The Rosenblatt bijection

One diagram, computed live: laplace is a change of coordinates, and the coordinates it changes to are boring.

Every tick, laplace maps the arriving point through its own predictive cdf: z_t = Φ⁻¹(F_t(y_t)). That map is causal and invertible — a bijection on paths (Rosenblatt, 1952). Under a calibrated forecaster its image is iid N(0,1): whatever trends, cycles and volatility bursts live upstairs, downstairs is white noise in a fixed band. Forecasting downstairs is trivial — and the inverse map carries that trivial forecast back up into the rich, widening, correctly-shaped band upstairs. The diagram commutes:

upstairs: raw series y, with laplace's band and 8-step fan

downstairs: the same points through the bijection — z = Φ⁻¹(F_t(y_t))

Toggle regimes upstairs and watch downstairs not care: the dots stay in the same ±1.96 band, because the forecaster absorbs the structure into the map. The orange fan upstairs is the image of the flat fan downstairs under F_t⁻¹ — one trivial forecast, correctly reshaped. This is the JavaScript twin of the Python package running live, not a recording.

Why this matters: the map does the work

Because the bijection is exact, log-likelihoods convert between coordinate systems by an exact change of variables — so any external forecaster can be run downstairs and scored fairly upstairs. Measured on 30 FRED series (one-step, held out, rolling refits): every opponent improves on essentially every series, and each converges to laplace-plus-a-few-hundredths-of-a-nat — the map had already extracted nearly everything they know how to model.

opponent	raw	laplace-fronted	mean lift (nats/pt)	series won
ETS	1.64	3.68	+2.04	30/30
AutoARIMA	1.65	3.72	+2.07	30/30
GARCH(1,1)	1.73	3.70	+1.97	30/30
Prophet	1.64	3.70	+2.07	30/30
laplace alone		3.67

The same coordinates lift anomaly detectors even harder (DSPOT 5.2×, RRCF 1.8× on the UCR archive) — that story lives at timemachines, the decision layer built on these surprise streams.