Bayesian Updating
What is Bayesian Updating?
Bayesian updating is the rule for revising a probability after new evidence, weighting each possibility by how likely that evidence would be if it were true.
Bayes' rule says the revised probability is the prior probability times the likelihood of the evidence under that hypothesis, divided by the total probability of seeing that evidence at all. Two inputs therefore matter equally: how common the hypothesis was before you looked, and how diagnostic the evidence is. Economics treats the rule as the benchmark for rational learning, and it sits underneath models of signaling, screening, reputation and asymmetric information, where an employer or a lender revises a belief after observing a costly signal. Behavioral economics uses the same rule as a measuring stick for how people actually update: base rate neglect means ignoring the prior, conservatism means shifting too little toward the evidence, and confirmation bias means treating agreeable evidence as more diagnostic than it is. It describes real behavior poorly when likelihoods are hard to estimate, when the same evidence gets counted twice, or when someone holds a prior of zero, because no quantity of evidence can move a belief away from zero.
Bayesian Updating: a worked example
Suppose 1 percent of firms in a sector are committing fraud, an audit flags 90 percent of fraudulent firms, and it also flags 5 percent of honest ones. Work in percentage points of the whole population: the fraud branch contributes 1 x 0.90 = 0.9 points, and the honest branch contributes 99 x 0.05 = 4.95 points. Flags of either kind therefore cover 0.9 + 4.95 = 5.85 percent of all firms, so the probability of fraud given a flag is 0.9 / 5.85 = 0.154, roughly 15 percent. The audit is genuinely informative, since 15 percent is about fifteen times the prior of 1 percent, yet a flagged firm is still far more likely to be honest than fraudulent because honest firms outnumber fraudulent ones so heavily.
The mistake students make with bayesian updating
The standard error is answering 90 percent because the audit catches 90 percent of fraud. That confuses the probability of a flag given fraud with the probability of fraud given a flag, and the gap between them is set by the base rate: the rarer fraud is, the further the second falls below the first. The fix is to count groups instead of juggling percentages. Out of 10,000 firms, 100 are fraudulent and 90 of those get flagged, while 9,900 are honest and 9,900 x 0.05 = 495 of those get flagged, so 90 of the 585 flags are true and the rest are false alarms.
Bayesian Updating questions
What is the difference between a prior and a posterior probability?
The prior is what you believed before seeing the new evidence, often just the base rate in the population. The posterior is the revised belief once the evidence is taken into account. Bayes' rule is the arithmetic that converts one into the other, and today's posterior becomes tomorrow's prior when the next piece of evidence arrives.
If people do not update like Bayesians, why do economists use the rule?
It supplies a precise standard to measure behavior against, which is what turns a bias into something testable rather than a vague complaint. Saying that someone shows base rate neglect only means something if there is a correct answer to compare against, and Bayes' rule provides it. Standard models also assume Bayesian agents so that when a prediction fails, the belief assumption is visible and can be blamed explicitly.
Can two rational people see the same evidence and still disagree?
Yes, if they started from different priors, since Bayes' rule maps a prior plus evidence into a posterior and different priors give different posteriors. As independent evidence accumulates their beliefs usually converge, because the likelihood term comes to outweigh the starting point. The exception is a prior of exactly zero or one, which no amount of evidence can shift.
Formula / Example
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