Bayesian spectroscopy of synthesized soft X-ray absorption spectra showing magnetic circular dichroism at the Ni-L3, -L2 edges

ABSTRACT We proposed a data-driven science study by Bayesian spectroscopy to analyze an X-ray magnetic circular dichroism (XMCD) spectrum and magnetic moments. In Bayesian spectroscopy, model selection becomes available to estimate the number of spectral components by using Bayes free energy. To demonstrate such advantage, we decomposed synthesized X-ray absorption (XA) and XMCD spectra at the -absorption edge of the Ni ion in NiFe O . In the synthesized helicity XA spectra, random noise was superimposed and a finite spectral width was convolved to mimic measured spectra. From these XA spectra, spectral components having intensity beyond the noise level were successfully extracted without excess or deficiency although transition components in close proximity within the spectral width were merged. From the XMCD spectrum, we also succeeded in extracting separately the original helicity components although an additional helicity component comparing with the case of XA spectra was extracted in the model selection. This additional component was extracted to explain an asymmetric XMCD spectral structure at the edge, and this result demonstrates that Bayesian spectroscopy can fully exploit the advantage of XMCD measurements being superior for detecting close transition components of opposite helicities. In addition, we proposed to use posterior probability distributions obtained by Bayesian spectroscopy for estimating magnetic moments through samplings of the spectral intensities for the separately decomposed helicity components on the -absorption edge. GRAPHICAL ABSTRACT


Introduction
Bayesian spectroscopy [1,2] is a data-driven science approach of that we apply Bayesian inference [3] onto spectral analysis to achieve highly challenging spectral decomposition. In the Bayesian inference of the datadriven science [4], Bayes' theorem [3] is applied to the joint probability of causes and results in the causality, and evaluation of posterior probability distributions for causes becomes available based on the resultant data. Such evaluation is one of major advantages in Bayesian spectroscopy and it makes possible to provide a statistical guarantee for a particularly-difficult spectral decomposition. Such advantage has been illustrated by the Bayesian spectroscopy of admixed photoluminescence spectra with exciton, biexciton and electron-hole droplet (EHD) states in a highly-excited GaAs/AlAs type-II superlattice [5], where we have demonstrated that the EHD state becomes stable from the evaluation of the posterior probability distributions for a chemical potential of the EHD, an energy of excitonic system and their effective temperature [6].
Another important advantage of Bayesian spectroscopy is the ability to select an optimal model for explaining the data without preconceptions by using Bayes free energy [1] (BFE) as an information criterion. This model selection is a key data-driven methodology in analyses for various measured data, and Bayesian Hamiltonian selection is also available to explain physical phenomena [7]. This model selection also works properly on decomposing complicated spectra [2,[8][9][10][11][12]. Applying Bayesian spectroscopy, we have succeeded in decomposing weak pre-edge structures from an X-ray absorption near edge structure (XANES) spectrum of α-Fe 2 O 3 [13].
In this study, we apply Bayesian spectroscopy to decompose X-ray absorption (XA) spectra and an X-ray magnetic circular dichroism [14][15][16] (XMCD) spectrum, which were synthesized to imitate the L 3;2absorption edges of Ni ions in NiFe 2 O 4 [17] under magnetic fields with � helicities. This material belongs to the class of strongly correlated 3d system under an octahedral crystal field, and so that, versatile split X-ray transitions are expected in the respective � helicity polarized XA spectra. In addition, at the L 3;2 edges of such magnetic materials having d-electrons, it is possible to obtain the spin, orbital and total magnetic moments from the spectral integration of the XMCD spectrum based on the sum rule [18].
To sensitively detect changes on opposite helicities, we have always obtained the XMCD spectrum as a difference spectrum of � helicity XA spectra by eliminating non-magnetic spectral components. From differential spectrum by modulation technique, one can detect spectral changes with high sensitivity [19]. However, in the case of the spectrum obtained by subtraction (difference spectrum), the data point number becomes half of that in the original spectra ( � helicity XA spectra), and the noise variance included in the difference spectrum becomes the sum of the noise variances superimposed on these original spectra. Therefore, it is concerned that information missing might occur on extracting physical properties for the target material.
In order to overcome such difficulties in difference spectral analyses, we apply the Bayesian spectroscopy with the BFE-based model selection [1,20] to decompose the original � helicity spectral components from an XMCD spectrum, and we demonstrate the advantage of the Bayesian spectroscopy comparing the decomposition results between the XMCD and individual � helicity XA spectra. In addition, we evaluate the estimation accuracies of the spin, orbital and total magnetic moments obtained by the decomposed spectral components based on the posterior probability distributions of the decomposed spectral components.

Synthesized spectra
In this section, we detail the synthesized spectra to analyze by Bayesian spectroscopy. The target spectra was prepared to imitate an XMCD spectrum of the L 2;3 -absorption edges of Ni 2þ ions in NiFe 2 O 4 [17,21].

Effective Hamiltonian
The XMCD spectrum was calculated as a difference spectra of � helicity XA spectra, while the XA spectra were synthesized on the basis of the X-ray transitions at the L 2;3 edges among electronic states derived from an effective Hamiltonian in Equation (1) [22].
Ni 2þ in NiFe 2 O 4 is located in the O h crystal field where oxygen atoms are octahedrally coordinated, and the number of d-electrons is d 8 . In this effective Hamiltonian, we considered X-ray transitions from d 8 where d nþλ L λ means charge transfer of λ electrons from the p-orbitals in the ligand (L) oxygen to d-orbitals in Ni [22]. In Equation (1), the first, second, and third terms are electron energies of the Ni-3d, Ni-2p and the oxygen 2p orbitals, respectively; the fourth term is a hybridization energy between the Ni-3d and oxygen 2p-orbitals; the fifth term is a Coulomb repulsion energy among the Ni-3d orbital electrons; and the sixth term is a Coulomb binding energy between holes in Ni-2p orbital and electrons in Ni-3d orbital; and the last term H multiplet means effects of crystal field and spin-orbit interaction. By using the physical quantities in a previous study for NiFe 2 O 4 [17], 23 and 31 transition components are obtained for � helicities, respectively. Vertical lines in Figure 1(a,b) denote these transition components for the respective helicities, where ordinates are in logarithmic scale and the ordinate for þ helicity is inverted upside down.

Synthesized spectra
Synthesized spectra were prepared on the basis of the transition components in Figure 1(a,b) with spectral convolution of Lorentz shapes Lðx; Θ j Þ to mimic measured XA spectra. The Lðx; Θ j Þ is expressed in Equation (2) as a function of energy x and is characterized by a parameter set Θ j ( :¼ I j ; E j ; Γ 0 � � ) with integral intensity I j , transition energy E j and fullwidth at half maximum Γ 0 .
where F � ðx i ; fΘ � j gÞ are the true XA spectra for � helicities and they consist of J À ¼ 23 and J þ ¼ 31 transition components, respectively, as described in Section 2.1. Energies x i were prepared from À 30 to þ 30 eV with N ¼ 601 (i ¼ 1; � � � ; N) data points in an arithmetic series, and all Lorentz shapes were given a common 1.00 eV spectral width (Γ 0 ). The respective blue and red lines in Figure 1(a,b) are the true spectra F � ðx i ; fΘ � j gÞ for � helicity XA spectra. N i ð0; σ XA noise Þ in Equation (3) means random noises in a normal distribution with mean and standard deviation being 0 and σ XA noise , respectively. The gray areas in Figure 1(a,b) denote the � helicity XA spectra D � to analyze, and in D � , normal noises with σ XA noise ¼ 5:00 � 10 À 4 in absorption intensity scale were superimposed independently. As a result, the standard deviation of noises superimposed on the XMCD spectrum becomes ffi ffi ffi 2 p σ XA noise and is larger than that of the � helicity XA spectra, since the XMCD spectrum was obtained by Equation (4). The synthesized spectra D À , D þ and D XMCD are shown by gray spectra in Figure 1(c-e), respectively, where the ordinate in Figure 1(d) is inverted for the þ helicity XA spectrum.

Bayesian spectroscopy
In this section, we outline the formulation of Bayesian spectroscopy and a replica exchange Monte Carlo method [23] used for sampling.

Bayesian spectroscopy
be a spectral dataset to analyze and f K ðx i ; ΘÞ be a phenomenological model for describing D, then a posterior probability distribution PðΘjD; K; bÞ for the parameter set Θ in the model f K ðx i ; ΘÞ can be expand as Equation (6) where K is a model identifier and b is quasi-inverse temperature [20] defined as b :¼ σ À 2 noise with the standard deviation σ noise of the superimposed noises in y i . PðΘjK; bÞ is a prior probability for Θ, and PðDjΘ; K; bÞ is a conditional probability of D under the condition Θ given, and the denominator PðDjK; bÞ is a normalization factor.
When the noises in D are distributed in a normal distribution with the inverse variance b, PðDjΘ; K; bÞ in Equation (6) can be written as Equation (7).
where E K ðΘÞ is an error function of f K ðx i ; ΘÞ for D as follows: The denominator PðDjK; bÞ in Equation (6) [20].   (1). Their intensity is displayed in logarithmic scales and þ helicity components are shown upside down. Blue and red curves indicate the true XA spectra for � helicity without normal noises. Gray spectra are � helicity XA spectra (D � ) to analyze, and dashed lines mean the standard deviation of superimposed normal noises in the synthesized XA spectra. Gray spectra in (c), (d) and (e) are � helicity XA spectra (D � ) and an XMCD spectrum (D XMCD ) to analyze. The solid curves indicate reproduced spectra obtained by Bayesian spectroscopy.
ðK;bÞ ¼ argmin FðK; bÞ; (8) where K is the model identifier for selected one, and a standard deviation σ noise of the normal noises in D is estimated by σ noise :¼b À 1=2 . Furthermore, the posterior probability of each candidate model can be evaluated. Although PðKjDÞ is expanded as PðKjDÞ ¼ PðDjKÞPðKÞ=PðDÞ on the basis of the Bayes' theorem [3], in unbiased model selection, PðKjDÞ is simply proportional to PðDjKÞ because the prior probabilities PðKÞ of all candidate models should be the same. Since the conditional probability PðDjKÞ can be evaluated by marginalization of PðDjK; bÞ with b, the PðKjDÞ is obtained as follows: Finally, the posterior probability distribution of the parameter set Θ in the selected model K can be sampled with the estimated quasi-inverse temperature b by the following equation: PðΘjD;K;bÞ / exp ÀbNEKðΘÞ h i PðΘjK;bÞ: (10)

Replica exchange Monte Carlo method
To perform model selection [1,20], a replica exchange Monte Carlo (RXMC) method is necessary to sample the parameter space (Θ) on multiple replicas with different quasi-inverse temperatures b. In the RXMC method, a state of Θ is updated by a Metropolis method [24][25][26] in each replica, and simultaneously the state Θ is exchanged stochastically between neighbor replicas. Although many replicas are prepared for a wide range of b, convexo-concave structures of the error function E K ðΘÞ are suppressed on the replicas with small b since b is multiplied on E K ðΘÞ in the exponential function of Equation (7). As a result, rapid search in the wide parameter space is realized on the replica having small b, and through state exchanges, release of the state Θ from local minima of E K ðΘÞ is also realized. Consequently, rapid and effective samplings become possible nearby the global optimal solution by the RXMC. We prepared commonly eighty replicas (L :¼ 80) for both analyses of the � helicity XA and XMCD spectra. Although the smallest b being b 1 ¼ 0, other replicas were prepared with geometric sequences on b , (, ¼ 2; � � � ; L). Although this range of b , should be adjusted to include the inverse variance of the normal noises in D, we set b 2 ¼ 7:2 � 10 3 and b L ¼ 7:2 � 10 6 for the XA spectra and b 2 ¼ 2:0 � 10 4 and b L ¼ 2:0 � 10 7 for the XMCD spectrum, respectively. The corresponding inverse-variances of the superimposed noises are 4:0 � 10 6 [ ¼ ðσ XA noise Þ À 2 ] for the � helicity XA spectra and 2:0 � 10 6 [ ¼ ð ffi ffi ffi 2 p σ XA noise Þ À 2 ] for the XMCD spectrum, and they are included in the respective ranges of b , (, ¼ 2; � � � ; L). In both cases, the ratio b ,þ1 =b , between neighbor replicas is 1.0926, and since it is close to unity, rapid and wide-range searching on the replicas having small b will be promoted through highly efficient state exchange between neighbor replicas. The RXMC samplings were performed at 200,000 times after sufficient burn-in phase of 200,000 times.

Results
Our purpose is to demonstrate the effectiveness of Bayesian spectroscopy in difference spectral analysis such as the XMCD spectra. While the spectral structure of XMCD becomes more complex, there is a concern that some components may be missing when one try to extract the original � helicity components from the XMCD spectrum. Therefore, in this section, we first describe the model selection on the individual � helicity XA spectra. After that, the result of model selection for the XMCD spectrum is presented. In the last part, we show posterior probability distributions of spin, orbital, and total magnetic moments to evaluate their estimation accuracies.

Model selections for � helicity XA spectra
The gray spectra in Figure 1(a-d) are � helicity XA spectra, respectively. To perform the spectral decomposition of these spectra, we employ phenomenological models f � ðx i ; Θ � Þ with the Lorentz shapes Lðx; ΘÞ as follows: where K � are the numbers of spectral components in the � helicity XA spectra. In this case, the model selection based on Equation (8) is equivalent to estimating the number of spectral components K � in the model f � ðx i ; Θ � Þ. The parameter sets for the respective Lorentz shapes are Θ k � : where I k � , E k � and Γ k � mean integral intensity, transition energy and spectral width, respectively, and for I k � and Γ k � , we introduce non-negative constraints. The spectral widths Γ k � were free parameters for each spectral component to estimate independently although the spectral widths of all Lorentz shapes were set to a common width ( ¼ 1:0eV) in the synthesized spectra.
We perform model selection to estimate K � in the respective XA spectra for � helicities using BFE as an information criterion, and the results of the � helicity XA spectra are shown in Figure 2(a,b), respectively. Line graphs connecting open squares show the variations of the minimized BFE [ ¼ FðK � ;b K � Þ] with the numbers K � of spectral components, in which the minimized values are determined from the variation of BFE with respect to b K � . The values of b K � that minimize BFE are b K À ¼8 ¼ 4:23 � 10 6 , b K þ ¼8 ¼ 3:55 � 10 6 for the � helicity XA spectra, respectively, and they hardly change in the whole ranges of K À ¼ 7,9 and K þ ¼ 6,8. From b K � , it is possible to estimate the standard deviations of the noises superimposed on the target spectra, and these standard deviations are 4:86 � 10 À 4 and 5:31 � 10 À 4 in signal intensity scale for the � helicity XA spectra, respectively, which are in close agreement with the standard deviation σ XA noise ( ¼ 5:00 � 10 À 4 ) of the noises in the synthesized XA spectra.
On the other hand, bar graphs in Figure 2(a,b) show the posterior probabilities PðK � jD � Þ for the model selection of the respective models ( where PðK � jD � Þ are calculated by Equation (9) and displayed in logarithmic scales. As seen in these figures, models containing eight spectral components are coincidentally selected in the both spectral decompositions of � helicity XA spectra, and it is obvious that the selected models (K À ¼ 8, K þ ¼ 8) have high posterior probabilities PðK � jD � Þ for model selection.
Although 23 and 31 transition components were derived for the � helicity components from the effective Hamiltonian, respectively, 8 spectral components (K � ¼ 8) were extracted for both helicities in the spectral decomposition of the respective � helicity XA spectra. However, these decreases in the number of decomposed components are reasonable under consideration of that the merging of the near transition components within the spectral width and the competition between the peak intensity of each spectral component and the superimposed noise intensity. A detailed discussion will be given in Section 5.1.
In the selected models (K À ¼ 8, K þ ¼ 8), we obtained the posterior probability distributions PðΘ k � jD � ;K � ;bK�Þ according to Equation (10) through the RXMC samplings, and evaluated the mean values Θ k � of the respective parameters and the standard deviations σ Θ k � of their posterior probability distributions. These results are summarized in the center-column group of Table 1 in the form of Θ k � � σ Θ k � , in which (a) and (b) show the results for the � helicity XA spectra, respectively. As categorized in the first column of Table 1, the spectral components having negative transition energies (E k � ) are the L 3absorption edge, and the subsequent components are the L 2 -absorption edge.
Solid curves in Figure 1(c,d) show reproduced � helicity XA spectra with the mean values Θ k � in Table 1, respectively, and it is found that they explain well the respective gray spectra D � . Root-mean-square deviations (RMSDs) of these reproduced spectra to the synthesized spectra D � are shown in the legends of Figure 1(c,d). Although the RMSDs are sufficiently small, they are about 11,38% larger than the standard deviation σ XA noise ( ¼ 5:00 � 10 À 4 ) of the normal noises superimposed on D � . This increase in RMSDs is considered to be due to the fact that the synthesized spectra contain weaker transition components than the magnitude of noises, while Bayesian spectroscopy may lose such weak components. Table 1. Decomposed spectral components from the � helicity XA spectra are summarized in the center-column group, in which mean values Θ and standard deviations σ Θ of the posterior probability distributions [PðΘjD;K;bÞ in Equation (10)] are presented. The right-column group means the corresponding transition components (discussed in Section 5.1). In the left column, L 2 and L 3 of absorption edges are categorized, and (a) and (b) are the results for � helicity XA spectra, respectively.  Blue and red curves denote the regressive XA spectra for � helicities, respectively. Dashed curves are the spectral components Lðx i ; Θ � k Þ included in these regressive spectra. (c) and (d) Blue and red curves denote the � helicity XA spectra synthesized with the respective parameter sets fΘ À κ g and fΘ þ κ g, which are obtained by the spectral decomposition of the XMCD spectrum. Dashed curves are the spectral components Lðx i ; Θ � κ Þ. The ordinate is displayed in a logarithmic scale, and horizontal dashed lines mean the standard deviation of the superimposed normal noises in the respective spectra. Transition components and � helicity spectra (D � ) are also indicated in the same manner with Figure 1(a) and (b).
To show the respective spectral components decomposed by Bayesian spectroscopy, Figure 3 was prepared. In this figure, the spectral intensities are shown on the logarithmic scale, and the superimposed noise intensities σ XA noise are depicted by horizontal dashed lines. The spectra depicted by dashed curves are the respective spectral components Lðx i ; Θ k � Þ, and Figure 3(a,b) are for the XA spectra of � helicities, respectively. The spectra shown in gray area are the � helicity XA spectra with noise superimposed, and are the same as those shown in Figure 1(a,b). Considering the noise intensity σ XA noise , the reproduced spectra indicated in blue and red curves are good reproductions of the � helicity XA spectra, respectively.
In Section 5, we will discuss the mapping of these decomposed spectral components by Bayesian spectroscopy to the transition components derived from the effective Hamiltonian in Equation (1). Here, we would like to focus the values of the estimated spectral widths. In Table 1, it is found that the spectral widths Γ k � for the spectral components having intense transition intensities (I k � > 10 � 10 À 3 ) are around 1:0eV, and the ground truth of the broadening factor Γ 0 ( ¼ 1:00 eV) in the XA spectra is estimated correctly.

Model selections for XMCD
In this study, we attempt to extract the original � helicity XA spectral components from the XMCD spectrum.
To realize this, we employ a phenomenological model in Equation (11) for the XMCD spectrum D XMCD .
where K À and K þ are the numbers of À helicity and þ helicity spectral components needed to describe the XMCD spectrum, and in this case, the model selection means the simultaneous estimation both of K À and K þ , and where we used the notations of K � and κ � for the numbers of spectral components and the indexes for the respective components, respectively, to avoid confusion with the case of individual XA spectra. In the model of Equation (11), we introduce nonnegative (I κ À > 0) and negative (I κ þ < 0) constrains on the integral intensities to distinguish � helicity spectral components. The other settings of the prior probabilities are the same as in Section 4.1.
On the basis of Equation (11), we perform the model selection to estimate K À and K þ using the BFE FðK þ ; K À ;b K þ ;K À Þ as the information criterion, and then, the model of K þ ¼ 8 and K À ¼ 9 is selected, in which the quasi-inverse temperature minimizing the BFE is bK þ ;K À ¼ 2:00 � 10 6 . Figure 2(c) shows a heat map of the selection posterior probabilities PðK þ ; K À jD XMCD Þ in the range of K þ ¼ 6,8 and K À ¼ 7,9, and it is confirmed that the selected model (K þ ¼ 8, K À ¼ 9) has the highest posterior probability. From the value of bK þ ;K À , the standard deviation of the superimposed noises in the XMCD spectrum is estimated to be 7:07 � 10 À 1 in the XMCD signal intensity scale, and it is completely coincide with ffi ffi ffi 2 p σ XA noise (see Section 2.2). Compared to the model selection results (K þ ¼ 8, K À ¼ 8) for individual XA spectra in Section 4.1, the selected model in the XMCD spectrum has one more À helicity component. Although this result seems strange at first glance, we will discuss the details in Section 5.2.
As similar with Section 4.1, we sampled the posterior probability distributions of Θ XMCD , and in the center-column group of Table 2, their mean values Θ κ � and the standard deviations σ Θ κ � of the posterior probability distributions are summarized in the form of Θ κ � � σ Θ κ � . Table 2(a,b) are for the � helicity spectral components, respectively, and the L 3 -and L 2absorption edges are also categorized in the first column of Table 2.
A reproduced spectrum using Θ κ � is depicted by a solid curve in Figure 1(e), and it explains well the graycolored XMCD spectrum. The RMSD of the reproduced spectrum to D XMCD is indicated in the legend of Figure 1 (e). Although the RMSD is larger than the RMSDs for � helicity XA spectra [see Figure 1(c,d)], in the case of the XMCD spectrum, it is reasonable because the standard deviation of the superimposed normal noises is ffi ffi ffi 2 p times larger ( ¼ 7:07 � 10 À 4 ) than the respective XA spectra, as described in Section 2.2.

Posterior probability distributions of magnetic moments
Another advantage of Bayesian spectroscopy is that, only from one dataset, we can evaluate posterior probability distributions of physical quantities. According to the sum rule [18], measurement of XMCD spectra covering the L 3;2 -absorption edge allow us to evaluate orbital m orb: , spin m spin and total m tot: magnetic moments as well as their ratio m ratio ( :¼ m orb: =m spin ) on the basis of spectral integral of the L 3;2 edge. However, the conventional method by spectral integration is point estimation, and the evaluation of estimation accuracy is quit difficult.
On the other hand, as demonstrated in Section 4.2, Bayesian spectroscopy enables us to decompose separately the � helicity spectral components from the XMCD spectrum, and samplings of the posterior probability distributions Pðm x jD XMCD Þ (x = orb., spin, tot., and ratio) become available from the integral intensities (I κ À and I κ þ ) of the decomposed � helicity spectral components since the magnetic moments m orb: and m spin can be evaluated by Equations (12) and (13) [18] in a unit of μ B /atom, respectively.
where n h is the occupancy number of 3d-holes of Ni ions and n h ¼ 1:832 [27], since the effective occupation number of 3d-electrons is evaluated to be 8:168 from the effective Hamiltonian in Equation (1).
Posterior probability distributions of m orb: , m spin , m tot: and m ratio are shown by color-filled graphs in Figure 4(a-d), respectively. Black lines in Figure 4 indicate the true values of these quantities, which are obtained from the effective Hamiltonian. On the other hand, red lines and error bars in Figure 4 are the mean values of these quantities and the standard deviations of their posterior probability distributions, and it is found that the true values of all these quantities are contained within the respective posterior probability distributions. This result demonstrates that the Table 2. Decomposed spectral components from the XMCD spectrum are summarized in the center-column group. The rightcolumn group means the corresponding transition components (discussed in Section 5.1). In the left column, L 2 and L 3 of absorption edges are categorized, and (a) and (b) are the � helicity spectral components in the XMCD spectrum, respectively.   magnetic moments are correctly estimated, and the evaluation of estimation accuracy can be realized with the distribution widths of their posterior probabilities. Such evaluation of the posterior probability distributions is meaningful when measurements with a high signal-to-noise (S/N) ratio are difficult. Although XMCD microscopic measurements [28], which capture magnetic domain in magnetic materials, become available with recent developments [29] in synchrotron radiation measurement techniques, it is still difficult to obtain a high S/N ratio for in situ measurements that capture the dynamics of magnetic domain formation. Of course, although the estimation accuracy will degrade with bad S/N ratios, the evaluation of posterior probability distributions by Bayesian spectroscopy becomes more important in such situations.

Discussion
In this section, we describe the attribution of the decomposed spectral components paying attention both to the noise intensity and the spectral widths in the target spectra. Subsequently, we will discuss the difference in the spectral decomposition results between the individual XA and XMCD spectra.

Attribution of decomposed spectral components
When performing spectral decomposition to extract weak transition components and to separate components with close transition energies, the magnitude of noises superimposed on the target spectrum and the spectral width of each transition component become obstacles. In fact, as described in Section 2.2, normally distributed noises with a standard deviation of σ XA noise ¼ 5:00 � 10 À 4 were superimposed on the synthesized � helicity XA spectra, and a spectral width of Γ 0 ¼ 1:00eV was convolved with each transition component derived from the effective Hamiltonian.
The first issue is the noise intensity, and competition with the noise intensity is considered to be characterized by the peak intensity of each spectral component. In the case of the Lorentzian shape [see Equation (2)] with a width of Γ 0 , the spectral component with an integral intensity of I k � has a peak intensity of 2I k � =ðπ Γ 0 Þ. From all transition components derived from the effective Hamiltonian, the components having peak intensities beyond the noise intensity were summarized selectively in the right-column group of Table 1, where the transition intensity I j 0� being I j 0� > ðπ Γ 0 =2Þ � σ XA noise ( � 7:85 � 10 À 4 ). The horizontal dashed lines in Figure 1(a,b) are the levels of transition intensities corresponding to the noise intensity. Although the index j in Equation (5) is scanned on all transition components derived from the effective Hamiltonian, and here after, the index j 0 is used for the subset that satisfies this condition.
The second issue is that each transition component has a finite spectral width. Although, as seen in the right-column group of Table 1, there are thirteen (j 0 À ¼ 1,13) and fourteen (j 0 þ ¼ 1,14) transition components in � helicities, respectively, it is found that there are components whose transition energies E j 0� are in close proximity within the spectral width Γ 0 . Such adjacent transition components are considered to be merged to one spectral component in the spectral decomposition by Bayesian spectroscopy, such merging is also shown in Table 1. For example, in Table 1(a), the transition components of j 0 À ¼ 1; 2 are merged to the spectral component of k À ¼ i at the L 3 -absorption edge. In order to distinguish between the transition components specified by index j 0 , the spectral components decomposed by Bayesian spectroscopy are numbered using Roman numerals in Table 1 and in Table 2 (discussed later). Columns of P I j 0� and E j 0� in Table 1 indicate the merged transition intensities and the weighted-mean values of the transition energies based on the transition intensity.
In Table 1, comparing the columns of I k � � σ I k � and P I j 0� , and of E k � � σ E k � and E j 0� , it is clear that P I j 0� and E j 0� are included within the respective posterior probability distributions of I k � and E k � of each spectral component decomposed by Bayesian spectroscopy. This result demonstrates that Bayesian spectroscopy is able to extract the transition components without excess or deficiency under consideration of the noise intensity and finite spectral width.
Such precise spectral decomposition by Bayesian spectroscopy is also realized by the spectral decomposition of XMCD although the noise intensity in the XMCD spectrum is ffi ffi ffi 2 p times larger than that of the XA spectra. In Table 2, taking the noise intensity into account, we summarized the attribution of the decomposed spectral components to the transition components obtained from the effective Hamiltonian. This result also demonstrates that, from the XMCD spectrum, original spectral components of � helicities are extracted separately without excess or deficiency by Bayesian spectroscopy. Figure 5 demonstrates the attributions of the decomposed spectral components by Bayesian spectroscopy and the transition components derived from the effective Hamiltonian. Figure 5(a,b) are for the results of the individual � helicity XA spectra, respectively, and Figure 5(c,d) are for the � helicity components decomposed from the XMCD spectrum, respectively, where the ordinates are in logarithmic scales and are inverted upside down for þ helicity ones. The horizontal dashed lines indicate the noiseintensity levels in the integral intensity scale.
Vertical lines in Figure 5 mean the merged transition components in the right-column group of Tables  1 and Tables 2, which are displayed with E j 0� and P I j 0� . Open squares in Figure 5(a,b) are the points of fE k � ; I k � g for the decomposed � helicity components from the � helicity XA spectra, respectively. On the other hand, the fE κ � ; I κ � g are plotted by open lozenges in Figure 5(c,d) for the respective � helicity components decomposed from the XMCD spectrum. The error bars accompanying those open marks are standard deviations of the respective posterior probability distributions and indicate the estimation accuracy of each mean value. In Figure 5, one can confirm that most of the points indicated by the vertical lines are included in the ranges of the respective posterior probability distributions. This result clearly implies that, even in the case of XMCD, the original � helicity components can be extracted separately without excess or deficiency. Such challenging spectral decomposition can be realize with Bayesian spectroscopy based on the model selection using BFE as the information criterion.
The results of such challenging spectral decomposition can be confirmed in Figure 3(c,d). The dashed curves are the respective spectral components Lðx i ; Θ κ � Þ for � helicities decomposed from the XMCD spectrum, and the blue and red spectra are the spectra where we attempt to reconstruct the � helicity XA spectra as the sum of those components. Amazingly, it is found that the blue and red reconstructed spectra well explain the original � helicity XA spectra, which are depicted by gray spectra for the guides to the eyes, under consideration of the superimposed noise intensity σ XMCD noise . This result implies that Bayesian spectroscopy has successfully reproduced the original -/+helicity spectra from the XMCD spectrum, even though they were hidden from the human eye in the XMCD spectrum. However, comparing the upper and lower panels in Figure 3, it is found that the spectra (XMCD) in the lower panel do not completely reproduce the upper ones ( � XA). Although this difference might be due to the difference in noise intensities (σ XMCD noise , σ XA noise ), in order to solve this problem, Bayesian-integration [30] study of � helicity XA and XMCD spectra is expected in the future.
The decomposition of XMCD spectra and the evaluation of magnetic moments based on Bayesian spectroscopy are expected to be widely applied to various magnetic materials such as nickel ferrite [17]. As described in this section, spectral components weaker than the noise intensity are impossible to be decomposed even with Bayesian spectroscopy, and a low signal-to-noise ratio results in a broadening of the posterior probability distribution of magnetic moments and a decrease in estimation accuracy. Conversely, however, it can be claimed that Bayesian spectroscopy is the method to decompose the spectral components without deficiency when it is difficult to improve the signal-to-noise ratio such as the case of in situ and operand measurements. The same analysis was performed using synthesized data with twice the noise intensity, and the spectral components exceeding the noise intensity were appropriately decomposed, and this result confirms the advantage of Bayesian spectroscopy for XMCD spectral analyses.