The experiment learns what to measure next from what it measured last.
Experimental design for linear dynamical systems with process and measurement noise (arXiv:2603.21370): the Fisher information matrix quantifies how informative a design is — which measurements, at which times, reveal the most about the unknown parameters. But the Fisher matrix depends on the unknown parameters. You need to know the answer to design the best experiment for finding it.
Two strategies combined: Bayesian design averages the Fisher information across a prior distribution (hedging against uncertainty), and adaptive design updates the information as measurements arrive (learning from accumulating data). The experiment starts with a Bayesian hedge, then adapts as each measurement sharpens the parameter estimate.
The Kalman filter enters as the natural computational engine. It already processes noisy measurements of dynamical systems, updating state estimates sequentially. The same recursive structure that estimates the state can be extended to estimate the Fisher information — tracking how much each new measurement contributes to parameter identifiability.
The structural insight: the best experiment is not a fixed protocol. It's a policy — a rule that selects the next measurement based on everything learned so far. Static designs optimize against expected uncertainty. Adaptive designs optimize against actual uncertainty, which decreases with each measurement. The gap between static and adaptive grows with the experiment's complexity — the more measurements you take, the more the adaptive design can exploit what it's learned. The first measurement is approximately Bayesian. The last measurement is approximately optimal. The path between is the learning.