Bayesian Updating: Change Your Mind by Degrees
Start from the base rate. Move in proportion to the evidence. Don't overreact.
Don't flip your beliefs all at once. Start from a base rate, weigh how well the new evidence fits, and land on a calibrated new belief instead of overreacting to the latest headline — the engine that turns priors plus evidence into a sharper estimate.
You already know the two ways people get it wrong. One person clings to their first belief no matter what evidence piles up — the world has clearly changed and they’re still defending last year’s opinion. The other flips completely at the first new headline, certain yesterday and equally certain of the opposite today. Both are mishandling the same thing: how much should a single piece of evidence move what you believe?
Bayesian updating is the precise answer. Named after the eighteenth-century minister Thomas Bayes, it’s the discipline of changing your mind by degrees — you start from a prior (how likely the thing was before the new evidence, which is usually just the base rate you met in the last course), you weigh the likelihood (how well the new evidence fits each possibility), and you combine them into a posterior (your sharper, updated belief). Strong, surprising evidence moves you a lot. Weak or ambiguous evidence moves you barely at all. Neither stubbornness nor whiplash — a proportionate update.
This course assumes you’ve done Thinking in Probabilities: you can put a number on a belief, you know a base rate is the outside view’s anchor, and you’ve seen the rare-disease test where a positive result still meant probably fine. Bayesian updating is the machinery under that result. We build it from the ground up: prior, likelihood, posterior in plain language; the canonical medical-test worked example done with real numbers and a natural-frequency tree (you’ll drive an interactive posterior calculator); Bayes’ theorem itself, in both its formula and its much friendlier odds form; likelihood ratios and the difference between the strength of evidence and the weight of your prior; and the rule that falls out of all of it — extraordinary claims require extraordinary evidence. By the end, “what’s my prior, and how much does this evidence really move it?” will be a reflex — the habit that separates calibrated updaters from headline-chasers, and the doorway to the expert tier of fat tails and calibration.
In this topic
- 1 Bayesian Updating: Change Your Mind by Degrees Two doctors see the same positive test. One panics, one shrugs — and the calm one is right, because a positive test on a rare disease is mostly a false alarm. Learning exactly how far evidence should move a belief is the whole course. 7 min
- 2 Prior, Likelihood, Posterior: The Three Ingredients Every Bayesian update is built from three pieces — what you believed before, how well the new evidence fits, and the sharper belief you get by combining them. Name them once and the whole machine clicks into place. 9 min
- 3 The Medical Test: Why a Positive Isn't a Verdict The canonical Bayesian puzzle, worked in full: a rare disease, a 99%-accurate test, a positive result — and a true chance of only about 9%. Build it with a natural-frequency tree and an interactive posterior calculator until the answer stops feeling like a trick. 11 min
- 4 Bayes' Theorem: The Equation and Its Friendlier Twin Lift the formula straight out of the natural-frequency tree, then meet the odds form — prior odds times a likelihood ratio equals posterior odds — that lets you update in your head without a calculator. 12 min
- 5 Strength of Evidence: The Likelihood Ratio How much should a clue move you? One number answers it: the likelihood ratio — how many times more likely the evidence is if your hypothesis is true than if it's false. Weak clues whisper, strong clues shout, and extraordinary claims need extraordinary evidence. 12 min
- 6 Where Updating Goes Wrong: The Five Failure Modes Bayes is simple; using it under pressure is not. The five ways smart people botch an update — dropping the prior, flipping the conditional, double-counting evidence, refusing to budge, and reading tea leaves — and the habit that fixes each. 11 min
- 7 Final Exam: Bayesian Updating A graded, one-way final exam on Bayesian updating — prior, likelihood and posterior; the medical-test paradox; Bayes' theorem and the odds form; likelihood ratios and strength of evidence; and the five ways updating goes wrong. Pass mark 70%. 22 min
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