Why Use Any Statistical Inference?

The community of researchers using measurements of the spin polarization of electron beams has long had available the estimator in equation 5.16, for obtaining the spin polarization of an electron beam, from the measured electron arrival rates at a Mott polarimeter's two detectors; this estimator eliminates systematic errors due to misalignment of the electron beam entering the polarimeter, and due to unequal sensitivities of the two detectors. By a happy coincidence, the systematic error that may be introduced by a stray field, which reverses with the sample magnetization, is also multiplicative, and is, therefore, also eliminated by this polarization estimator.

It was established, following a published derivation [5], in section 3.3.1 that, in the absence of random errors, the estimator is equal to the true polarization. Why, therefore, cannot the estimator, along with published values of the Sherman function $S$, be used to calculate the reflected beam polarization, and then equations 2.8 and 2.10 inverted to give the electro-magnetic characteristics of the sample, avoiding any need for a statistical process?

A footnote to section 3.3 alludes to the answer, which lies in the seemingly innocuous phrase ``in the absence of random errors.'' Real experiments are characterized by the presence, not the absence, of random errors, in particular, in this case, in the measured electron arrival rates. When there are random errors, the electron arrival rate $f_{i,j}$, for given true values of the intensity and polarization of the beam entering the polarimeter, $G_{\textrm{true}}$ and $P_{\textrm{true}}$, is characterized^5.5 by a probability density distribution $P(f_{i,j}\vert G_{\textrm{true}},P_{\textrm{true}})$. On the four-dimensional space of the $f_{i,j}$, one can define both the joint probability density $\prod_{i,j}P(f_{i,j}\vert G_{\textrm{true}},P_{\textrm{true}})$, and the polarization estimator $P_{\textrm{formula}}(\mathbf{f}) = \frac{\sqrt{f_{1,+}f_{2,-}}-\sqrt{f_{1,-}f_{2,+}}}{\sqrt{f_{1,+}f_{2,-}}+\sqrt{f_{1,-}f_{2,+}}}$. One can then, in principle, obtain the expectation value $\int_{\textrm{All space}}P_{\textrm{formula}}(\mathbf{f})\prod_{i,j}P(f_{i,j}\vert G_{\textrm{true}},P_{\textrm{true}})\mathrm{d}f_{i,j}$ of the estimator, given $G_{\textrm{true}}$ and $P_{\textrm{true}}$. If this expectation value is equal to $P_{\textrm{true}}$, the estimator is unbiased. If, on the other hand, the expectation value is not equal to $P_{\textrm{true}}$, it is a biased estimator; in other words, despite its success in eliminating systematic errors due to certain physical conditions, the estimator has created a new systematic error of its own, out of the random errors in the electron arrival rate measurements; this is the systematic error which, it was suggested above, may have been responsible for the non-zero polarization estimates obtained from bare copper.

Unfortunately, because the estimator is non-linear in the raw measurements, it is very likely that its expectation value will not be equal to $P_{\textrm{true}}$, and that such an artificial systematic error will arise. Worse still, evaluating the integral, to quantify the size of this systematic error (and possibly correct it,) is beyond the author's analytical capabilities; it could be integrated numerically, but this would sacrifice the advantage of mathematical simplicity, which was the motivation for examining this estimator. This is what justifies turning to statistical inference.