// Update beliefs systematically based on new evidence using probabilistic reasoning. Use when estimating probabilities, learning from data, or making decisions under uncertainty.
| name | thinking-bayesian |
| description | Update beliefs systematically based on new evidence using probabilistic reasoning. Use when estimating probabilities, learning from data, or making decisions under uncertainty. |
Bayesian thinking provides a framework for updating beliefs based on new evidence. Rather than treating beliefs as binary (true/false), it recognizes degrees of confidence that should shift as evidence accumulates. This approach, rooted in Bayes' Theorem, helps avoid both overconfidence and underreaction to new information.
Core Principle: Beliefs are probabilities that should update incrementally as evidence arrives. Strong priors require strong evidence to shift.
Decision flow:
Uncertain about something? → yes → Have prior belief? → yes → New evidence? → APPLY BAYESIAN UPDATE
↘ no → Establish base rate first
↘ no → Standard analysis may suffice
Your belief BEFORE seeing new evidence:
P(H) = probability that hypothesis H is true
Example: Before any symptoms, what's the probability someone has disease X?
Use base rate: If 1 in 1000 people have it, P(disease) = 0.001
How probable is the evidence IF the hypothesis is true?
P(E|H) = probability of seeing evidence E, given H is true
Example: If someone HAS the disease, what's the probability of a positive test?
If test is 99% sensitive: P(positive|disease) = 0.99
Your belief AFTER seeing the evidence:
P(H|E) = updated probability of H, given you observed E
This is what Bayes' Theorem calculates.
P(E|H) × P(H)
P(H|E) = ─────────────────────────
P(E)
Where:
P(H|E) = posterior (what we want)
P(E|H) = likelihood (how expected is evidence if H true)
P(H) = prior (initial belief)
P(E) = total probability of evidence
Posterior odds = Prior odds × Likelihood ratio
If evidence is 10x more likely under H than under not-H,
your odds should shift by factor of 10.
What did you believe before this evidence?
Question: Will this feature increase conversion?
Prior: Based on similar features, ~30% succeed significantly
P(success) = 0.30
How strong is this evidence? Consider:
Evidence: Early A/B test shows 5% lift (p=0.08)
P(this result | feature works) = 0.60 (moderately expected)
P(this result | feature doesn't work) = 0.15 (possible but less likely)
Likelihood ratio = 0.60 / 0.15 = 4x
Apply the likelihood ratio to your prior:
Prior odds: 0.30 / 0.70 = 0.43
Likelihood ratio: 4x
Posterior odds: 0.43 × 4 = 1.72
Posterior probability: 1.72 / (1 + 1.72) = 0.63
Updated belief: 63% confidence feature will succeed
(up from 30% prior)
Yesterday's posterior becomes today's prior:
New evidence: Week 2 shows lift holding at 4.5%
Prior (from step 3): 0.63
[Repeat update process]
New posterior: 0.78
Scenario: Test for rare disease (1 in 10,000 prevalence)
Test: 99% sensitive, 99% specific
Prior: P(disease) = 0.0001
If positive test:
P(positive|disease) = 0.99
P(positive|no disease) = 0.01
P(positive) = 0.99 × 0.0001 + 0.01 × 0.9999 ≈ 0.0101
Posterior: P(disease|positive) = (0.99 × 0.0001) / 0.0101 ≈ 0.0098
Even with 99% accurate test, positive result only means ~1% chance of disease!
Base rate dominates when condition is rare.
Bug report: Users see error X
Prior beliefs:
P(database issue) = 0.20
P(network issue) = 0.30
P(code bug) = 0.40
P(user error) = 0.10
Evidence: Error happens only on mobile
P(mobile-only | database) = 0.05
P(mobile-only | network) = 0.30
P(mobile-only | code bug) = 0.60
P(mobile-only | user error) = 0.40
Update: Code bug becomes most likely (posterior ~0.55)
Next step: Investigate mobile-specific code paths
Prior: Based on similar projects, P(on-time) = 0.40
Evidence 1: Team is experienced with this stack
Likelihood ratio: 1.5x → Posterior: 0.50
Evidence 2: Requirements are unclear
Likelihood ratio: 0.6x → Posterior: 0.38
Evidence 3: Critical dependency has risk
Likelihood ratio: 0.7x → Posterior: 0.30
Final estimate: 30% chance of on-time delivery
| Evidence Type | Typical Likelihood Ratio |
|---|---|
| Definitive proof | 100x+ |
| Strong evidence | 10-100x |
| Moderate evidence | 3-10x |
| Weak evidence | 1.5-3x |
| Noise | ~1x (no update) |
Update strongly when:
Update weakly when:
Common error: Ignoring prior probability when evidence arrives
Wrong: "Positive test = probably have disease"
Right: "Positive test shifts probability, but base rate matters"
Track predictions and outcomes:
| Stated Confidence | Should Mean |
|---|---|
| 50% | Coin flip |
| 70% | Would bet 2:1 |
| 90% | Would bet 9:1 |
| 99% | Would bet 99:1 |
"People tend to assess the relative importance of issues by the ease with which they are retrieved from memory—and this is largely determined by the extent of coverage in the media."
Don't let vivid evidence override base rates. A plane crash doesn't make flying more dangerous than driving, even though it feels that way.
Recognize Senge's Systems Archetypes to diagnose recurring organizational and technical problems, identify why fixes keep failing, and design interventions that address root structure.
Apply Herbert Simon's Bounded Rationality and satisficing to make good-enough decisions under real-world constraints. Use for design decisions under time pressure, recognizing cognitive limits, and setting appropriate stopping criteria.
Know the boundaries of your expertise and operate within them. Use when evaluating opportunities, making decisions outside your domain, or assessing when to defer to experts.
Classify problems by complexity domain (clear, complicated, complex, chaotic) and match approach to domain. Use for choosing methodologies, problem framing, and process design.
Systematic checklist to identify and counteract cognitive biases in decision-making. Use before major decisions, when evaluating recommendations, or when stakes are high.
Apply Kahneman's Dual-Process Theory to recognize when to trust intuition vs engage deliberate analysis. Use for high-stakes decisions, error-prone contexts, or when balancing speed vs accuracy.