Bayes' Formula

The formula rearranges the definition of conditional probability. The joint probability of an event and the evidence can be written two ways: prior times the likelihood of the evidence given the event, or unconditional probability of the evidence times the posterior. Setting these equal and dividing gives the revision rule.

In practice the denominator does the heavy lifting, because the unconditional probability of the evidence is rarely handed to you directly. It is built with the total probability rule: sum, across every mutually exclusive and exhaustive state, the probability of that state multiplied by the probability of observing the evidence in it. So the posterior is one weighted branch divided by the sum of all branches that produce the same observation. That structure is why the posteriors across all states must add to one, and why a large posterior for one state necessarily shrinks the others.

Consider a screening model applied to a universe where two percent of firms default. The model flags ninety percent of the firms that eventually default, and also flags ten percent of the firms that do not. Among a thousand firms, roughly eighteen true defaulters are flagged and roughly ninety-eight healthy firms are flagged as well. A flagged firm therefore carries a posterior default probability near fifteen percent, not ninety percent.

The reason is scale: the non-event group is so much larger that even a modest false-positive rate generates more flags than the event group can. This is the core lesson of Bayesian revision for analysts. High sensitivity is worthless as evidence if the base rate is small and the false-positive rate is not correspondingly tiny. Evidence shifts a probability; it does not replace it.

Credit work is the clearest case: rating migration studies and default screens supply the likelihoods, while historical default frequencies by rating supply the prior. Equity analysts use the same logic when interpreting an earnings surprise, an insider purchase, or a short-interest spike, all of which occur in both good and bad outcome states. Performance evaluation is another application, where the question is the probability that a manager is genuinely skilled given a run of outperformance, knowing that unskilled managers also produce good runs by chance.

Macro signals work identically. Any indicator that precedes downturns but also fires without them demands the full calculation rather than a reading of the signal's hit rate alone. Bayesian updating is also the conceptual bridge to inference from sample evidence and to scenario-weighted valuation.

Draw a tree. The first split holds the states with their prior probabilities; the second split holds the evidence outcomes with their conditional likelihoods. Multiply along each path to get joint probabilities, then divide the path you care about by the total of all paths ending in the observed evidence. A two-by-two table of joint probabilities achieves the same result and is faster when the problem gives counts rather than percentages.

Be able to identify which conditional probability the question supplies and which it wants, since exam items frequently invert them. Be able to reach the unconditional probability of the evidence when it is not stated. And recognise that updating is sequential: today's posterior becomes tomorrow's prior when a second, independent piece of evidence arrives, and the order of updates does not change the final answer.