Thomas Bayes, an English Presbyterian minister, described in a 1763 posthumous paper the mathematical relationship between a prior probability, new evidence, and the resulting posterior probability. The formula is simple: P(H|E) = P(E|H) times P(H) divided by P(E). The prior probability of a hypothesis, multiplied by the likelihood of the observed evidence given that hypothesis, divided by the probability of the evidence overall, yields the updated probability. It is the mathematics of updating belief in proportion to evidence.¹

Humans are spectacularly bad at Bayesian updating. Kahneman and Tversky demonstrated in the 1970s that people systematically neglect base rates. If a disease affects 1 in 10,000 people and a test has a 5 percent false positive rate, most people (including most doctors) dramatically overestimate the probability that a positive test indicates actual disease. The correct answer, approximately 0.2 percent, is counterintuitive because it requires properly weighting the very low base rate against the apparently alarming positive test.

Computational systems have no such bias. A Bayesian updating algorithm applied to a prediction system starts with a prior probability derived from the constraint graph simulation or from historical base rates. When a new fact arrives, the system computes the likelihood of that fact under the hypothesis that the prediction is correct versus the hypothesis that it is incorrect. The posterior probability, the updated prediction confidence, follows from the mathematics.

In practice, the updating in Crystal Ball is implemented through the confidence trajectory system described in Article 15. Each new fact triggers a reassessment. If Kazatomprom announces production below target, the constraint graph's depletion node is updated, the simulation is re-run, and the prediction confidence adjusts. The adjustment is proportional to the significance of the new evidence and the strength of its connection to the prediction. A major production shortfall announcement shifts confidence by 5-10 points. A minor policy statement shifts it by 1-2 points. The accumulation of small updates, what Tetlock calls granular probability estimation, produces well-calibrated forecasts over time.²

The key insight is that calibration is not a static property. It is the result of disciplined updating. A system that updates frequently, in small increments, based on evidence rather than narrative, will converge on well-calibrated probabilities over time. A system that updates rarely, in large jumps, based on dramatic events, will oscillate between overconfidence and panic. The mathematics guarantees the former if the likelihood ratios are correctly computed. The architecture of the constraint graph and fact pipeline enforces this by linking every update to a specific, measurable input.