Prior Probability Distributions

Bayesian estimation of the parameters in a model, on the basis of experimental data, requires [89,85,52] the explicit statement of prior probability distributions, representing the beliefs that are held about the parameters, before the experimental data are examined. The author has made some remarks about the importance of this requirement elsewhere [10]. The prior probability distributions for this problem are presented in tables 5.6 and 5.7.

Justifications for these prior probability distributions follow.

• For the acceptance probability $a$, a uniform distribution over the range from $0$ to $1$ seems sensible, given that we know very little about the properties of this system, other than that the energy loss window at the front of the polarimeter is (table A.1) held constant throughout the experiments.
• The author and collaborators performed (section 5.5) experiments, before those described here, which provided a rough estimate of the extent of the drift in the channeltron detection efficiencies. The magnitude of the drift was found to be $\sim{}10\%$ peak-to-peak, on a typical time-scale of the order of a few hours. The time-scale is represented in the division of the data set into chronologically contiguous subsets, which, by the design of the present experiment, coincide with single film thicknesses, and the magnitude in the standard deviation of the prior probability distributions over the time-dependent perturbations $p_{ij}$.
• The scattering probability $\Gamma$ at the thorium foil must be in the range between zero and one, leading to the top-hat form for the distribution over $\Gamma_0$.
• In the distributions over $\Gamma_k$, $k > 0$, the typical range of variation of the scattering probability over the energy scale of the experiments, due to any one term in the Taylor expansion, is asserted to be one.
• The Sherman function $S$ at the thorium foil must be in the range between $-1$ and one, leading to the top-hat form for the distribution over $S_0$.
• In the distributions over $S_k$, $k > 0$, the typical range of variation of the Sherman function over the energy scale of the experiments, due to any one term in the Taylor expansion, is asserted to be two.
• Because the front ends of the channeltrons are earthed, the electrons reach them with the same kinetic energy that they have at the sample. This means that the energies of the electrons, on arrival at the channeltrons, will range from $250\,\mathrm{eV}$ to $1500\,\mathrm{eV}$. The detection efficiencies of channeltrons of the type used, in this energy range, are [66] between $0.78$ and $0.89$. Therefore, a top-hat probability distribution, with these as its limits, is used for $\eta_i$. The author does not believe that it is worthwhile to create an energy-dependent model, since the uncertainty in the efficiency, at a particular energy, is [97,66] comparable with its variation with energy, over this range.
• It can be assumed that, in as much as the sample is believed to be bulk, the bulk material in question is, for $j > 0$, cobalt, or, for the clean sample $j = 0$, copper, because the meaning of ``bulk'' is ``topmost layer too thick for the electron beam to penetrate,'' i.e. of a thickness that is large compared with $1\,\mathrm{nm}$ [83,51]. Therefore, the electrostatic potential $V_j$ is expected to be the difference in expected work function between stainless steel, which is the material from which the front end of the electron gun, which sets the zero of potential for the electrons, is constructed, and copper for $j = 0$, or cobalt for other film thickness indices $j$. The width of the distribution of $V_j$ can be estimated by combining in quadrature the random errors $\Delta\Phi_l = 0.05\Phi_l$ [44] in measured work functions for the materials. The work functions are $\Phi_{SS} = 4.1\,\mathrm{V}$ for stainless steel [49], $\Phi_{Cu} = 4.65\,\mathrm{V}$ for copper [50], and $\Phi_{Co} = 5\,\mathrm{V}$ for cobalt [50].
• For the copper surface $j = 0$ in the main model, and for all film thickness indices $j$ in the null model, there are assumed to be no magnetic effects; the magnetic flux densities $B_j$ acting on the electrons, in these cases, are certainly zero.
• The magnetic flux density $B_j$ that affects the electrons is expected to be the same effective field, originating in the exchange interaction [31,32], which creates ferro-magnetism. For the samples with a cobalt surface, $j > 0$, in the main model, a prior expectation of zero will be used, with the width of the prior probability distribution, i.e. the typical size of the flux density, being given by the order-of-magnitude estimate $\frac{k_Bm_eT_c}{e\hbar}$ of the strength of this Weiss field that can [44] be obtained from the Curie temperature $T_C$. This is the only difference between the null and main models.

The parameters are taken, a priori, to be independent. Therefore, the prior probability density of the parameter vector

$$\mathbf{Q} = (Q_1,Q_2,\ldots{},Q_{47})\protect$$

$$= (a,p_{10},\ldots{},p_{17},p_{20},\ldots{},p_{27},\Gamma_0,\ldots{},\Gamma_5,S_0,\ldots{},S_5,\eta_1,\eta_2,\protect$$

$$V_0,\ldots{},V_7,B_0,\ldots{},B_7)$$ (5.33)

is

$$P(\mathbf{Q}\vert M_n) = \prod_{m=1}^{47}P(Q_m\vert M_n)\protect$$

$$= P(a\vert M_n)\left(\prod_{i=1}^2\prod_{j=0}^7P(p_{ij}\vert M_n)P(\eta_i\vert M_n)P(V_j\vert M_n)P(B_j\vert M_n)\right)\protect$$

$$\times\left(\prod_{k=0}^5P(\Gamma_k\vert M_n)P(S_k\vert M_n)\right)\textrm{.}$$ (5.34)

There are also prior probabilities of each model, with any set of parameter values. These are taken to be

$$P(M_N) = P(M_M)\protect$$

$$= \frac{1}{2}\textrm{.}$$ (5.35)