Five observations, a different accuracy limit.
The same observation budget need not provide the same guaranteed accuracy. In a model with shared bias and bounded noise, we prove that five adaptive observations can attain the optimal worst-case accuracy of all nine. The error ratio between the best fixed five-observation strategy and the adaptive strategy can grow without bound.
FBest fixed five
Square-root error order
AOptimal adaptive five
Linear error order
F and A denote optimal worst-case absolute recovery errors. The noise bound ε tends to zero within the proved parameter region, with the other model conditions fixed.
Decide where to measure before taking any observations. Or use the values already obtained to choose what to measure next. How much can this difference change the accuracy attainable from the same number of observations?
ChoiceProof studies this question in a mathematical model with bias shared across observations, calibration uncertainty, and bounded measurement noise. The state being observed is continuous. We evaluate the worst-case absolute recovery error: the error bound that holds for every state and error consistent with the assumed conditions.
Five observations reach the optimum of all nine.
For a model with nine observable entries, we construct a strategy that uses the first four readings to choose the fifth entry to observe.
Within an explicit continuous parameter region, these five observations attain the optimal worst-case error achievable by observing all nine entries. Five observations reach an accuracy limit that even full observation cannot improve.
Information that guides the next observation is already present in the readings obtained so far. Using that relationship makes it possible to achieve the same guaranteed accuracy as full observation with fewer acquisitions.
The same observation budget can change the rate at which error decreases.
The comparison is with the best fixed five-observation strategy: the most accurate of all strategies that choose five locations in advance.
As noise decreases while the model’s other conditions remain fixed, the optimal error for the fixed strategy decreases on the order of the square root of the noise level. The adaptive strategy’s error decreases linearly with the noise level.
Both errors approach zero, but at different rates. As a result, the fixed strategy’s error divided by the adaptive strategy’s error eventually exceeds any constant factor. We derive this separation from achievable upper bounds and lower bounds that rule out further improvement.
Establishing the limits of recovery accuracy.
For the rank-one model studied, we also characterize the optimal recovery accuracy attainable with full observation across the model’s entire parameter range.
By matching an attaining method with a lower bound that prevents improvement, we determine how accurately the state can be recovered from observations. Alongside the mathematical proofs, we carry out verification using independent checkers.
Within a limited observation budget, what should be acquired, and when can acquisition stop? ChoiceProof studies the mathematical foundations for designing these choices with guarantees. This research considers the value of observations through both their number and the way acquired information changes the next choice.