Failure of sign recovery from magnitude-only calibration data

Failure of sign recovery from magnitude-only calibration data occurs when opposite signed calibration models induce exactly the same distribution of every recorded magnitude. Under this symmetry, more repetitions can estimate the magnitude distribution accurately without deciding the missing sign. This is the limitation relevant to sign-sensitive measurements that resolve a covariance-axis ambiguity.

Let $Z=sm+ϵ$ with sign $s\in \{-1,+1\}$, known $m>0$ and symmetric noise $ϵ$ having density $f(x)=f(-x)$. If the recorded value is $Y=|Z|$, then for $y>0$Symmetry gives $g(y\mid +1)=f(y-m)+f(y+m)=g(y\mid -1)$. Independent repetitions multiply identical densities, so their likelihood ratio remains one. With equal sign priors and equal decision costs, no decision rule based solely on these magnitudes can have average sign-error probability below1/2.

For normal distribution noise with standard deviation $\sigma$, the magnitude has a folded normal distribution:where $\varphi$ is the standard normal density. The formula depends on $|m|$, not its sign. The pinned numerical reference provides the equivalent standardized folded-normal density and its scale transformation.^[ref]

For the original calibration scale $m=0.20\mathrm{V}$ and $\sigma =0.050\mathrm{V}$, recording a signed value can distinguish the two models strongly, but recording only its magnitude cannot do so under the symmetric model. This is an information loss in the measurement map, rather than a shortage of averaging trials.

The symmetry assumption is essential. As an original counterexample, let noise equal−1 with probability2/3 and+2 with probability1/3, giving zero mean but an asymmetric distribution. With $m=1$, the positive-sign magnitudes are0 or3, while the negative-sign magnitudes are2 or1. Their disjoint supports permit sign recovery if this asymmetric noise model is independently established. The calibrated asymmetry itself supplies sign-reference information absent from the symmetric case.

Similarly, a genuinely signed projection or a known phase reference can break the original ambiguity. An unverified readout polarity or unknown amplitude sign can preserve it. Specify which signed information the apparatus actually records before interpreting a magnitude fit as an oriented calibration.