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State Bayes' rule and use it to update probabilities.

FRM® Part I · Quantitative Analysis · Fundamentals of Probability · FRM1.QA.1.g

The idea

Bayes' rule reverses a conditional probability, so the chance of A given the evidence B is: Posterior = P(B | A) × P(A) ÷ P(B). Here P(A) is the prior, our belief before the evidence. P(B | A) is the likelihood of the evidence if A is true, and P(A | B) is the posterior. The denominator P(B) is the total probability of the evidence, summing likelihood times prior over every hypothesis: Evidence = P(B | A) × P(A) + P(B | not A) × P(not A). With several hypotheses, divide each likelihood-times-prior product by the total of all of them. When the prior is small, even an accurate default test produces a modest posterior, because most of its flags come from the large group of non-defaulters. The usual mistake is to treat the likelihood, such as the share of defaulters flagged, as the posterior. On the exam, write the prior, the two likelihoods and the total before dividing.

Formulas

Bayes' rule with the evidence expanded

P(A | B) = P(B | A) × P(A) ÷ [P(B | A) × P(A) + P(B | not A) × P(not A)]

P(A)
the prior probability of A
P(not A)
one minus the prior
P(B | A)
the likelihood of the evidence if A is true
P(B | not A)
the likelihood of the evidence if A is false

Use when the total probability of the evidence is not given. With a small prior the posterior stays modest even for an accurate test; do not read the likelihood as the posterior.

A practice question

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

Which of the following correctly states Bayes' rule for events A and B with P(B) > 0?

  1. P(A | B) = P(B | A) x P(B) / P(A)Why not: Swaps P(A) and P(B); the prior P(A) belongs in the numerator and the evidence P(B) in the denominator.
  2. P(A | B) = P(B | A) x P(A) / P(B) - correct
  3. P(A | B) = P(B | A)Why not: Treats the two conditional probabilities as equal; reversing a conditional requires the prior and the evidence terms.
  4. P(A | B) = P(A) x P(B) / P(B | A)Why not: Inverts the likelihood; P(B | A) must multiply the prior in the numerator.

Because P(A and B) = P(B | A) P(A) = P(A | B) P(B), dividing by P(B) gives P(A | B) = P(B | A) x P(A) / P(B).

A harder one

A model flags 90% of the firms that later default and also flags 10% of the firms that do not default. The unconditional default rate is 2%. A firm is flagged. What is the probability that it defaults?

  1. 15.5% - correct
  2. 90.0%Why not: Reports the likelihood P(flag | default) as if it were the posterior P(default | flag).
  3. 1.8%Why not: Reports only the numerator, the joint probability of default and flag, without dividing by P(flag).
  4. 18.4%Why not: Divides by the false-positive term only and omits the true-positive term from P(flag).

P(flag) = 0.02 x 0.90 + 0.98 x 0.10 = 0.116. P(default | flag) = 0.018 / 0.116 = 15.5%. Most flagged firms are non-defaulters because they are so numerous.

The hardest tier

A screening model flags 70% of firms that go on to default and 10% of firms that do not. The unconditional default rate is 5%. A firm is not flagged. What is the probability that it defaults?

  1. 1.72% - correct
  2. 30.0%Why not: Reports the miss rate P(not flagged | default) as if it were the posterior P(default | not flagged).
  3. 5.00%Why not: Reports the prior and assumes that the absence of a flag carries no information.
  4. 13.6%Why not: Uses the false-positive rate 10% where the true-negative rate 90% belongs in the not-flagged denominator.

Not flagged and default: 0.05 x 0.30 = 0.015. Not flagged and no default: 0.95 x 0.90 = 0.855. Posterior = 0.015 / 0.870 = 1.72%.

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