The Update

You have a hunch — a 50/50 feeling about whether a new restaurant will still be open when you arrive. No strong reason to believe either way. Then a friend texts: “I drove past it twenty minutes ago. Looked packed.”

How much should that move your belief? Double it? Triple it? Your gut has an answer, but your gut doesn't show its work. The machinery of belief update is invisible to introspection — and that's exactly why it goes wrong so often.

Look at the interactive panel. You're starting at 50%. The colored bar is your belief, and the ghost behind it is where you started. Every slider change rewrites the bar in real time.

The Immovable Prior

Drag the Prior Belief slider down to 1%. Now set Evidence Strength to 10 — that's overwhelming evidence, ten times more likely under the hypothesis than against it.

Watch the bar. It barely moves. A 1% prior absorbs a 10× likelihood ratio and lands somewhere around 9%. You threw your strongest evidence at near-certainty in the other direction, and the needle hardly budged.

This is why extraordinary claims require extraordinary evidence. When your starting belief is close to zero, even powerful data can't drag it far. The prior is an anchor, and the evidence is a rope — not a catapult.

The Ratchet

Set the prior back to 50% and leave evidence strength at 3×. Now slowly drag Evidence Pieces from 1 up to 10.

Watch the posterior climb. Each piece of evidence pushes the log-odds by the same fixed amount — it's additive. Like a ratchet: each click advances the mechanism by an identical step, regardless of where it already sits. Three pieces at 3× have the same log-odds impact as one piece at 27×.

This is the deep insight of the log-odds form. On a probability scale, each update feels different — the jump from 50% to 75% looks bigger than 90% to 97%. But in log-odds space, both are the same sized step. The ratchet doesn't care where it started.

The Equation

Toggle the equation overlay. The log-odds form makes belief update beautifully simple: take your prior in log-odds, add n times the log of the likelihood ratio, and convert back. Addition, not multiplication. Linearity, not cascading fractions.

Drag each slider and watch its corresponding term light up. The Prior sets the starting point. The Likelihood Ratio determines how much each piece of evidence is worth. The Evidence Pieces multiply that contribution — independent observations stack linearly.

This is Bayes' theorem in its most operational form — not a fraction to memorize, but a machine you feed evidence into one piece at a time.

When the Noise Isn't Random

This is the same equation wearing a new disguise. In the previous chapters, you saw what happens when noise is statistical — false positives, detection thresholds, overlapping distributions. The update rule absorbed all of that cleanly.

But what happens when the noise isn't random — when it's financial? When the evidence is a stock price, the likelihood ratio is a market signal, and the prior is the collective belief of every trader on the floor? The math still works. But the stakes change.

In the next chapter, you'll feed real price data into this same update machinery — and discover why even perfect Bayesian reasoning can't save you from a market that's already priced in the signal.