Posterior Probability

Bayes's theorem states that the posterior probability of a hypothesis given new evidence equals the prior probability multiplied by the likelihood of the evidence under that hypothesis, divided by the marginal probability of the evidence. In practice, forecasters rarely compute this formally; instead, they reason about how much a piece of news should shift their estimate. A strong, reliable signal should produce a large shift from Prior Probability toward the direction implied by the evidence. A weak or ambiguous signal warrants only a small update. Proper Bayesian updating is central to Calibration: forecasters who update correctly are neither too slow nor too fast, and their stated probabilities align with actual outcome frequencies.

In a Prediction Market, contract prices move continuously as traders incorporate new information, with each new equilibrium price representing a market-consensus posterior probability. When an unexpected event occurs, sharp traders update their models quickly and trade before slower participants do, driving the price toward the new posterior. Platforms like Polymarket and Kalshi show this process in real time: an unexpected poll result may shift a political contract from 45 cents to 60 cents within minutes. The Wisdom of the Crowd mechanism ensures that the aggregated posterior, reflected in market prices, often outperforms individual forecasters using the same evidence. Good Forecast Aggregation methods exploit this by weighting contributions from better-calibrated forecasters more heavily.