Bayesian Updating Questions in Trading Interviews

Trading interviews test Bayesian updating with fast puzzles rather than definitions, because pricing a market is updating a belief under a clock.

Bayesian updating questions in trading interviews test how quickly and correctly a candidate revises a probability once new information arrives, not whether the candidate can recite Bayes' theorem from memory. The two skills sound similar and are not, and the gap between them is exactly what separates a candidate who freezes on an urn problem from one who prices it in seconds.

The updated, or posterior, probability of a hypothesis after seeing some evidence is proportional to the probability of that evidence occurring under the hypothesis, multiplied by the probability the hypothesis was true before any evidence arrived, the prior. That product is then normalized, divided by the total probability of seeing the evidence at all across every hypothesis under consideration, so that the resulting probabilities across all hypotheses sum to one.

Written this way, without any fraction notation, the theorem is a statement about weighting: how much a piece of evidence should move a belief depends both on how strongly that evidence favors the hypothesis and on how likely the hypothesis was to begin with. Ignoring the second part is the single most common error candidates make under time pressure.

Consider a market maker who assigns a prior probability of 10 percent that an arriving order comes from an informed counterparty, someone trading on information the market maker does not have. Historically, the specific order pattern just observed appears 40 percent of the time when the counterparty is informed, and 8 percent of the time when the counterparty is uninformed.

The posterior probability that this counterparty is informed is calculated as follows. The numerator is the probability of the observation under the informed hypothesis, 0.40, multiplied by the prior probability of that hypothesis, 0.10, giving 0.04. The denominator is the total probability of the observation across both hypotheses: the same 0.04 from the informed case, plus the probability of the observation under the uninformed hypothesis, 0.08, multiplied by the prior probability of that hypothesis, 0.90, giving 0.072. Adding 0.04 and 0.072 gives 0.112. Dividing the numerator by that total, 0.04 divided by 0.112, gives approximately 0.357. The probability that this counterparty is…