Bayesian updating
A Socratic walk-through of Bayesian updating — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #How should new evidence change the probability we assign to a belief?
A test for a disease is right 99% of the time and yours comes back positive. What is the chance you have the disease? Almost everyone answers "99%". Notice what that answer quietly does: it reports how good the test is and calls it a fact about you. Are those the same question?
Reasoning it through
REASONING #Ask what evidence can and cannot do. It never tells you how likely a claim is on its own; it can only say how much more likely one explanation has become than its rivals. So updating needs two ingredients: how surprising the evidence would be if the belief were true, and how surprising if it were false.
Make it concrete. Imagine ten thousand people, of whom 1% — a hundred — have the disease. The test catches 99 of them. Among the 9,900 healthy, a test with 95% specificity wrongly flags 5%, which is 495. Count the positives: 99 true and 495 false, 594 in all. How many are genuine? Ninety-nine out of 594 — about one in six, or 17%.
Sit with how far that is from 99%. Nothing was wrong with the test. What the intuition ignored was the size of the two pools the evidence had to sort: a rare condition offers few true positives, while even a small error rate harvests many false ones from a vast healthy pool.
Now generalise that arithmetic. Bayes' rule is cleanest in odds: prior odds multiplied by the likelihood ratio — how much more often the evidence appears when the belief is true than when false — give the posterior odds. Here the ratio is 0.99 divided by 0.05, near 20; prior odds of 1 to 99, multiplied by 20, give roughly 1 to 5. One in six again. So the evidence contributes one fixed multiplier whatever you believed beforehand, and the prior decides where it lands.
The analogy
THE ANALOGY #Think of a belief as scales already loaded before the evidence arrives, the prior sitting in one pan. Evidence does not empty the pans and start again; it multiplies what is in each pan by a fixed factor. A strong result is a large factor — but a large factor applied to a nearly empty pan still leaves it light.
Scales have two pans, while many real questions have several rivals at once; and a scale has no memory, whereas a posterior becomes the next prior, so yesterday's conclusion is the load already in the pan today.
Clarifying the model
THE MODEL #Two corrections. A prior is not a guess to be embarrassed about — ignoring it does not make you neutral, it silently sets it to something arbitrary, which is exactly the error in the 99% answer. And updating is only as honest as the likelihoods: if you cannot say how probable the evidence would be were you wrong, you cannot update at all. Evidence that fits your belief but fits every rival equally well has a ratio of 1, and should move you not an inch.
A picture of it
THE PICTURE #How to readThe horizontal axis is what you believed before the test, the vertical what you should believe after. Every point comes from the same positive result — the test never changes, only the prior does. At the far left, a prior of one in a thousand leaves you near 2%, still more likely healthy than not; at the right, the identical result carries you above 95%. The steep middle is where evidence does its real work.
What became clearer
WHAT CLEARED #Evidence does not deliver a probability; it supplies a multiplier. What you should believe afterwards depends jointly on how diagnostic the evidence is and how plausible the claim was beforehand — and neglecting the second half is how a 99% test yields a 17% conclusion.
Where to go next
ONWARD #- How likelihood ratios from several tests combine, and why independence is fragile.
- What happens when a prior is set to zero, and why that belief can never be updated.
- Why a p-value answers a different question and cannot be read as this posterior.
Key terms
TERMS #| Term | What it means |
|---|---|
| Prior | the probability assigned to a belief before the new evidence is considered. |
| Posterior | the revised probability after the evidence, which becomes the prior for the next update. |
| Likelihood ratio | how much more probable the evidence is when the belief is true than when false; the multiplier the evidence supplies. |
Every term the collection defines is gathered in the glossary.