# O’ R Inflation in $F$-term Supergravity

Sibo Zheng  Department of Physics, Chongqing University, Chongqing 401331, P.
R. China

(October, 2016)

###### Abstract

The supersymmetric realization of inflation in $F$-term supergravity is
usually plagued by the well known “$\eta$” problem. In this paper, a broad
class of small-field examples is realized by employing general O’ Raifeartaigh
superpotentials, where the moduli is identified as the massless inflaton. For
illustration we present the simplest example in detail, which can be
considered as a generalization of hybrid inflation.

##  I Introduction

Inflation is an extensively studied and widely accepted scenario to solve both
the flatness and horizon problems in the early Universe. For a review, see,
e.g. 0907.5424 . On the other hand, Supersymmetry (SUSY) is a scenario to deal
with the divergent problem of Standard Model Higgs mass, which is inferred to
probably play an important role in the high energy region (far above the weak
scale) from both the null results at particle colliders (e.g., LHC) as well as
dark matter direct detection experiments (e.g., LUX). Therefore, it is natural
to ask the question whether inflation and SUSY are connected.

When we explore high energy physics near the Planck mass scale $m_{P}$,
gravitational effect has to be taken into account. This implies that one
should discuss the realization of such kind of inflation in the context of
Supergravity. Given the structure of potential
$V_{\text{local}}=V^{F}_{\text{local}}+V^{D}_{\text{local}}$ in supergravity,
studies on the SUSY realizations of inflation can be divided into two classes
- the F-term and D-term supergravity. In the former one, in unit of Planck
mass

$\displaystyle{}V^{F}_{\text{local}}=e^{K}\left[D_{i}WK^{-1}_{ij^{*}}D_{j^{*}}W^{*}-3\mid
W\mid^{2}\right]$ (1)

where $K(\Phi_{i})$ and $W(\Phi_{i})$ is the Kahler and superpotential,
respectively, and covariant derivative $D_{i}W=\partial_{i}W+K_{i}W$, with
index $i$ referring to a chiral superfield $\Phi_{i}$. The expotential factor
$e^{K}$ introduces the well known “$\eta$” problem to $F$-term Supergravity.
For reviews, see, e.g, 9807278 ; 1101.2488 . Progresses along this line are
significantly improved by Kallosh et al 1011.5945 , where the authors gave a
prescription to construct a general potential of inflation, but with the price
of some specific shift symmetry as required in the Kahler potential.

There is no such “$\eta$” problem in $D$-term supergravity 9405389 . Along
this line, it was firstly proposed by Halyo 9606423  that a positive and non-
zero $D$-term naturally realizes inflation. Unfortunately, by following the
insights in 0904.1159 , the Fayet-Iliopoulos term itself is not consistent
with the supergravity.

In this paper, we focus on the realization of inflation in $F$-term
supergravity. Unlike in 1011.5945 , we turn to the canonical Kahler potential
111$\eta$ problem which may be induced by operators with mass dimension higher
than four in the Kahler potential will not be discussed here. without any
specific shift symmetry. In order to keep our calculations perturbatively
valid, we restrict to the small-field inflation222 For examples of the large-
field inflation such as chaotic inflation, see, e.g., 0004243 ; 1008.3375 ..
Previous studies under these same assumptions in the literature include hybrid
inflation 9703209 , new inflation 9608359  and single-field inflation
1605.01934 .

The plan of this paper is organized as follows. In Sec.II we give a
prescription to construct a general superpotential without the $\eta$ problem
for canonical Kahler potential. We find that it is actually valid for O’
Raifeartaigh (O’ R) superpotential. In Sec.III we present a new example with
the simplest O’ R superpotential for illustration. This model obviously
differs from the hybrid inflation in the sense that there is no need of a
critical mass scale $\sigma_{c}$ to end inflation although it gives rise to a
spectral index similar to the hybrid inflation. Finally, we conclude in
Sec.IV.

##  II O’ R superpotential without $\eta$ problem

This section is devoted to a prescription, from which there is no $\eta$
problem for a renormalizable superpotential $W$ and canonical Kahler potential
$K$. In order to provide inflation the potential $V$ in Eq.(1) must be
positive and non-zero. Consider that for the small-field inflation the local
potential is well approximated to

$\displaystyle{}V^{F}_{\text{local}}\approx
V^{F}_{\text{global}}+\mathcal{O}(1/m^{2}_{P}).$ (2)

This requirement is equivalent to that the global SUSY should be broken $\mid
F_{i}\mid^{2}\neq 0$. Meanwhile, the inflaton mass should be always small
9401011 ; 1307.4343  (in compared with the Hubble parameter $H$) in order to
evade the $\eta$ problem, and the others scalars’ masses should be comparable
with $H$ if one considers a realization of single-field inflation 333Note that
$H^{2}\approx V_{\text{global}}/3$ in unit of Planck mass.. This requirement
is equivalent to that there exists a flat direction with the modulus
identified as the inflaton superfield $X$.

In the global version of SUSY the two requirements above can be satisfied
simultaneously by choosing general O’ R superpotential,

$\displaystyle{}W(X,\chi_{i})=X(\mu^{2}+f(\chi_{i}))+g(\chi_{i}),$ (3)

where $X$ and $\chi_{i}$ are chiral superfields, and functions $f(\chi_{i})$
and $g(\chi_{i})$ are defined as 444In principle, linear terms like
$a_{i}\chi_{i}$ and $b_{i}\chi_{i}$ may appear in $f(\chi_{i})$ and
$g(\chi_{i})$, respectively. However, the former probably introduce dangerous
mixing between $X$ and $\chi_{i}$, and the later introduce new linear
structure except $X$ in the superpotential. These terms can be forbidden by
adjusting their $R$ charges.,

$\displaystyle{}f(\chi_{i})$ $\displaystyle=$ $\displaystyle
a_{ij}\chi_{i}\chi_{j},$ $\displaystyle g(\chi_{i})$ $\displaystyle=$
$\displaystyle b_{ij}\chi_{i}\chi_{j}+b_{ijk}\chi_{i}\chi_{j}\chi_{k}.$ (4)

The coefficients $a_{ij}$, $b_{ij}$, etc, are assumed to be real for
simplicity, and $\mu$ is the SUSY breaking scale. The superpotential is only
linear function of $X$, from which with suitable $R$ charge assignments the
minimal value of $V$ is determined by 0703196 ,

$\displaystyle{}F_{X}\mid_{\left<\chi_{i}\right>}=\mu^{2},~{}~{}~{}~{}F_{\chi_{i}}\mid_{\left<\chi_{i}\right>}=0,~{}~{}~{}~{}\text{and}~{}~{}~{}~{}X~{}\text{arbitrary}$
(5)

Now, let us examine whether there is indeed no $\eta$ problem in the local
version in Eq.(1) for superpotential in Eq.(3) with the structure of vacuum in
Eq.(5). After a detailed calculation we find that 555Keep in mind that
operators with mass dimensions higher than 4 are compensated by the correct
powers of Planck mass.,

$\displaystyle{}V^{F}_{\text{local}}=e^{K}\cdot\left\\{A_{0}+[A_{1}X+h.c]+A_{2}\mid
X\mid^{2}+[A_{3}X\mid X\mid^{2}+h.c]+A_{4}\mid X\mid^{4}\right\\},$ (6)

where

$\displaystyle{}A_{0}$ $\displaystyle=$ $\displaystyle\mid F_{X}\mid^{2}+\mid
F_{\chi_{i}}\mid^{2}+\mid
g\mid^{2}(\mid\chi_{i}\mid^{2}-3)+[g^{*}F_{\chi_{i}}\chi_{i}+h.c]$
$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\left(F^{*}_{\chi_{i}}\chi^{*}_{i}+g^{*}\mid\chi_{i}\mid^{2}-2g^{*}\right)F_{X}$
$\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle-\mid F_{X}\mid^{2}+\mid
F_{X}\mid^{2}\mid\chi_{i}\mid^{2}+\mid g\mid^{2}$ $\displaystyle A_{3}$
$\displaystyle=$ $\displaystyle g^{*}F_{X}$ $\displaystyle A_{4}$
$\displaystyle=$ $\displaystyle\mid F_{X}\mid^{4}.$ (7)

Eq.(6) is organized in powers of $X$, the benefit of which is that the terms
of mass squared for $X$ can be easily extracted from individual terms in
Eq.(6) after one expands the expotential factor $e^{K}\approx 1+\mid
X\mid^{2}+\mid\chi_{i}\mid^{2}$.

Under the small-field approximation ($\mid X\mid<<1$ etc.) we obtain the final
expression for $V^{F}_{\text{local}}$,

$\displaystyle{}V^{F}_{\text{local}}$ $\displaystyle\approx$
$\displaystyle\tilde{A}_{0}+[\tilde{A}_{1}X+h.c]+\tilde{A}_{2}\mid
X\mid^{2}+[\tilde{A}_{3}X\mid X\mid^{2}+h.c]+\tilde{A}_{4}\mid X\mid^{4}$ (8)
$\displaystyle+$ $\displaystyle[\tilde{A}_{5}X\mid
X\mid^{4}+h.c]+\tilde{A}_{6}\mid X\mid^{6}+\cdots$

where we have ignored higher-dimensional operators suppressed by $m_{P}$.
Here, functions $\tilde{A}_{i}$ are given by,

$\displaystyle{}\tilde{A}_{0}$ $\displaystyle=$
$\displaystyle(1+\mid\chi_{i}\mid^{2})A_{0},~{}~{}~{}~{}~{}~{}\tilde{A}_{1}=(1+\mid\chi_{i}\mid^{2})A_{1}$
$\displaystyle\tilde{A}_{2}$ $\displaystyle=$ $\displaystyle
A_{0}+(1+\mid\chi_{i}\mid^{2})A_{2},~{}~{}\tilde{A}_{3}=A_{1}+(1+\mid\chi_{i}\mid^{2})A_{3}$
$\displaystyle\tilde{A}_{4}$ $\displaystyle=$ $\displaystyle
A_{2}+(1+\mid\chi_{i}\mid^{2})A_{4},~{}~{}\tilde{A}_{5}=A_{3},~{}~{}\tilde{A}_{6}=A_{4}$
(9)

From Eq.(8) the mass squared $m^{2}_{X}$ is read as,

$\displaystyle{}m^{2}_{X}=\left[\mid
F_{X}\mid^{2}+\cdots\right]+(1+\cdots)\left[-\mid
F_{X}\mid^{2}+\cdots\right]+\mathcal{O}(\mid X\mid/m_{P})$ (10)

where we have used $F_{X}\mid_{\left<\chi_{i}\right>}=\mu^{2}$ and $\cdots$
represent contributions from high-dimensional operators. Clearly,
$m^{2}_{X}=0$ at the leading order, and its smallness in compared with $H^{2}$
still holds as long as we restrict to the small-field inflation.

One may also verify that all the masses squared for $\chi_{i}$ are of order
$H^{2}$ from $\tilde{A}_{0}$ in Eq.(II) and Eq.(II). It results from the fact
that there is only one non-zero $F$-term in the minimal of $V$.

In summary, for small-field approximation our statements about masses for $X$
and $\chi_{i}$ are always true as long as the mass scales of coefficients
$a_{ij}$, $b_{ij}$, etc, in Eq.(II) are all of the order $\mu$ for O’ R
superpotential. In the light of our results it is easy to understand why there
is no $\eta$ problem in some simple inflation models such as hybrid inflation
($g=0$, $f=\bar{\chi}\chi$) in the literature. We refer this broad class of
small-field inflation models as O’ R inflation.

##  III An Example

By following the results in the previous section, in this section we propose a
concrete and simple example for illustration. As we will see, this new
inflation model obviously differs from the hybrid inflation in the sense that
there is no tachyon mass problem, although it gives rises to a similar
spectral index.

###  III.1 Inflaton effective potential

We begin with the simplest O’ R model OR  in the class of superpotentials
defined in Eq.(3),

$\displaystyle{}W=X(\mu^{2}+\frac{1}{2}h\chi^{2}_{1})+m\chi_{1}\chi_{2}.$ (11)

In this model the SUSY-breaking vacuum is described by $\chi_{1,2}=0$ and
arbitrary $X$, and $V^{F}_{\text{global}}=\mid F_{X}\mid^{2}=\mu^{4}$. After
substituting $\left<f\right>\mid_{\chi_{1,2}=0}=0$ and
$\left<g\right>\mid_{\chi_{1,2}=0}=0$ into Eq.(8) one can verify that
$m_{X}=0$. So, $X$ is the candidate of inflaton. Note that the structure of
superpotential in Eq.(11) allows a $Z_{2}$ symmetry. In order to avoid
possible production of domain wall DW  due to this $Z_{2}$ symmetry, we simply
assume that it has been explicitly broken by higher dimensional operators
0006168 . It is expected that the domain wall problem can be resolved in more
general O’ R model without the aid of higher dimensional operators.

In order to derive the effective potential for $X$ we firstly calculate the
mass spectral for the chiral superfields $\chi_{1,2}$. By virtue of the
standard formulas, the scalar masses squared $\mathcal{M}^{2}_{B}$ and the
fermion masses squared $\mathcal{M}^{2}_{F}$ are given by, respectively,

$\displaystyle{}\mathcal{M}^{2}_{B}=m^{2}\cdot\left(\begin{array}[]{cccc}\mid\epsilon_{X}\mid^{2}+1&\epsilon_{X}&\epsilon_{\mu}&0\\\
\epsilon^{*}_{X}&1&0&0\\\
\epsilon_{\mu}&0&\mid\epsilon_{X}\mid^{2}+1&\epsilon^{*}_{X}\\\
0&0&\epsilon_{X}&1\\\ \end{array}\right)$ (16)

and

$\displaystyle{}\mathcal{M}^{2}_{F}=m^{2}\cdot\left(\begin{array}[]{cccc}\mid\epsilon_{X}\mid^{2}+1&\epsilon_{X}&0&0\\\
\epsilon^{*}_{X}&1&0&0\\\ 0&0&\mid\epsilon_{X}\mid^{2}+1&\epsilon^{*}_{X}\\\
0&0&\epsilon_{X}&1\\\ \end{array}\right),$ (21)

where $\epsilon_{X}=h\mid X\mid/m$ and $\epsilon_{\mu}=h\mu^{2}/m^{2}$.

Here a few comments are in oder regarding the magnitude of $\epsilon_{X}$ and
$\epsilon_{\mu}$. First, if $\epsilon_{\mu}<1$ fields $\chi_{i}$ would be
decoupled from field $X$, inflation will nerve end. In the following analysis
we will impose $\epsilon_{\mu}\geq 1$. Second, for $\mu$ is far below the
Planck mass the ratio $\epsilon_{X}/\epsilon^{1/2}_{\mu}=h^{1/2}\mid
X\mid/\mu$ is larger than unity, which implies that the magnitude of
$\epsilon_{X}$ is larger than unity as well. It can be verified that for such
$\epsilon_{X}$ and $\epsilon_{\mu}$ the signs of determinant of
$\mathcal{M}^{2}_{B}$ in Eq.(16) and that of $\mathcal{M}^{2}_{F}$ in Eq.(21)
are both positive, and the effective potential for the inflaton ($\sigma=\mid
X\mid$) can be approximated as,

$\displaystyle{}V_{\text{eff}}(\sigma)=\mu^{4}\left[1+\frac{h^{2}}{16\pi^{2}}\log\left(\frac{h\sigma}{\Lambda}\right)+\sigma^{6}+\cdots\right],$
(22)

where $\Lambda$ denotes the renormalizable scale in the model, and non-
renormalizable terms with mass dimensions higher than $\sigma^{6}$ are
neglected.

###  III.2 Fit to inflationary parameters

Now we discuss the implications in the simplest O’ R inflation model. First,
we count the total number of e-folds during inflation,

$\displaystyle{}N_{tot}=\int^{\sigma_{i}}_{\sigma_{end}}\frac{V}{V^{\prime}}d\sigma=\int^{\sigma_{i}}_{\sigma_{d}}\frac{d\sigma}{6\sigma^{5}}+\int^{\sigma_{d}}_{\sigma_{end}}\frac{16\pi^{2}}{h^{2}}\sigma
d\sigma\simeq\pi^{4/3}h^{-4/3}\left[\left(\frac{1}{144}\right)^{1/3}+\left(\frac{16}{3}\right)^{1/3}\right],$

where $\sigma_{i}$ and $\sigma_{end}$ is the initial and end value of inflaton
(in unit of Planck mass), respectively, and
$\sigma_{d}\simeq(h^{2}/96\pi^{2})^{1/6}$ being the critical value above
(below) which the non-renormalizable term (log-term) dominates the inflation.
Eq.(III.2) shows that the period between $\sigma_{in}$ and $\sigma_{d}$ gives
rise to $\sim 10\%$ contribution to $N_{tot}$. From Eq.(III.2) the requirement
of the e-fold number $N_{tot}=50(60)$ then leads to $h=0.27(0.24)$.

Second, consider the period between $\sigma_{d}$ and $\sigma_{end}$. The slow
roll parameters are given by,

$\displaystyle{}\epsilon$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\simeq\frac{h^{2}}{64\pi^{2}N},$
$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{V^{\prime\prime}}{V}\simeq-\frac{1}{2N},$ (24)

where $N$ is the e-fold number corresponding to the inflaton value
$\sigma_{N}=\sqrt{Nh^{2}/8\pi^{2}}$ during inflation. From Eq.(III.2) one
finds that $i)$ the magnitude of $\mid\epsilon\mid$ is small in comparison
with $\mid\eta\mid$ due to the one-loop factor suppression. $ii)$, Unlike the
hybrid inflation where $\sigma_{end}$ is approximately determined by the
critical value $\sigma_{c}$, for our model where there is no critical mass
scale corresponding to tachyon mass $\sigma_{end}$ is determined to be
$h/4\pi$ by taking $\mid\eta_{end}\mid=1$. To summarize, $\sigma_{d}\simeq
0.21$ and $\sigma_{end}\simeq 0.02$ for $N_{tot}=50$.

Figure 1: Parameter space in the plane of $h-N$. the curves of $r$ and $n_{s}$
are shown in green and red, respectively. Only Regions between the right-hand
side of the red line $n_{s}=0.98$ and left-hand side of the red line
$n_{s}=0.982$ are allowed.

In terms of the slow roll parameters in Eq.(III.2) the main inflationary
parameters measured by experiments are given by,

$\displaystyle{}n_{s}$ $\displaystyle=$ $\displaystyle 1-6\epsilon+2\eta\simeq
1-\frac{1}{N},$ $\displaystyle\frac{dn_{s}}{d\text{ln k}}$
$\displaystyle\simeq$ $\displaystyle
16\epsilon\eta-24\epsilon^{2}\simeq-\frac{h^{2}}{8\pi^{2}N^{2}},$
$\displaystyle r$ $\displaystyle=$ $\displaystyle
16\epsilon\simeq\frac{h^{2}}{4\pi^{2}N},$ $\displaystyle\mathcal{R}_{s}$
$\displaystyle=$
$\displaystyle\frac{1}{24\pi^{2}}\frac{V}{\epsilon}\simeq\frac{8\mu^{4}N}{3h^{2}},$
(25)

where $n_{s}$ is the spectral index, $dn_{s}/d\text{ln k}$ is the running, $r$
is the tensor-to-scalar ratio, and $\mathcal{R}_{s}$ is the amplitude of
primordial fluctuations. In Fig.1 we show the parameter space by fitting to
the experimental constraints on the inflationary parameters in Table.1. It
shows that only regions between the right-hand side of the solid red line
$n_{s}=0.98$ and left-hand side of the dotted red line $n_{s}=0.982$ are
allowed. We would like to mention that the spectral index in this model is
mainly sensitive to the total number of e-folds during inflation. Typically,
we have $n_{s}\sim 0.98$, $r\sim 10^{-6}$ and $dn_{s}/d\text{ln k}\sim
10^{-7}$.

| $\Lambda\text{CDM+r+nrun}$
---|---
$n_{s}$ |  $0.9721\pm 0.011$ (68% CL)
$\frac{dn_{s}}{dlnk}$ |  $-0.0038\pm 0.0068$ (68% CL)
$r_{0.01}$ |  $\leq 0.075$ (95% CL)
$\text{ln}(10^{10}\mathcal{R}_{s})$ |  $3.117\pm 0.021$ (68% CL)
Table 1: The latest experimental limits at 68% CL on the inflationary
parameters in the cosmological model $\Lambda\text{CDM+r+nrun}$ from
$P15+BK14+BAO15$ data combination 1512.07769 .

Let us examine the constraints on magnitude of $h$. Substituting the
experimental value $\mathcal{R}\simeq 2.44\times 10^{-9}$ in Table 1 into the
last formula in Eq.(III.2) gives rise to $\mu\simeq 2.0\times 10^{-3}h^{1/2}$
for $N=50$, which together with $\epsilon_{X}\geq 0.1$ and $\epsilon_{\mu}\geq
1$ implies $\sigma_{end}\geq 2.5\times 10^{-4}$. At the meantime, $h$ is fixed
to $0.24-0.27$ as required by $N_{tot}=50-60$, which is consistent with this
bound.

Finally, we would like to mention that there remain a few issues to be
addressed for a complete phenomenological study. First, it is likely to tune
the value of $n_{s}$ from the expectation value of hybrid inflation ($\sim
0.98$) to the favored value of Planck data ($\sim 0.974$ Planck ) by taking
the effect of reheating into account. See, e.g., 1606.06677 . Second, the
tuning of initial condition and gravitino overproduction problem usually
plague SUSY driven inflation model. See Refs. 1007.5152 ; 1404.1832  for
similar discussions in hybrid inflation. These aspects are beyond the scope of
this paper.

##  IV Conclusion

A prescription to construct a general superpotential without the $\eta$
problem for canonical Kahler potential has been addressed in the $F$-term
supergravity. We have verified that it is valid for O’ R superpotential at
least for the small-field inflation. Also, the simplest O’ R model is studied
in detail for illustration, which is found to be similar to the hybrid
inflation. Since there is no need of a critical mass scale $\sigma_{c}$ to end
the inflation, it can be considered as a generalization of hybrid inflation.
Although it is not a concrete statement, the constraint on the spectral index
in the simplest O’ R model may be relaxed in some non-minimal examples.

###### Acknowledgements.

The author would like to thank the referee for comments. This work is
supported in part by the Fundamental Research Funds for the Central
Universities with project No. 106112016CDJXY300004.

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# Counterfactual Explanations of Black-box Machine Learning Models using
Causal Discovery with Applications to Credit Rating
* ††thanks:

Daisuke Takahashi  Graduate School of Data Science,
Shiga University, Hikone, Japan
<EMAIL_ADDRESS>Shohei Shimizu  Graduate School of Data Science,
Shiga University, Hikone, Japan
RIKEN Center for Advanced
Intelligence Project, Tokyo, Japan
<EMAIL_ADDRESS>Takuma Tanaka  Graduate School of Data
Science,
Shiga University, Hikone, Japan
<EMAIL_ADDRESS>
###### Abstract

Explainable artificial intelligence (XAI) has helped elucidate the internal
mechanisms of machine learning algorithms, bolstering their reliability by
demonstrating the basis of their predictions. Several XAI models consider
causal relationships to explain models by examining the input-output
relationships of prediction models and the dependencies between features. The
majority of these models have been based their explanations on counterfactual
probabilities, assuming that the causal graph is known. However, this
assumption complicates the application of such models to real data, given that
the causal relationships between features are unknown in most cases. Thus,
this study proposed a novel XAI framework that relaxed the constraint that the
causal graph is known. This framework leveraged counterfactual probabilities
and additional prior information on causal structure, facilitating the
integration of a causal graph estimated through causal discovery methods and a
black-box classification model. Furthermore, explanatory scores were estimated
based on counterfactual probabilities. Numerical experiments conducted
employing artificial data confirmed the possibility of estimating the
explanatory score more accurately than in the absence of a causal graph.
Finally, as an application to real data, we constructed a classification model
of credit ratings assigned by Shiga Bank, Shiga prefecture, Japan. We
demonstrated the effectiveness of the proposed method in cases where the
causal graph is unknown.

###### Index Terms:

Counterfactual explanations, explainable machine learning, causal discovery,
rating classification

##  I Introduction

In recent years, the use of artificial intelligence (AI) has rapidly increased
owing to its revolutionary impact on various aspects of society and industry.
This is primarily attributed to advances in deep learning [1] and ensemble
learning algorithms such as LightGBM [2]. However, it is impossible for humans
to understand the basis for the predictions by these models due to the complex
calculations and internal structures. Consequently, people are hesitant to
utilize machine learning for tasks that require accountability (bank loans,
medical diagnosis) [3]. To address this challenge, numerous methods have been
developed in explainable artificial intelligence (XAI) to improve the
explainability of black-box models [4].

Among the most promising approaches of XAI is the explanation of predictive
models using causal inference [5]. Causal inference-based XAI considers the
correlation between the input and output and the dependence of features; thus,
it can efficiently explain the desired predicted value. In particular, the
estimation by existing methods, such as decision tree-based feature importance
[6] and SHAP [7], is based on the correlation of variables, which may be
pseudo-correlation [8]. Most XAI methods based on causal inference employ
counterfactual explanations, which elucidate a prediction by computing the
change in the individual’s features to modify the prediction to the desired
class [9]. A previous study [10] presented an XAI system called LEWIS, which
leveraged causal models and counterfactual probabilities. This study employed
loan approval as an example to estimate a counterfactual probability score for
a binary classification. Based on this explanatory score, this method rendered
the features requiring change as explicit to modify the prediction if the loan
for a customer was predicted to be rejected.

However, LEWIS requires complete knowledge of the causal graph of features.
When building machine learning models, background knowledge such as causal
relationships are rarely completely known. This is a major challenge when
applying this method to real data. When the causal graph is unknown, it can be
estimated by causal discovery [11]; however, its performance in artificial and
real-world data has not been explored.

Figure 1: Framework of counterfactual probability explanations using causal
structure information

This study proposed a new XAI framework that relaxed the constraint of the
causal graph being known. As shown in Figure 1, a causal discovery can be
possibly performed with prior knowledge on the causal structure, and the
resulting causal graph can be used to explain the features using
counterfactual probabilities. Our contributions are summarized as follows:

  * •

We conducted numerical experiments to analyze the variation of explanatory
scores with causal structure and the proposal of useful prior information on
causal structure.

  * •

Our artificial data experiments implied that the combination of causal
discovery with certain prior information proposed above recovered the
estimates of the explanatory score better than previous methods without causal
discovery or graph assumption.

  * •

By applying the method to real world financial data from the Shiga bank, Ltd.,
which is the largest regional bank in Shiga prefecture of Japan, we
demonstrated that useful explanations could be made from the graph estimated
by causal discovery. To the best of our knowledge, this is the first real-
world example of counterfactual probability explanations in case that the
causal structure is unknown and is estimated by causal discovery.

The remainder of this paper is organized as follows. Section II provides an
overview of related research and defines the symbols used in this paper.
Section III analyzes the effects of causal structures on explanation scores
and proposes the use of useful prior information on the causal structure. In
Section IV, using artificial data, we provide counterfactual explanations by
combining prior information on the causal structure and causal exploration.
Consequently, we examine whether the estimated explanatory score can restore
the estimate of the true explanatory score. Section V demonstrates that the
proposed method is useful even when the causal graph is unknown using real
data. Finally, Section VI summarizes the results and provides future
perspectives.

##  II Background

This section introduces the mathematical symbols and formulas used in this
paper and outlines the related methods.

###  II-A Structural Causal Model

Structural causal model (SCM) [12] is a mathematical framework for handling
causal relationships. SCM comprises a set of endogenous variables
$\mathbf{V}=\\{V_{1},V_{2},...,V_{p}\\}$, a set of exogenous variables
$\mathbf{U}=\\{U_{1},U_{2},...,U_{q}\\}$, which are variables not determined
by the endogenous variables, and a set of functions
$\mathbf{F}=\\{f_{1},f_{2},...,f_{p}\\}$, which determines the values of
endogenous variables from those of other endogenous and exogenous variables.
In addition, we define the parent set of endogenous variables as
$\mathrm{pa}(V_{i})$ and we assume variables in $\mathbf{U}$ are independent.
Consequently, if SCM is specified by

$V_{i}=f_{i}(\mathrm{pa}(V_{i}),U_{i})$ (1)

and the system is assumed to be autonomous, Equation (1) is called a
structural causal model, wherein the quantitative causal relations of
variables are expressed by a directed graph called a causal graph. Here,
autonomy implies that changing any function or distribution of variables does
not change the distribution of other functions or distributions.

###  II-B LEWIS

LEWIS [10] is an XAI method that provides counterfactual explanations for
machine learning predictions based on structural causal models. Consider the
following binary classification problem. The explanatory variable is
$X\in\mathbf{V}$ and the value of the explanatory variable is
$\\{x,x^{\prime}\\}\in X$ $(x>x^{\prime})$, wherein all the values of the
explanatory variables are discretized. Let $O\in\\{o,o^{\prime}\\}$ be the
predicted value corresponding to the explanatory variable, where $o$ is the
positive predicted value and $o^{\prime}$ is the negative predicted value. The
counterfactual in any causal model $M=\langle V,U,F\rangle$ can be defined for
a certain individual $u\in U$ as [12]

$X(u)=x\Rightarrow Y_{X=x}(u)=Y_{M_{x}}(u),$ (2)

where $Y_{X=x}$ is the counterfactual of the value of $Y$ if the value of $X$
were $x$, with $Y_{M_{x}}$ being that of $Y$ if the value of $X$ were $x$ in
the causal model $M$. Further, the counterfactual probability $P(Y_{X=x})$ can
be expressed as $P(Y|\mathit{do}(X=x))$ using the intervention symbol
$\it{do}$. It represents the conditional probability of $Y$ if the value of
$X$ were changed to $x$ and can be computed based on a causal graph.

In this setting, LEWIS defines the following three explanatory scores as the
probability of a counterfactual, necessity score (Nec)

$\mathrm{Nec}(x,x^{\prime})=P(o^{\prime}_{X=x^{\prime}}|x,o),$ (3)

sufficiency score (Suf)

$\mathrm{Suf}(x,x^{\prime})=P(o_{X=x}|x^{\prime},o^{\prime}),$ (4)

and necessity and sufficiency score (Nesuf)

$\mathrm{Nesuf}(x,x^{\prime})=P(o^{\prime}_{X=x^{\prime}},o_{X=x}).$ (5)

The necessity score and sufficiency score (called reversal probability)
calculate the probability of the degree of change in the predicted value if
the explanatory variable had a different value, given the value and predicted
value of a given explanatory variable. In particular, the necessity score
represents the probability that the predicted value would change if the value
of the explanatory variable decreased, whereas the sufficiency score
represents the probability that the predicted value would change if the value
increased. The necessity and sufficiency score is a score that balances the
necessity score and sufficiency score and can be considered as a feature
importance in machine learning [10].

In [10], LEWIS was used to examine a binary classification model that
determines whether a loan was approved or rejected. For example,
$\text{Nec}=0.1$ for a feature implies that if a person whose loan is
predicted to be approved decreased the value of a feature, there would be at
most a 10% chance of a prediction for the loan to be rejected. In addition,
$\text{Suf}=0.8$ implies that if a person whose loan is predicted to be
rejected increased the value of a certain feature, there is a maximum
probability of 80% that the loan would be predicted to be approved.

These three scores are expressed as

$\displaystyle\mathrm{Nec}(x,x^{\prime})=\frac{P(o^{\prime}|\mathit{do}(X=x^{\prime}))-P(o^{\prime}|x)}{P(o|x)},$
(6)
$\displaystyle\mathrm{Suf}(x,x^{\prime})=\frac{P(o|\mathit{do}(X=x))-P(o|x^{\prime})}{P(o^{\prime}|x^{\prime})},$
(7)
$\displaystyle\mathrm{Nesuf}(x,x^{\prime})=P(o^{\prime}|\mathit{do}(X=x^{\prime}))-P(o^{\prime}|\mathit{do}(X=x)),$
(8)

if the causal graph $G$ corresponding to the causal model $M$ is known and the
monotonicity $(x>x^{\prime}\Rightarrow O_{x}>O_{x^{\prime}})$ is satisfied. In
this case, the global explanation score of LEWIS, $\mathrm{maxNesuf}(X)$, is
given by the maximum value of $\mathrm{Nesuf}(x,x’)$ for all pairs of the
value of the explanatory variable, $(x,x’)$. Finally, if no causal graph is
provided to LEWIS, it assumes that $P(O|\mathit{do}(X))=P(O|X)$ under the
assumption that there are no confounding factors, which is likely to be
violated.

##  III Numerical experiment of exploring useful prior knowledge on causal
structure

In this section, we analyze the effects of causal structures on explanation
scores and propose useful prior information on the causal structure.

###  III-A Analysis of the influence of causal structure on explanation scores

####  III-A1 Experimental design

We built a machine learning model leveraging data generated from a three-
variable causal structure and analyzed the characteristics of the LEWIS
explanation score estimated for the predicted value. Data were generated from
the five causal graphs in Figure 2. Structures A, B, and E are termed the
collider, chain, and fork paths, respectively, and each exhibited its own
unique independence [12]. Structure C exhibited confounding behavior, where
the variable $X$ was the confounding variable. Structure D contained a
variable $X$ that was independent of variable $Y$. The coefficients of each
structural equation were all equal for each variable in structures B–E, and
the coefficients of A were set to 1 and 1.5 for $X$ and $Z$, respectively.
After generating data from the five causal structures and building a machine
learning model using that data, we estimated the Nesuf score of Equation (5)
with LEWIS. This was iterated 100 times and the characteristics of the average
estimated value were analyzed.

TABLE I: Experimental conditions Function | {linear, nonlinear}
---|---
Probability of error variables | uniform distribution of $[0,1]$
Model | catboost
Sample size | 5000
Discretization method | equal-width discretization
Number of discretization bins | 10

Table I presents the detailed experimental settings. In each repetition, error
variables were generated from a uniform distribution of $[0,1]$ using a linear
or nonlinear (second-order monomial) function system for the causal structures
A–E. The generated data were discretized using the same discretization bin
number, and catboost was used as the classifier. Herein, the target variable
$Y$ was also discretized to binary values using equally spaced discretization.
The sample size of the data was $5\,000$, which Nesuf was estimated for the
predictions of.

Figure 2: Causal graph used in analysis. The values on the directed edges
represent the coefficients of the respective structural equations.

####  III-A2 Experimental results

The estimates of Nesuf are presented in Tables II and III. The results
suggested two possible pieces of prior information on causal structure to
determine Nesuf based on a linear causal structure. First, in case of multiple
variables that are direct parents of a target variable such as structure A,
variables with strong correlations would be more important. In structure A,
the correlation coefficients with $Y$ of variables $Z$ and $X$ were 0.8075 and
0.4779 on average, respectively, indicating that variable $Z$ exhibited a
stronger correlation than $X$ and was therefore more important. Second, if the
target variable is independent and has no directed edge from the variable $X$,
such as in the causal structure D or E, the importance will be zero. Thus, if
an explanatory variable is independent of the target variable, the LEWIS
explanatory score is zero because intervention on that explanatory variable
cannot change the target variable. A similar argument is possible in the case
of nonlinear causal structures to determine Nesuf.

TABLE II: Mean of Nesuf in the case of linear causal structure | A | B | C | D | E
---|---|---|---|---|---
$x$ | 0.367 | 0.498 | 0.999 | 0 | 0
$z$ | 0.999 | 0.999 | 0.325 | 1 | 1
TABLE III: Mean of Nesuf in the case of nonlinear causal structure | A | B | C | D | E
---|---|---|---|---|---
$x$ | 0.377 | 0.556 | 0.999 | 0 | 0
$z$ | 0.945 | 0.999 | 0.949 | 1 | 1

###  III-B Useful prior information on the causal structure

In Section III-A, we observed that the score differed greatly depending on
whether the explanatory variable could yield the predictive variable. In
particular, all explanatory scores will be 0 for the explanatory variables
that are independent of the target variable. In addition, in case of a
directed edge, or reverse causation, from the target variable to the
explanatory variable, the target variable and explanatory variable are
dependent. However, even if the value of the explanatory variables were
changed, the value of the target variable would not be changed; thus, so the
value of Nesuf becomes 0, and Suf and Nec provide no information. Therefore,
we proposed the following prior information on the causal structure to compute
the explanation scores:

  1. (a)

Target variable is the sink variable, where a sink variable is a variable that
does not cause any variable.

  2. (b)

Target variable has the direct parent-child relationship with all explanatory
variables, that is, there is a direct causal path from the explanatory
variables to the target variable.

Prior information (a) (Figure 3a) helps eliminate reverse causality from the
target variable to the explanatory variables when performing causal discovery
including the target variable. With prior information (b) (Figure 3b), the
target variable is dependent on all explanatory variables. This facilitates
the performing of causal discovery using only explanatory variables.

Figure 3: Prior information of causal structure. (a): Target variable Y is the
sink variable. (b): Target variable Y has the direct parent-child relationship
with all explanatory variables.

##  IV Numerical experiment and evaluation

This section describes the simulation and experimental results of explanations
based on counterfactual probabilities through causal discovery using
artificial data.

###  IV-A Numerical experiment setting

In the numerical experiment, we generated artificial data from the 8-variable
causal graph shown in Figure 4, performed causal discovery based on prior
information on the causal structure proposed in the previous section, and
estimated the Nesuf for the predicted values. We performed evaluations
considering the order of estimated the Nesuf and that of true Nesuf and the
error of the estimated Nesuf value. We compared these results with the value
of Nesuf estimated assuming that $P(Y|\mathit{do}(X))=P(Y|X)$, similar to that
in case of [10]. The artificial data to be generated were continuous and mixed
data. Continuous variable values were generated from linear or non-linear
function systems. For mixed data, only a linear system was used, the
probability distribution of the error variables $X_{1}$ and $X_{7}$ was a
Bernoulli distribution, and the probability distribution of the other error
variables were a uniform or Gaussian distribution. The model, sample size,
discretization method, and the number of discretization bins are presented in
Table I.

Figure 4: Causal graph in artificial data experiments

For the evaluation, we calculated the mean absolute error (MAE) with respect
to the true Nesuf estimate. We used $\overline{\mathrm{MAE}}$, which is the
average of the MAE

$\mathrm{MAE}=\frac{1}{N}\sum_{i=1}^{N}|\mathrm{maxNesuf}_{X_{i},\mathrm{true}}–\mathrm{maxNesuf}_{X_{i},\mathrm{est}}|$
(9)

for seven variables, $X_{1},\,\ldots,\,X_{7}$, where $N$ is the number of
experimental trials, $\mathrm{maxNesuf}_{X_{i},\mathrm{true}}$ is the true
value of $\mathrm{maxNesuf}(X_{i})$, and
$\mathrm{maxNesuf}_{x_{i},\mathrm{est}}$ is the estimated value of
$\mathrm{maxNesuf}(X_{i})$. The standard error of $\mathrm{maxNesuf}(X_{i})$
was also evaluated.

Here, we assume that $\mathbf{a}=\\{a_{1},\ldots,a_{n}\\}$ and
$\mathbf{b}=\\{b_{1},\ldots,b_{n}\\}$ convert the value of the true Nesuf and
the estimated Nesuf into a rank for $n$ variables, respectively. Defining
$\mathbf{d}=\\{d_{1},\ldots,d_{n}\\}$ by $d_{i}=a_{i}-b_{i}$, we can calculate
Spearman’s rank correlation coefficient [13] with

$\mathrm{SPR}=1-\frac{6\sum_{i=1}^{n}d_{i}^{2}}{n(n^{2}-1)},$ (10)

where $\mathrm{SPR}$ attains the value $-1\leqq\mathrm{SPR}\leqq 1$. The
closer it is to 1, the more the order of the estimated Nesuf matches the true
order of feature importance. Moreover, the closer it is to $-1$, the more the
order relationship is reversed. In this experiment, the evaluation was based
on the average MAE, the standard error and the average
$\overline{\mathrm{SPR}}$ of 100 trials.

Next, for causal discovery, we used four methods tailored to specific
functional forms. We implemented the causal discovery algorithm in Python. We
used the causal-learn [14] method for PC and linear non-Gaussian acyclic model
(LiNGAM) package [15] for DirectLiNGAM, RESIT, and linear mixed (LiM).

  * •

PC [16]: The PC algorithm is a causal discovery algorithm based on conditional
independence. The estimation algorithm first removes undirected edges by
testing conditional independence. The direction of the causal relationship is
determined by using v-structures [12] and orientation rule [17] for the
remaining undirected skeleton. When using a PC to orient undirected edges, the
directions may not be determined in the end, and the graph may be estimated as
a partially oriented graph. In that case, our evaluation was based on the
causal graph with all directed edges derived from the partially oriented graph
that had the maximum (PC_Max) or minimum (PC_Min) Spearman’s rank correlation
coefficient. We used the Fisher-$z$ test to investigate the conditional
independence.

  * •

DirectLiNGAM [18]: DirectLiNGAM is a causal discovery algorithm that expresses
causal relationships using a linear structural equation model called LiNGAM
[19], which assumes that the graph is acyclic and that the probability
distribution of the error terms is non-Gaussian. DirectLiNGAM estimates the
causal structure by repeated regression and evaluating the independence of
residuals of each variable.

  * •

RESIT [20]: RESIT is a causal discovery algorithm used when the causal
structure is nonlinear and the error variables are additive (Additive Noise
Model: ANM). Similar to DirectLiNGAM, the algorithm estimates the causal
direction by evaluating the independence of each variable and the residual of
nonlinear regressions.

  * •

LiM [21]: LiM causal discovery algorithm extends LiNGAM to handle the mixed
data that comprises both continuous and discrete variables. The estimation is
performed by first globally optimizing the log-likelihood function on the
joint distribution of data with the acyclicity constraint and then applying a
local combinatorial search to output a causal graph.

The following estimates were based on either prior information (a), prior
information (b), or no prior information, denoted by (a), (b), and (0),
respectively. PC and DirectLiNGAM can incorporate prior information (a) and
(b). However, RESIT and LiM cannot incorporate prior information (a). In
addition, we evaluated cases wherein causal discovery was performed without a
causal graph (No graph). Table IV presents the experimental settings used in
the numerical experiment.

TABLE IV: Experimental settings for artificial data

Data | {continuous, mix}
---|---
Function | {linear, nonlinear}
Probability of error variables | {uniform, Gaussian}
Sample size | 5000
Discretization method | equal-width discretization
Number of discretization bins | 10
Model | catboost
Causal discovery | {PC, DirectLiNGAM, RESIT, LiM}
Prior knowledge | {(0), (a), (b), (No graph)}

###  IV-B Results of experiment

The experimental results of $\overline{\mathrm{MAE}}$ $\pm$ standard error and
$\overline{\mathrm{SPR}}$ are presented in Tables V–X. First, regarding the
linear causal structure in Tables V and VI, when the causal discovery matched
the true graph, the $\overline{\mathrm{MAE}}$ approached 0 and the
$\overline{\mathrm{SPR}}$ approached 1. DirectLiNGAM assumed that the
functional relationship was linear and that the probability distribution of
the error terms was non-Gaussian. In such cases, $\overline{\mathrm{MAE}}$ was
smaller and the $\overline{\mathrm{SPR}}$ was greater than that in the case No
graph (Table V). As shown in Table VI, when a Gaussian distribution was
assumed with no prior information or No graph, the $\overline{\mathrm{MAE}}$
was large and the $\overline{\mathrm{SPR}}$ was small for DirectLiNGAM.
However, these scores were improved with prior information (a) and (b).
Furthermore, the PC algorithm consistently achieved high accuracy for both
distributions even in the absence of causal discovery. In all cases with a
linear structure, we could estimate the true value and order of Nesuf scores
better than that case when No graph was assumed.

TABLE V: Results of linear and uniform distribution

Method and prior information | $\overline{\mathrm{MAE}}$ | $\overline{\mathrm{SPR}}$
---|---|---
DirectLiNGAM (0) | 0.0032 ± 0.0197 | 0.9889
DirectLiNGAM (a) | 0.0032 ± 0.0197 | 0.9889
DirectLiNGAM (b) | 0.1700 ± 0.0023 | 0.6140
PC_Max (0) | 0.0633 ± 0.0595 | 0.9007
PC_Min (0) | 0.0953 ± 0.1127 | 0.8507
PC_Max (a) | 0.0092 ± 0.0249 | 0.9817
PC_Min (a) | 0.0703 ± 0.0656 | 0.9082
PC_Max (b) | 0.1709 ± 0.0030 | 0.6117
PC_Min (b) | 0.1709 ± 0.0030 | 0.6117
No graph | 0.1709 ± 0.0030 | 0.6117

Values presented in bold indicate the best results.

TABLE VI: Results of linear and Gaussian distribution Method and prior information | $\overline{\mathrm{MAE}}$ | $\overline{\mathrm{SPR}}$
---|---|---
DirectLiNGAM (0) | 0.3605 ± 0.1290 | 0.3307
DirectLiNGAM (a) | 0.2846 ± 0.1080 | 0.7403
DirectLiNGAM (b) | 0.1910 ± 0.0100 | 0.5920
PC_Max (0) | 0.0981 ± 0.0461 | 0.8796
PC_Min (0) | 0.1186 ± 0.0920 | 0.8492
PC_Max (a) | 0.0088 ± 0.0287 | 0.9764
PC_Min (a) | 0.1013 ± 0.0584 | 0.8971
PC_Max (b) | 0.1912 ± 0.0097 | 0.5550
PC_Min (b) | 0.1912 ± 0.0097 | 0.5550
No graph | 0.1912 ± 0.0097 | 0.5550

Tables VII and VIII present the results in the case of a nonlinear causal
structure. When the error term was nonlinear and had a uniform distribution,
RESIT and PC yielded smaller $\overline{\mathrm{MAE}}$ values and greater
$\overline{\mathrm{SPR}}$ values than that for No graph in Table VII. However,
as shown in Table VIII, no improvement in accuracy was observed with each
prior information of causal structure for Gaussian distributions.

TABLE VII: Results of nonlinear and uniform distribution Method and prior information | $\overline{\mathrm{MAE}}$ | $\overline{\mathrm{SPR}}$
---|---|---
RESIT (0) | 0.0250 ± 0.0190 | 0.8114
RESIT (b) | 0.0334 ± 0.0059 | 0.7285
PC_Max (0) | 0.0398 ± 0.0273 | 0.5540
PC_Min (0) | 0.0398 ± 0.0273 | 0.5540
PC_Max (a) | 0.0183 ± 0.0226 | 0.7781
PC_Min (a) | 0.0183 ± 0.0226 | 0.7781
PC_Max (b) | 0.0337 ± 0.0064 | 0.7781
PC_Min (b) | 0.0337 ± 0.0064 | 0.7781
No graph | 0.0337 ± 0.0064 | 0.7067
TABLE VIII: Results of nonlinear and Gaussian distribution Method and prior information | $\overline{\mathrm{MAE}}$ | $\overline{\mathrm{SPR}}$
---|---|---
RESIT (0) | 0.0292 ± 0.0115 | 0.3296
RESIT (b) | 0.0377 ± 0.0128 | 0.3235
PC_Max (0) | 0.0323 ± 0.0177 | 0.1853
PC_Min (0) | 0.0325 ± 0.0179 | 0.1810
PC_Max (a) | 0.0325 ± 0.0179 | 0.1810
PC_Min (a) | 0.0325 ± 0.0179 | 0.1810
PC_Max (b) | 0.0372 ± 0.0128 | 0.3317
PC_Min (b) | 0.0372 ± 0.0128 | 0.3317
No graph | 0.0372 ± 0.0128 | 0.3317

Finally, Tables IX and X present the results of the mixed data. In Table IX,
the assumptions of LiM were satisfied, thus, the $\overline{\mathrm{MAE}}$ was
smaller and the $\overline{\mathrm{SPR}}$ was larger than that when no causal
discovery was performed. In addition, in Table X, the LiM assumption that the
probability distribution of the error terms was non-Gaussian was not
satisfied, the score was the same as the case of No graph. Even in the case of
linear mixed data, it can be said that the true Nesuf can be partially
restored using the proposed prior information of causal structure.

TABLE IX: Results of linear and uniform distribution in mix data Method and prior information | $\overline{\mathrm{MAE}}$ | $\overline{\mathrm{SPR}}$
---|---|---
LiM (0) | 0.1234 ± 0.0287 | 0.6571
LiM (b) | 0.1380 ± 0.0195 | 0.6892
No graph | 0.1741 ± 0.0145 | 0.4292
TABLE X: Results of linear and Gaussian distribution in mix data Method and prior information | $\overline{\mathrm{MAE}}$ | $\overline{\mathrm{SPR}}$
---|---|---
LiM (0) | 0.1726 ± 0.0184 | 0.4350
LiM (b) | 0.1540 ± 0.0234 | 0.4635
No graph | 0.1546 ± 0.0028 | 0.4603

##  V Application to real data

In this section, we demonstrate the effectiveness of this framework for the
case where the causal graph is unknown by applying our method to real data.

###  V-A Dataset and preprocessing

We applied our method to the anonymized credit rating data of $14\,018$
business customers provided by the Shiga Bank, Ltd. A credit rating was
assigned by a bank to a debtor based on the analysis of its financial
statements [22]. Although there were several grades in the credit rating, we
simplified the grades to high (excellent) and low (poor), which facilitated
its modeling as a binary classification problem. We used the industry type,
amount of capital stock, number of employees, most recent annual sales, and
total liabilities and equity as the explanatory variables. We performed equal-
frequency discretization with 10 bins because the capital stock, number of
employees, most recent annual sales, and total liabilities and equity had
highly skewed distributions. These discretized variables are ordinal measures
with enough many levels and can be seen as continuous variables. As a machine
learning model, we used Random Forest, which is an algorithm that performs
classification by combining the results of multiple decision trees constructed
from randomly selected learning data and explanatory variables; it is an
ensemble learning algorithm [23]. Random Forest was expected to have
sufficient predictive accuracy for real data in this study.

Although the data involved mixed data where only the industry type is discrete
variable and the others are regarded as continuous variables, the industry
type could not be caused by the other explanatory variables. Thus, we used
DirectLiNGAM for continuous variables, assuming that industry type was an
exogenous discrete variable that affected all other variables. In this case,
DirectLiNGAM can handle discrete variable similarity to the conventional
structural equation modeling [24]. Because we aimed at a quantitative analysis
of all explanatory variables, we conducted our analysis assuming that the
target variable exhibited a direct parent-child relationship with all
explanatory variables in the causal structure, that is, prior information (b).

###  V-B Analysis results and discussion

The black lines of Figure 5 show the results with DirectLiNGAM for the entire
dataset. The interpretation of the causal direction of the results is as
follows. The causal direction from capital stock to total liabilities and
equity was consistent with domain knowledge because the total liabilities and
equity are the sum of debt and equity on the balance sheet. In addition, the
causal direction from capital stock to sales was consistent with the domain
knowledge that companies conducted business activities based on capital and
that this influenced sales. These variables can affect the credit rating on
the basis of prior information of the causal structure in Figure 5 (red
lines).

Figure 5: Causal graph estimated by DirectLiNGAM (black lines). Prior
information on the causal structure (red lines).

Next, Nesuf estimated using the causal graph and No graph is shown in Figure
6. The importance ranking of each of these variables obtained by LEWIS can be
explained as follows using the causal graph shown in Figure 5. The industry
type was an exogenous variable that affected the other four variables on the
causal graph. If the industry type were to be changed, the values in the other
four variables would also change, which is likely to affect the final
predicted value. Thus, its importance in LEWIS is the greatest. Similarly, the
value of capital stock changed the value of the sales and total liabilities
and equity, and the total liabilities and equity changed the value of sales.
The number of employees and sales were less important because changing these
values did not change the values of other variables. However, the Nesuf scores
estimated with No graph were smaller, and those of the capital stock in
particular were significantly smaller, which contradicted domain knowledge.

Figure 6: Nesuf estimated from causal graph and No graph

The LEWIS reversal probability score is shown in Figure 7. Overall, the Suf
score (orange) was high, and there was a high probability that changing the
value of the variable would change the company from a low-rated one to a high-
rated one. Industry type also significantly affected the predicted value if it
were changed. Reversal probabilities can assist the decision-making about
which variables are likely to change the rating for a low-rated company and
can quantify the company’s strengths and weaknesses based on each score.

Figure 7: Reversal probability score estimated from the causal graph. Nec
(blue) is the probability that the prediction would change by lowering the
value of that variable for a company whose rating is predicted to be high. Suf
(orange) is the probability that the prediction would change by increasing the
value of the variable for a company whose rating is predicted to be low.

##  VI Conclusion

This study proposed a new causal XAI framework that combined causal structure
information and causal discovery without the knowledge of the causal graph. We
analyzed the global explanation scores by using counterfactual explanations
based on the causal structure and proposed prior information on the causal
structure. Numerical experiments demonstrated the possibility of estimating
the global explanatory score and the order of the true feature importance even
if the causal graph was not fully known. By applying our method to real data,
we demonstrated the usefulness of the proposed framework even if the causal
graph is unknown.

As an extension of LEWIS, a method for multi-class classification was proposed
[10]. However, whether proposed prior information on causal structure is valid
even for the multi-class classification remains an open question.

## Acknowledgment

We would like to express our gratitude to Shiga Bank, Ltd. for their valuable
comments on this study and for providing us with the data used.

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