Adapt the experiment while keeping an honest budget

Let $h_{r-1}$ denote previous outcomes and controls, and choose setting $u_r$ from that history before the next measurement. The likelihood of the full record factors into conditional outcome probabilities at those actual settings. Bayesian updating multiplies the prior by each new conditional probability and renormalizes. A likelihood estimator uses the same record without requiring a prior distribution. In either method, changing controls during data collection is part of the statistical model and must be logged rather than averaged away as though the whole run used one fixed probability.

The control rule must not use the unknown true parameter as an oracle. It can use calibrated constants, a declared initial range, and previously observed data. If feedback is randomized, its known rule must also be represented consistently. Information from a complete adaptive experiment includes averages over possible histories, so summing only the most favorable realized settings can overstate expected performance. Model checks should include occasional incorrect branch choices and unusually misleading early outcomes, because rare large errors can dominate mean-squared error.

A probe experiencing k coherent applications of the same phase acquires $k\phi$, making its local phase information proportional to $k^2$ in the ideal unit-contrast model. The resulting fringe also repeats every $2\pi/k$. A coarse single-pass stage can distinguish intervals that a high-pass stage cannot. Additional readout phases resolve sine-versus-cosine sign ambiguities. A complete protocol allocates enough resources to both coarse identification and fine estimation rather than quoting the sharpest peak of a many-peaked likelihood.

The resource is not merely the number of detected particles. A four-pass particle interacts with the parameter four times and often needs roughly four times the coherent exposure of a single-pass particle. Long interrogation can similarly trade elapsed time for phase response. Count failed preparation attempts, discarded detections, pulse operations, and reference calibrations when those constrain the experiment. Different budgets can select different optimal protocols, so an advantage under a particle-detection budget should not silently become an advantage under a total-time or specimen-dose budget.

A constant field can be estimated accurately while rapid variations are averaged out. With a rectangular free-evolution window, accumulated phase depends on the time integral of the field, so a sinusoidal signal has a frequency-dependent response with sinc-like zeros. The complete cycle sets the interval between reported samples, including dead time. Uniform cycle sampling has a Nyquist limit, but that limit alone does not guarantee usable amplitude response below it. Filtering, feedback delay, and irregular adaptive sampling need their own temporal-response analysis.

For repeated independent cycles, sensitivity per square root observation time is a useful operating figure. State its signal frequency, duty cycle, contrast, probe number, and definition of uncertainty. Evaluate a proposed improvement using an optimized reference experiment with the same constraints, and retain accuracy contributions such as timing, coupling calibration, and drift. Adaptive control can keep the readout informative and resolve ambiguity; its value should appear in verified final error statistics, not only in a favorable local bound calculated after the true parameter is known.