calculator. For complex cases (multiple hypotheses) → Study resources/​methodology.md.

Step 5: Calibrate and document

Check calibration (over/​underconfident?), validate assumptions (are likelihoods reasonable?), perform sensitivity analysis, create bayesian-reasoning-calibration.md, and note limitations. Self-check using resources/​evaluators/​rubric_bayesian_reasoning_calibration.json: verify prior based on base rates, likelihoods justified, evidence diagnostic (LR ≠ 1), calculation correct, posterior calibrated, assumptions stated, sensitivity noted. Minimum standard: Score ≥ 3.5.

Common Patterns

For forecasting:

• Use base rates as starting point
• Update incrementally as evidence arrives
• Track forecast accuracy over time
• Calibrate by comparing predictions to outcomes

For hypothesis testing:

• State competing hypotheses explicitly
• Calculate likelihood ratio for evidence
• Update belief proportionally to evidence strength
• Don't claim certainty unless LR is extreme

For risk assessment:

• Consider multiple scenarios (not just binary)
• Update risks as new data arrives
• Use ranges when uncertain about likelihoods
• Perform sensitivity analysis

For avoiding bias:

• Force explicit priors (prevents anchoring to evidence)
• Use reference classes (prevents ignoring base rates)
• Calculate mathematically (prevents motivated reasoning)
• Document before seeing outcome (enables calibration)

Guardrails

Do:

• State priors explicitly before seeing all evidence
• Use base rates and reference classes
• Estimate likelihoods with justification
• Update incrementally as evidence arrives
• Be honest about uncertainty
• Perform sensitivity analysis
• Track forecasts for calibration
• Acknowledge limits of the model

Don't:

• Use extreme priors (1%, 99%) without exceptional justification
• Ignore base rates (common bias)
• Treat all evidence as equally diagnostic
• Update to 100% certainty (almost never justified)
• Cherry-pick evidence
• Skip documenting reasoning
• Forget to calibrate (compare predictions to outcomes)
• Apply to questions where probability is meaningless

Quick Reference

• Standard template: resources/​template.md
• Multiple hypotheses: resources/​methodology.md
• Examples: resources/​examples/​product-launch.md, resources/​examples/​medical-diagnosis.md
• Quality rubric: resources/​evaluators/​rubric_bayesian_reasoning_calibration.json

Bayesian Formula (Odds Form):

Posterior Odds = Prior Odds × Likelihood Ratio

Likelihood Ratio:

LR = P(Evidence | Hypothesis True) / P(Evidence | Hypothesis False)

Output naming: bayesian-reasoning-calibration.md or {topic}-forecast.md