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Bayesian Theory

Bayesian Updating

Financial Crime

Epistemic Capture

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WHAT TO DO

EXACT CELL

WHAT TO WATCH FOR

CONCEPT

🌀 Core Concept

Don't change anything yet. Just read the two posterior values in column E. Which hypothesis has a higher posterior after seeing one heads flip?

Core Concept, col E (rows 11-12)

The biased coin should win — even though priors were equal. Why?

Prior × Likelihood = Joint

Change the prior for 'Coin is FAIR' (C7) to 0.90. You now strongly believe the coin is fair before seeing any evidence.

Core Concept, C7 change to: 0.90

Does the fair coin still lose after one heads? At what prior does it start winning?

Strong prior resists evidence

Set both priors back to 0.50. Now change the likelihood for 'Coin is BIASED' (D8) to 0.55 (only slightly biased). Compare posteriors.

Core Concept, D8 change to: 0.55

The posterior gap should narrow dramatically. Evidence quality matters as much as prior.

Likelihood strength

Set D8 back to 0.90. Now set it to 0.51 — barely biased. At what point does the evidence become too weak to shift your belief?

Core Concept, D8 try: 0.51

When P(E|biased) ≈ P(E|fair), the posterior ≈ the prior. Evidence is uninformative.

Uninformative evidence

🏥 Doctor Diagnosis

Look at the current 'most likely diagnosis' cell at the bottom. It should say Flu. Now read the prior column (C). Common Cold has the highest prior.

Doctor Diagnosis — just read the output cell

Flu wins despite lower prior because its likelihood of causing fever is much higher.

Evidence overrides prior

Set the Flu prior (C7) to 0.05 — you're in a region where flu is rare. Does the diagnosis change? What wins now?

Doctor Diagnosis, C7 change to: 0.05

Even strong evidence can be overcome by a very low prior. This is base rate neglect in reverse.

Base rate matters

Set all four likelihoods (D column) to the same value — try 0.50 for all. What do the posteriors equal now?

Doctor Diagnosis, D7:D10 all to: 0.50

Posteriors = priors exactly. When all likelihoods are equal, evidence has zero power.

Likelihood equality = null

Reset likelihoods to originals. Now set Bacterial Infection likelihood (D9) to 0.99. Does it become the top diagnosis even with a lower prior?

Doctor Diagnosis, D9 change to: 0.99

Very strong evidence for one hypothesis can overcome a weaker starting position.

Extreme evidence can flip rank

🎲 Sequential Updating

Look at the chart. Notice how belief in 'fair coin' starts at 0.50 and moves with each flip. Find the flip where belief drops most sharply.

Sequential Updating — read the chart

A single tails flip (0) should cause a visible drop. Multiple heads in a row push it down gradually.

Sequential updating

Change all 10 observations (row 9, cols C-L) to 1 (all heads). How many flips before you're >90% certain the coin is biased?

Sequential Updating, row 9 all cells to: 1

With strong evidence repeating, certainty accumulates faster than you might expect.

Evidence accumulation

Set the starting prior (C5) to 0.95 — you're almost certain the coin is fair. Keep all observations as 1. How many heads does it take to change your mind?

Sequential Updating, C5 change to: 0.95

A strong prior is hard to shift. This is why preconceptions survive weak contradicting evidence.

Prior resistance

Set starting prior back to 0.50. Set observations to: 1,0,1,0,1,0,1,0,1,0 (alternating). What happens to belief over time?

Sequential Updating, row 9 alternate 1s and 0s

Belief oscillates but stays near 0.50. Mixed evidence keeps you uncertain — as it should.

Uncertainty under mixed evidence

⛅ Free Energy Preview

Read the Surprise column (D) in Part A. Find the event with highest surprise. Notice what probability produces a surprise of exactly 0.

Free Energy Preview, Part A just read col D

Surprise = 0 only when P = 1.0 (certainty). Everything uncertain = some surprise.

Surprise = -ln(P)

In Part B, set Q values (col D) to match P values (col C) exactly. What does the total KL divergence become?

Free Energy Preview, Part B set D = C for all rows

KL divergence = 0 when your beliefs perfectly match reality. That's the goal of inference.

KL = 0 means perfect belief

Now set Q for Outcome A (D) to 0.99 while P stays at 0.70. You're very confident about something that's only 70% true.

Free Energy Preview, D row A change to: 0.99

KL divergence spikes. Overconfidence is expensive — it means your model is far from reality.

Overconfidence costs free energy

🧾 Single Suspect

🏢 Multi-Suspect Lineup

🕸️ Crime Type Web

📈 Evidence Value

📊 Self-Fulfilling Prior

🎚️ Precision Manipulation

🔄 Niche Construction

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