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Use Bayes' formula to revise a probability

CFA® Level I · Quantitative Methods · Statistical Distributions for Financial Asset Prices and Returns · QM.4.c

The idea

Bayes' formula revises a probability when new information arrives: the updated (posterior) probability equals the likelihood of the evidence given the event, times the prior probability, divided by the total probability of the evidence. In words, it asks how much more consistent the evidence is with one state of the world than another, and moves the prior accordingly. Strong evidence moves a prior a long way; weak evidence barely moves it. Most reasoning errors here come from ignoring the prior altogether, which makes a rare event look common whenever a test for it is positive.

Formulas

Bayes' formula

P(A | B) = P(B | A) × P(A) ÷ P(B)

P(A)
the prior — what you believed before the evidence
P(B | A)
how likely the evidence is if A is true
P(B)
how likely the evidence is overall

Updating a belief with new information. The commonest error is ignoring the prior when it is small.

A practice question

From the question bank, with the reason each wrong option is wrong.

What does Bayes' formula do?

  1. Updates a prior probability in the light of new evidence — correct
  2. Calculates the variance of a distributionWhy not: Variance is a summary of a distribution you already have. Bayes' formula changes which distribution you hold.
  3. Converts a conditional probability into an unconditional oneWhy not: It converts a conditional probability into the *reverse* conditional — P(evidence given state) into P(state given evidence). Both remain conditional.

Bayes' formula combines a prior belief with the strength of new evidence to give a revised posterior probability. It is the formal version of changing your mind by the right amount.

A harder one

A screening test is positive. Why is the probability that the condition is present usually far below the test's accuracy?

  1. Because the test's accuracy is normally overstatedWhy not: Take the stated accuracy at face value and the result still holds. The gap comes from the base rate, not from a dishonest specification.
  2. Because false positives among a large healthy population can outnumber true positives — correct
  3. Because Bayes' formula does not apply to medical testingWhy not: It is the standard worked example of the formula. The arithmetic is exactly what produces the counterintuitive answer.

If the condition is rare, the small false-positive rate applies to a very large group and can swamp the true positives. Ignoring that base rate is the commonest error in probability reasoning.

The hardest tier

An analyst applies Bayes' formula but uses a prior of 50% for an event that historically occurs 5% of the time. What is the effect?

  1. None, since the evidence dominates the calculationWhy not: The posterior is proportional to the prior, so the prior never drops out. Evidence moves it; it does not replace it.
  2. The posterior will be too lowWhy not: The wrong direction. A prior ten times the base rate inflates the posterior, it does not depress it.
  3. The posterior will be far too high, because the prior is ten times the base rate — correct

The posterior is proportional to the prior, so starting from an inflated prior inflates the answer. Bayes' formula is only as good as the base rate fed into it.

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