And Statistics 2 — Probability
She introduced the : Var(Y) = E[Var(Y|X)] + Var(E[Y|X]) The fishermen scratched their heads. She explained: “The total uncertainty of your position comes from two things: the average internal chaos (the Drift’s random variance) plus the uncertainty in the Drift’s mean behavior.”
The Kalman filter, now robustified, predicted the Drift would reverse direction in 20 minutes. The fleet turned back. The mountain guild, still using their old periodic model, sailed into the surge. They survived, but their nets were shredded. That night, Elara addressed the city: probability and statistics 2
The Drift was a chaotic ocean current that changed speed randomly each hour, but its average behavior over a week was surprisingly predictable. The problem? The variance of the Drift’s speed wasn’t constant. Sometimes it was gentle (small variance), sometimes violent (large variance). The old methods failed. She introduced the : Var(Y) = E[Var(Y|X)] +
They ran a Gibbs sampler (a type of MCMC) overnight. By dawn, the chains had converged. The posterior distribution revealed that the Drift switched states every 3.2 days on average. Now they could build a real-time predictor. For the next hour’s Drift speed, they used a Kalman filter —a recursive algorithm that updates predictions as new data arrives. The mountain guild, still using their old periodic
