Bayes' Theorem in Quant Interviews

Every quant loop tests Bayesian updating, because updating beliefs on new information is the job. The formula everyone knows; the interview skill is executing updates fast and not falling for the base-rate traps.

Use the odds form

For interviews, the multiplicative form beats the textbook one:

$$\text{posterior odds} = \text{prior odds} \times \text{likelihood ratio}$$

Disease testing classic: 1% prevalence, test 99% sensitive, 95% specific. Prior odds 1:99; likelihood ratio $0.99/0.05 \approx 20$; posterior odds 20:99 → probability $\approx 17\%$. One line, no table — and immune to the trap (most people guess 99%; the base rate dominates).

The recurring interview shapes

• False-positive traps — rare event, imperfect test. The answer is always "lower than intuition"; the discipline is quantifying it.
• Witness/corroboration problems — "the witness is 80% reliable; how much do you believe the claim?" Worked here, and its two-witness extension here.
• Identify-the-coin/die — draw from one of several dice/coins, observe outcomes, infer which. Sequential updating, one likelihood ratio per observation. Worked here.
• Trading-flavored updates — "an informed trader would buy 70% of the time; you saw a buy" is the same math wearing a market. This is the bridge interviewers hope you make: quotes ARE posteriors.

Traps to not fall into

• Confusing $P(A|B)$ with $P(B|A)$ — the prosecutor's fallacy; naming it earns points.
• Forgetting the base rate — a strong likelihood ratio on a tiny prior is still small.
• Double-counting correlated evidence — two witnesses who colluded are one witness; independence assumptions must be stated.

Practice with solutions

• Witness reliability
• Truth given corroboration
• Bayesian die identification
• Bayesian coin selection + sequential betting — updating fused with bet sizing.