The Same 95% Interval Covers the Truth 95.3% of the Time in One World and 0.0% in Another, on Byte-Identical Data
python
dev.to
Every column in every dataset is a recording, and the recording is not the thing. Write w = x + u: you fit on w, the world runs on x. The textbook consequence is attenuation — the slope comes back multiplied by the reliability λ — and the textbook reaction is that it is conservative, so never mind. Fix the observed law instead. Standardise Var(w) = 1, write c = Cov(w, y) and v = Var(y), and a world is then one number: β = c / λ σ²ᵤ = 1 - λ σ²ε = v - c²/λ = 0 λ >