The Evidence
Correlated Witnesses
You're on a jury. Two eyewitnesses independently place the suspect at the scene of the crime. That sounds convincing — two separate people, telling the same story. Your gut says guilty.
But before you cast your vote, ask a question nobody else in the courtroom is asking: how independent are those witnesses, really? Did they see the same news coverage? Talk to the same detective? Sit in the same waiting room before testifying?
The answer changes everything. And the math will show you exactly why.
The Collapse
Drag the Independence slider to 0%. Watch the second branch of the tree collapse. The third node shrinks back toward the second — the posterior barely moves.
When witnesses aren't independent — when they talked before testifying, or saw the same news coverage, or were interviewed by the same detective who inadvertently suggested details — the second testimony adds nothing new. It's the same information, dressed up as confirmation.
Now drag it back to 100%. The tree opens up. The posterior jumps. Two truly independent witnesses are genuinely powerful. The difference between 0% and 100% independence isn't a detail — it's the entire case.
The Correlation Penalty
This is the hidden variable that changes everything. Two correlated witnesses are just one data point wearing a disguise. The courtroom counts two voices, but the math counts one piece of evidence.
Try adjusting the Witness Reliability slider. A more reliable witness produces a stronger likelihood ratio — the branch of the tree thickens. But notice: even with 95% reliability, correlation between the two witnesses can erase almost all the gain from having a second one.
The same pattern appears everywhere. Polls that sample the same demographics. Studies that cite the same source data. News articles that quote each other. The number of voices matters far less than their independence.
The Equation
Toggle the equation overlay above. The exponent on the second likelihood ratio is the independence parameter. At 100%, it's full strength — the second witness carries the same evidential weight as the first. At 0%, it collapses to 1 — no update at all.
The sigma function wraps everything in a logistic curve, keeping the posterior probability between 0 and 1 no matter how extreme the evidence. The log-odds form is the natural language of Bayesian updating — evidence adds, it doesn't multiply.
Drag each slider and watch its corresponding term light up. The Prior Guilt shifts the starting point. The Witness Reliability determines how loud each piece of evidence is. And Independence decides whether the second voice is truly new information — or just an echo.
The Pattern
Six chapters. Six disguises. One equation. The base rate fooled the doctor. Noise fooled the astronomer. Aggregation tamed the forecaster. Anchoring trapped the updater. Fat tails humbled the trader. And correlation deceived the jury.
In the final chapter, you'll watch all six move together — and see that the pattern was always the same.