# Dynamically Consistent Approximate Rational Solutions to the Thomas-Fermi
Equation

Ronald E. Mickens
Department of Physics
Clark Atlanta University
Atlanta, GA 30314 USA
Email<EMAIL_ADDRESS>
and
Isom H. Herron
Department of Mathematical Sciences
Rensselaer Polytechnic Institute
Troy, NY 12180 USA
Email<EMAIL_ADDRESS>
###### Abstract.

We construct two rational approximate solutions to the Thomas-Fermi (TF)
nonlinear differential equation. These expressions follow from an application
of the principle of dynamic consistency. In addition to examining differences
in the predicted numerical values of the two approximate solutions, we compare
these values with an accurate numerical solution obtained using a fourth-order
Runge-Kutta method. We also present several new integral relations satisfied
by the bounded solutions of the TF equation

###### Key words and phrases:

Thomas-Fermi equation; atomic structure; rational approximate solutions;
dynamic consistency

###### 2010 Mathematics Subject Classification:

34A45, 34B40, 34E05, 41A20, 81V45

##  1\. Introduction

The major purpose of this paper is to construct a new rational approximation
to the Thomas-Fermi equation

(1.1)
$\frac{d^{2}y\left(x\right)}{dx^{2}}=\frac{y\left(x\right)^{\frac{3}{2}}}{x^{\frac{1}{2}}},$

with boundary conditions

(1.2) $y\left(0\right)=1,y\left(\infty\right)=0.$

This equation arises in the modeling of certain phenomena in the atomic
structure of matter ([1], [2]).

Since it is expected that no exact analytical solution formula exists for the
general solution in terms of a finite combination of elementary functions, a
broad range of approximate solutions have been derived ([3], [4], [5], [6],
[7], [8]). The mathematical procedures used to obtain these expressions
include variational methods ([3], [6]), the use of iteration techniques [4],
application of the Adomian decomposition method [5], homotopy analysis [7],
and rational approximations [8]. The indicated references ([3], [4], [5], [6],
[7], [8]) are but a small subset of the hundreds of publications on this
subject. Note that the existence of such a substantial research literature is
due, in large part, to the fact that this thorny boundary-value problem has
drawn out a significant number of ideas as to how approximations to its
solutions should be done.

The purpose of this paper is to use a new methodology to construct approximate
solutions to the Thomas-Fermi (TF) equation. We do this within the framework
of the concept or principle of dynamic consistency [9] and the restrictions
that it imposes on any mathematical representation of the forms used for
rational approximations to a TF equation solution. We demonstrate that this is
rather easy to do.

In outline, Section 2 provides a summary of several of the general, exact, and
for the most part qualitative properties of the TF equation [10]. This section
includes several new results of the authors. Section 3 introduces and defines
the concept of dynamic consistency [9] and it is used in Section 4 to
construct two elementary rational approximations to the TF solution. Finally,
in the last section, we provide a summary of our results and discuss possible
extensions to this work.

##  2\. Preliminaries: Exact results

It turns out that even in the absence of knowledge of an explicit, exact
solution for the TF equation many of the essential properties of such a
solution can be derive. Below, we give a concise summary of these items and
refer the reader to the paper of Hille [10] for proofs of some of these
statements.

a) An exact solution to the TF equation is

(2.1) $y\left(x\right)\equiv y_{s}\left(x\right)=\frac{144}{x^{3}}.$

Note that it contains no arbitrary parameters and thus is not a special case
of the (unknown) general solution. We denote it as, $y_{s}\left(x\right)$ and
call it a singular or asymptotic solution in the sense that given a solution
$y\left(x\right)$ then [10],

(2.2) $\lim_{x\rightarrow\infty}(x^{3}y\left(x\right))=144.$

b) The curve $y\left(x\right)\equiv y_{s}\left(x\right)\ $separates the
bounded and unbounded solutions of the TF equation. This is evident from
inspection of the flow-space as indicated in Figure 1.

Figure 1. Singular solution $y_{s}$ separating unbounded and bounded solutions

Note that since $y\left(x\right)\equiv 0$, is a solution, it follows that all
solutions below the $y=y_{s}\left(x\right)$ curve are bounded and all
solutions above are unbounded. The bounded solutions have a finite value for
$y\left(0\right),$ while the unbounded have $y\left(0\right)=+\infty$.

c) The physical phenomena for which the TF equation was constructed requires

(2.3) $\displaystyle y\left(x\right)$
$\displaystyle>0,y^{\prime}\left(x\right)\equiv\frac{dy\left(x\right)}{dx}<0,0\leq
x\,<\infty,$ (2.4) $\displaystyle y\left(0\right)$ $\displaystyle=1,\
y\left(\infty\right)=0.$

Observe that since $y^{\prime\prime}\left(x\right)>0,0<x<\infty,$ then all
solutions are concave upward and this is consistent with the trajectory-flow
given in Figure 1.

The figure also provides an overview of the bounded and unbounded trajectories
of the TF equation.

d) For the bounded solutions

(2.5)
$0<y\left(x\right)<y\left(0\right)<\infty,y^{\prime}\left(0\right)<y^{\prime}\left(x\right)<0,y^{\prime\prime}\left(x\right)>0,0<x\,<\infty.$

Further we have

(2.6)
$y^{\prime\prime}\left(x\right)=O\left(\frac{1}{\sqrt{x}}\right),x\rightarrow
0^{+}.$

e) It is remarkable that $y^{\prime}\left(0\right)$ can be calculated to
essentially any decimal place of accuracy [11]; defining the constant $B$ as
follows,

(2.7) $y^{\prime}\left(0\right)=-B,$

we have [11]

(2.8) $B=1.588071022611375\ldots.$

f) For small $x$, $y\left(x\right)$ has the representation [4] for
$y\left(0\right)=1$ and $y^{\prime}\left(0\right)=-B,$

$\displaystyle y\left(x\right)$
$\displaystyle=\left[1-Bx+\left(\frac{1}{3}\right)x^{3}-\left(\frac{2B}{15}\right)x^{4}+\cdots\right]$
(2.9)
$\displaystyle\cdots+x^{3/2}\left[\frac{4}{3}-\left(\frac{2B}{5}\right)x+\left(\frac{3B^{2}}{70}\right)x^{2}\right.+\left.\left(\frac{2}{27}+\frac{B^{3}}{252}\right)x^{3}+\cdots\right].$

g) The following “sum-rules”, involving integrals of $y\left(x\right),$have
been derived by the authors. As far as we are aware, most of these are new
results pertaining to the solution (see Eqs. (1.1) and (1.2)) of the TF
equation:

(2.10) $\int_{0}^{\infty}\sqrt{x}y\left(x\right)^{3/2}dx=1,$

(was derived in [15])

(2.11)
$\int_{0}^{\infty}y\left(x\right)dx=\frac{1}{2}\int_{0}^{\infty}x^{\frac{3}{2}}y\left(x\right)^{\frac{3}{2}}dx,$
(2.12)
$B=\int_{0}^{\infty}\left[\left(\frac{dy}{dx}\right)^{2}+\frac{y^{\frac{5}{2}}}{\sqrt{x}}\right]dx$
(2.13) $B=\int_{0}^{\infty}\frac{y\left(x\right)^{3/2}}{\sqrt{x}}dx.$

Let us derive the result in Eq. (2.12). Multiply both sides of the TF equation
by $y\left(x\right):$

(2.14) $yy^{\prime\prime}=\frac{y^{\frac{5}{2}}}{\sqrt{x}}.$

Integrating once by parts and evaluating at $x=0$ and $x=\infty,$ gives

(2.15)
$y\left(\infty\right)y^{\prime}\left(\infty\right)-y\left(0\right)y^{\prime}\left(0\right)-\int_{0}^{\infty}\left(y^{\prime}\left(x\right)\right)^{2}dx=\int_{0}^{\infty}\frac{y^{\frac{5}{2}}}{\sqrt{x}}dx.$

But $y\left(0\right)=1,y^{\prime}\left(0\right)=-B$ and
$y\left(\infty\right)=y^{\prime}\left(\infty\right)=0;$ therefore, the result
listed in Eq. (2.12) is obtained. The other relations can be derived using
similar techniques.

Finally, we go back to Eq. (2.9) and demonstrate one possible way to derive
it. First, observe that for small $x,$ the TF equation is approximated by the
expression

(2.16) $y^{\prime\prime}\left(x\right)\simeq\frac{1}{\sqrt{x}},$

where $y\left(0\right)=1$ and $y^{\prime}\left(0\right)=-B.$ Integrating twice
and imposing the just stated initial conditions gives

(2.17) $y\left(x\right)\simeq 1-Bx+\left(\frac{4}{3}\right)x^{\frac{3}{2}},$

corresponding to the first three terms of (2.9).

Let us now construct an iteration scheme to obtain a small$\ x$ representation
for the TF equation. We assume it has the form

(2.18) $\displaystyle y_{N+1}^{\prime\prime}(x)$
$\displaystyle=\frac{\left[y_{N}\left(x\right)\right]^{\frac{3}{2}}}{\sqrt{x}},\
N=0,1,2,\ldots;$ (2.19) $\displaystyle y_{N}\left(0\right)$
$\displaystyle=1,y_{N}^{\prime}\left(0\right)=-B,\ $ (2.20) $\displaystyle
y_{0}\left(x\right)$ $\displaystyle=1.$

Note that $y_{1}\left(x\right)$ is the result given in Eq. (2.17).

To calculate $y_{2}\left(x\right)$, we start with

$\displaystyle y_{2}^{\prime\prime}(x)$
$\displaystyle=\frac{\left[y_{1}\left(x\right)\right]^{\frac{3}{2}}}{\sqrt{x}}$
(2.21)
$\displaystyle=\frac{\left[1-Bx+\left(\frac{4}{3}\right)x^{\frac{3}{2}}\right]^{\frac{3}{2}}}{\sqrt{x}}.$

Expanding the expression in the brackets and retaining the first two terms
gives

(2.22)
$\left[1-Bx+\left(\frac{4}{3}\right)x^{\frac{3}{2}}\right]^{\frac{3}{2}}=1-\left(\frac{3B}{2}\right)x+\cdots\
.$

Thus it follows that

(2.23)
$y_{2}^{\prime\prime}\left(x\right)=\frac{1}{\sqrt{x}}-\left(\frac{3B}{2}\right)\sqrt{x}.$

Integrating this equation twice, with the imposition of the initial conditions
$y_{2}\left(0\right)=1,y_{2}^{{}^{\prime}}\left(0\right)=-B,$ gives

(2.24)
$y\left(x\right)=1-Bx+\left(\frac{4}{3}\right)x^{\frac{3}{2}}-\left(\frac{2B}{5}\right)x^{\frac{5}{2}}.$

The result corresponds, respectively, to the first two terms in the brackets
of Eq. (2.21).

Finally, with careful attention to which terms to retain, at each step of the
iteration given by Eq. (2.18-2.20), the full result presented in Eq. (2.9) can
be produced.

##  3\. Dynamic consistency

Consider two systems $A$ and $B.$ Let system $A$ have the property $P.$ If
system $B$ also has property $P,$ then $B$ is said to be dynamic consistent to
$A$, with respect to property $P\ $[9].

Dynamic consistency (DC) has served as one of the fundamental principles of
the nonstandard, finite difference methodology for constructing improved
discrete models for the numerical integration of differential equations [9].
It serves as an assessment of the “closeness” or “fidelity” of two systems
based on their mutually shared properties. Note that the systems $A$ and $B$
may not be of the same type object, structure or thing. For example, $A$ might
be a physical system, while $B$ could be a mathematical model of $A.\
$Likewise, $A$ might be a system of differential equations and $B$ a
particular discretization of these equations. Thus DC is of general
applicability and is available for use in diverse situations.

In the next section, we provide two ansatzes for approximate solutions to the
TF equation, Our proposed solutions are to be DC with respect to the following
properties of the exact, but unknown, solution to the second-order, nonlinear
TF differential equation:

(3.1) $\text{(i)}\ \ y\left(0\right)=1,y^{\prime}\left(0\right)=-B;$ (3.2)
$\text{(ii)}\ \ 0<y\left(0\right)\leq 1,-B\leq
y^{\prime}\left(x\right)<0,0\leq x<\infty;$ (3.3) $\text{(iii)}\ \
\lim_{x\rightarrow\infty}\left(x^{3}y\left(x\right)\right)=144\ ;\ $

(iv) the approximate solutions $y_{a}\left(x\right)$ are to be taken as
rational functions of either $x$ or $\sqrt{x}$;

(v) the approximate solutions$,y_{a}\left(x\right)$, are to have one or the
other of the following forms for small $x$;

(3.4) $y_{a}\left(x\right)=\left\\{\begin{array}[c]{c}1-Bx,\ \ \text{rational
in }x;\\\ 1-Bx+\left(\frac{4}{3}\right)x^{3/2},\text{ rational in
}\sqrt{x}\end{array}\right.$

##  4\. Approximate solutions

We take the following two ansatzes to represent approximations to the solution
of the TF equation

(4.1)
$y_{a}^{\left(1\right)}(x)=\frac{1}{1+Bx+\left(\frac{1}{144}\right)x^{3}},$
(4.2) $\displaystyle y_{a}^{\left(2\right)}\left(x\right)$
$\displaystyle=\frac{1}{1+Bx-\left(\frac{4}{3}\right)x^{3/2}+Cx^{2}+\left(\frac{1}{144}\right)x^{3}},$
(4.3) $\displaystyle C$ $\displaystyle=\frac{B^{2}}{2},$

where $B=1.588071$, and as a consequence of (4.3), $C=1.260985.$

Close inspection of $\ y_{a}^{\left(1\right)}(x)$ and
$y_{a}^{\left(2\right)}\left(x\right)$ shows that they are DC with the five
conditions listed in section 3. Observe that
$y_{a}^{\left(1\right)}\left(x\right)$ is a rational function of $x$, while
$y_{a}^{\left(2\right)}\left(x\right)$ is a rational function of $\sqrt{x}.$

Figure 2. Plots of $y_{a}^{\left(1\right)}\left(x\right)$
$\left({}^{\\_\\_\\_\\_\\_}\right)\ $and
$y_{a}^{\left(2\right)}\left(x\right)\ \left(\cdot\cdot\cdot\cdot\right)$

Figure LABEL:fig:2 gives plots of $y_{a}^{\left(1\right)}(x)$ and
$y_{a}^{\left(2\right)}(x)$ vs. $x$. The two curves lie close to each other,
but do not coincide, i.e.

(4.4)
$\left\\{\begin{array}[c]{c}y_{a}^{\left(1\right)}(x)<y_{a}^{\left(2\right)}(x),\
0<x<x_{0},\\\ y_{a}^{\left(1\right)}(x)>y_{a}^{\left(2\right)}(x),\ \
x>x_{0}\end{array}\right.$

with $x_{0}\approx 1.$

##  5\. Discussion

In summary we have constructed two representations of approximate solutions to
the TF equation. However, it must be stated that this equation is itself an
approximation to an (unknown) equation describing atomic quantum phenomena
([1], [2]). In addition, all numerical procedures have errors associated with
their algorithms and the hardware on which they are implemented.

The methodology used to generate the ansatzes, given in Eqs. (4.1) and (4.2,
4.3), is based on constructing for $y\left(x\right)$, the exact solution,
rational functions such that the five conditions listed in section 3 are
satisfied. An important benefit of this procedure is that the process is
essentially algebraic and requires the introduction of no parameters which
must be determined later in the construct. The $y_{a}^{\left(1\right)}(x)$ and
$y_{a}^{\left(2\right)}(x)$ are constructed to be consistent with the
$x-$small and $x-$large asymptotics. Thus, these representations possess great
simplicity both in their manner of construction, as well as the lack of a need
to introduce new parameters.

In Table 1 we compare the numerics of a fourth-order Runge-Kutta method and
the values of $y\left(x\right)$ as calculated from our formulae for
$y_{a}^{\left(1\right)}(x)$ and $y_{a}^{\left(2\right)}(x)$. Clearly, both
analytic approximations and the numerical integration results are consistent,
and in fact are in very good qualitative agreement with each other.

Table - 1 Numerical values of $y\left(x\right)$ and $y_{a}\left(x\right).$
$x$ $y\left(x\right)$ numerical [12] $y_{a}^{\left(1\right)}\left(x\right)$
$0.0$ $1.0000$ $1.0000$ $0.5$ $0.6070$ $0.5571$ $1.0$ $0.4240$ $0.3853$ $2.0$
$0.2430$ $0.2363$ $4.0$ $0.1840$ $0.1283$ $5.0$ $0.0789$ $0.1020$ $10.0$
$0.0243$ $0.0420$ $25.0$ $0.0035$ $0.0067$ $40.0$ $0.0011$ $0.0020$
$y_{a}^{\left(2\right)}\left(x\right)$ $1.0000$ $0.6102$ $0.3964$ $0.1817$
$0.0578$ $0.0378$ $0.0093$ $0.0013$ $0.0005$

Since the main task has been achieved, i.e., to demonstrate through explicit
construction an alternative methodology for finding rational approximations
for the solution of the TF equation, we now can proceed to the next stage,
namely, determine a restriction or condition allowing us to access the overall
validity or accuracy of our solutions. To do this, we consider the integral
relations stated in Section 2. For our purposes, we select the sum-rule given
in Eq. (2.12), mainly because it takes the form of an energy type integral
[13], widely used throughout physics. In fact, inspection of Eq. (2.12)
suggests the following correspondence

$\left(\frac{dy}{dx}\right)^{2}\longrightarrow\text{ kinetic energy,}$
$\frac{y^{\frac{5}{2}}}{\sqrt{x}}\longrightarrow\ \text{potential energy.}$

Note that the result in Eq. (2.12) can be used as a consistency check, since
$B$ occurs in $y\left(x\right)$ which appears in the integral. Using
$y_{a}^{\left(1\right)}(x)$ and $y_{a}^{\left(2\right)}(x)$ and carrying out
the required numerical integration we find, respectively

$B_{1}=1.584744,\ \ B_{2}=1.592931,$

which are to be compared to the input value of

$B=1.588071.$

The fractional percentage errors are

$E_{1}=\left(\frac{B-B_{1}}{B}\right)\cdot 100=0.21\%,\
E_{2}=\left(\frac{B-B_{2}}{B}\right)\cdot 100=-0.27\%.$

Thus, with regard to the sum-rule, stated in Eq. (2.12), both rational
approximations are essentially the same in their ability to satisfy the sum-
rule.

Eventually, the next step in this research is to construct more complex
rational approximations for the solution of the TF equation which incorporate
further terms in the known asymptotic expansions for both $x-$small and
$x-$large using Padé approximants [14].

##

## References

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# Mitigating Health Disparities in EHR via Deconfounder

Zheng Liu<EMAIL_ADDRESS>University of Illinois at ChicagoChicagoUSA ,
Xiaohan Li<EMAIL_ADDRESS>University of Illinois at ChicagoChicagoUSA and
Philip Yu<EMAIL_ADDRESS>University of Illinois at ChicagoChicagoUSA

(2022)

###### Abstract.

Health disparities, or inequalities between different patient demographics,
are becoming a crucial issue in medical decision-making, especially in
Electronic Health Record (EHR) predictive modeling. In order to ensure the
fairness of sensitive attributes, conventional studies mainly adopt
calibration or re-weighting methods to balance the performance on among
different demographic groups. However, we argue that these methods have some
limitations. First, these methods usually mean making a trade-off between the
model’s performance and fairness. Second, many methods attribute the existence
of unfairness completely to the data collection process, which lacks
substantial evidence. In this paper, we provide an empirical study to discover
the possibility of using deconfounder to address the disparity issue in
healthcare. Our study can be summarized in two parts. The first part is a
pilot study demonstrating the exacerbation of disparity when unobserved
confounders exist. The second part proposed a novel framework, Parity Medical
Deconfounder (PriMeD), to deal with the disparity issue in healthcare
datasets. Inspired by the deconfounder theory, PriMeD adopts a Conditional
Variational Autoencoder (CVAE) to learn latent factors (substitute
confounders) for observational data, and extensive experiments are provided to
show its effectiveness.

health disparity, fairness, deconfounder, deep generative model

††copyright: acmcopyright††journalyear: 2022††copyright:
acmcopyright††conference: 13th ACM International Conference on Bioinformatics,
Computational Biology and Health Informatics; August 7–10, 2022; Northbrook,
IL, USA††booktitle: 13th ACM International Conference on Bioinformatics,
Computational Biology and Health Informatics (BCB ’22), August 7–10, 2022,
Northbrook, IL, USA††price: 15.00††doi: 10.1145/3535508.3545516††isbn:
978-1-4503-9386-7/22/08††submissionid: 127††ccs: Applied computing Health
informatics††ccs: Social and professional topics User characteristics††ccs:
Computing methodologies Machine learning

##  1\. Introduction

Machine learning models have demonstrated the promising potential on
Electronic Health Records (EHRs), such as risk prediction (Ma et al., 2018;
Cheng et al., 2016), auxiliary diagnosis (Rajkomar et al., 2018b), and
automated prescription (Zhang et al., 2017). However, recent evidence shows
these models also exacerbate bias and disparities in healthcare, which raises
considerable concern and criticism (Chen et al., 2021; Pfohl et al., 2021;
Chen et al., 2019, 2020b; Zhou et al., 2021; Mhasawade et al., 2021;
Gianfrancesco et al., 2018). Many studies have shown that a machine learning
model may provide disparate results for people with different backgrounds. For
example, for patients of different races, the accuracy and quality of a
machine learning model’s prediction may vary significantly (Obermeyer et al.,
2019). This disparity reduces the utility of machine learning models and is
especially detrimental to disadvantaged and underrepresented populations (Chen
et al., 2021; Pfohl et al., 2021).

To deal with this fairness issue, conventional studies adopt certain criteria
such as equalized odds (Hardt et al., 2016) or counterfactual fairness (Kusner
et al., 2017) to mitigate the disparities (Pfohl et al., 2019; Kamiran and
Calders, 2012; Pleiss et al., 2017). However, we argue that there are
limitations to these approaches. First, maintaining fairness usually means
sacrificing some precision of the model, especially when the model
penalizes/calibrates the majority of datapoints (Feldman et al., 2015; Pleiss
et al., 2017). If a model’s performance deteriorates after adopting certain
fairness criteria, the utility of the model does not necessarily improve.
Second, many methods attribute the existence of unfairness completely to the
data collection process, which lacks substantial evidence. For example, many
causality-based fairness methods assumes “no unobserved confounders”
assumption (Dwork et al., 2012; Kilbertus et al., 2020), which usually doesn’t
hold on EHR data.

To address these issues, we first focus on the causes of health disparity and
construct a fairness model on this basis. Consider a predictive task that
using the observation (including both sensitive attributes and other clinical
observations) to predict the outcomes of patients. In this paper we attribute
the cause of health disparity to two factors: the imbalance of sensitive
attributes and the existence of unobserved confounders (e.g., ethnicity,
social/economic status, etc.) that affects both clinical observations and
outcomes.

Then, we propose a novel framework, Parity Medical Deconfounder (PriMeD), to
deal with the above two factors. Inspired by the deconfounder theory (Wang and
Blei, 2019, 2020) and the Inverse Propensity Weighting (IPW) methods (Haukoos
and Lewis, 2015), PriMeD can provide accurate and fair prediction by
addressing the above two factors of health disparity. In detail, PriMeD is a
two-stage supervised framework that infers unobserved confounders in the first
stage and makes predictions in the second stage. In the first stage, PriMeD
resorts to a weighted Conditional Variational Auto-encoder (CVAE) (Sohn et
al., 2015) to learn fair latent representation for observational data, and the
weight of each datapoint is the probability of the sensitive attributes exist.
Therefore, the imbalance of sensitive attributes is addressed in this step,
and according to the deconfounder theory, we regard the fair representation as
substitute confounders. In the second stage, PriMeD adopts a self-attentive
deep neural network to predict medical outcomes based on the clinical
observations, sensitive attributes and the fair representation of the first
stage. The goal of this step is to achieve higher accuracy in prediction, and
theoretically any predictive model is eligible here. In this paper, we adopt
an attention neural network to achieve this goal.

In summary, in this paper our contribution is listed as follows:

  * •

To the best of our knowledge, this paper is the first one using deconfounder
theory to address health disparities in EHR analysis.

  * •

We propose two causes of health disparity and design a model based on this
judgement. Parity Medical Deconfounder (PriMeD), is the novel model we
proposed to address the health disparity in EHR modeling.

  * •

Extensive experiments are provided to show the superiority of PriMeD in
achieving equalized odds(Hardt et al., 2016) and making accurate decisions.

(a) The SCM of EHR. Circle with dotted line denotes unobserved variable.

(b) The architecture of PriMeD.

Figure 1. The causal graph and model architecture of PriMeD. $X$, $Y$, $S$ and
$Z$ denote clinical features, outcomes, sensitive attributes and confounders.
Rounded rectangles ($x$, $\hat{y}$, $b$, $z$) in Figure 1(b) denote
corresponding data vector.

##  2\. Method

###  2.1. Problem Definition

We consider an EHR with the latent structure shown in Figure 1(a). In this
causal diagram, an EHR consists of three parts: the sensitive attributes $S$
(e.g., gender, race, etc.), the clinical observations $X$ and the
outcome/predictor $Y$. We also assume the existence of unobserved confounders
$Z$ (e.g., living habits, social/economic status, genotypes, etc.) in the
diagram. Please note that although $Z$ may be associated with $B$ in some
scenarios, in this paper we don’t consider this effect.

In practice, we have an EHR dataset
$\mathcal{D}=\\{(\bm{x}_{i},\bm{s}_{i},y_{i})\\}_{i=1}^{N}$ where $\bm{x}$,
$\bm{s}$ and $y$ corresponds to $X$, $S$, and $Y$ in Figure 1(a). Our goal is
using $\bm{x}$ and $\bm{s}$ to predict $y$, while minimizing the the gap
between performances on different subgroups.

###  2.2. PriMeD

Inspired by the deconfounder theory, PriMeD consists of two stages. The first
stage is to address the disparity by learning fair representations, and the
second stage is to make accurate predictions.

#### Stage 1: weighted Conditional Variational Auto-encoder (CVAE)

In the first stage, PriMeD addresses the confounding effect of $Z$ and $S$ by
learning fair latent representations. Here we adopt a weighted Conditional
Variational Auto-encoder (CVAE) (Sohn et al., 2015) to achieve this goal. As a
variant of Variational Autoencoder (VAE) (Kingma and Welling, 2013), CVAE
regards the observations as consequences of a latent factor and a condition.
In EHR modeling, the clinical observations $X$ are the effect of latent factor
$Z$ and sensitive attributes $S$, which is identical to the CVAE setting.

CVAE introduces an variational posterior $q_{\phi}(\bm{z}|\bm{x},\bm{s})$ as a
recognition network, to deal with the intractability of true posterior
$p_{\theta}(\bm{z}|\bm{x},\bm{s})$ in maximum likelihood inference. It
exhibits an autoencoder structure, with the recognition network
$q_{\phi}(\bm{z}|\bm{x},\bm{s})$ as encoder and the generation network
$p_{\theta}(\bm{x}|\bm{z},\bm{s})$ as decoder. When optimizing CVAE, the
objective is designed to maximize the variational lower bound (ELBO) of the
log-likelihood, which is written as

(1) $\displaystyle\log p_{\theta}(\bm{x}|\bm{s})\geq$ $\displaystyle-
KL(q_{\phi}(\bm{z}|\bm{x},\bm{s})||p_{\theta}(\bm{z}))$
$\displaystyle+\mathbb{E}_{q_{\phi}(\bm{z}|\bm{x},\bm{s})}[\log
p_{\theta}(\bm{x}|\bm{z},\bm{s})].$

To approximate the second term, we draw $L$ samples
$\bm{z}^{(l)}(l=1,\cdots,L)$ from the recognition distribution
$q_{\phi}(\bm{z}|\bm{x},\bm{s})$, and rewrite the empirical objective for
datapoint $(\bm{x},\bm{s})$ as

(2) $\displaystyle\mathcal{L}(\bm{x},\bm{s};\theta,\phi)=$ $\displaystyle-
KL(q_{\phi}(\bm{z}|\bm{x},\bm{s})||p_{\theta}(\bm{z}))$
$\displaystyle+\frac{1}{L}\sum_{l=1}^{L}\log
p_{\theta}(\bm{x}|\bm{z}^{(l)},\bm{s})$

where the first term minimizes the difference between the posterior and the
prior of $\bm{z}$, and the second term minimizes the difference between the
input and output of CVAE. In this paper, we set $p_{\theta}(\bm{z})$ as
standard Gaussian distribution. By adopting such an architecture, the
confounding effect of $X$ and $S$ can be addressed.

Moreover, to deal with the imbalanced distribution of $S$, in the training
process we assign a weight $\omega_{\bm{s}}$ for each datapoint
$(\bm{x},\bm{s})$ in $\mathcal{D}$. The intuition of using the weight is
straightforward: since the distribution of sensitive attributes $\bm{s}$ in
the dataset is usually imbalanced, we would like to assign larger weights for
rarer $\bm{s}$ to mitigate this imbalance.

To obtain $\omega_{\bm{s}}$, suppose there are $J$ sensitive attributes
$\bm{s}=[s_{1},\cdots,s_{J}]$ for each patient in the dataset. The training
weight $\omega_{\bm{s}}$ for patient with sensitive attributes $\bm{s}$ is
defined as:

(3) $\omega_{\bm{s}}=\frac{1}{\prod_{j=1}^{J}freq(s_{j})}$

where $freq(s_{j})$ is the frequency of certain attribute that occurs in the
dataset. Hence, the more frequent one attribute occurs in the sensitive
attribute vector $\bm{s}$, the smaller the corresponding $\omega_{\bm{s}}$ is.
Since we usually won’t focus on too many sensitive attributes, such an
approximation of propensity score is enough to mitigate the imbalance of data
distribution. The objective for the whole dataset can be written as

(4)
$\mathcal{L}_{CVAE}(\mathcal{D};\theta,\phi)=\sum_{i=1}^{N}\omega_{\bm{s}}\mathcal{L}(\bm{x}_{i},\bm{s}_{i};\theta,\phi)$

where $(\bm{x}_{i},\bm{s}_{i})$ is $\bm{x}$ and $\bm{s}$ of the $i$-th
datapoint in dataset $\mathcal{D}$.

####  Stage 2: Prediction with Attention

Once the CVAE is trained successfully, the learned latent variable $\bm{z}$
can be viewed as substitute confounders and be used to derive accurate and
unbiased predictions. Theoretically, any model taking $\bm{x}$, $\bm{s}$ and
$z$ as inputs is eligible in this stage. Here we use a deep learning model
with attention layers to achieve higher prediction performance. Suppose the
length of sensitive attributes $\bm{z}$ is $K$, and the clinical features
$\bm{x}=[x_{1},x_{2},\cdots,x_{M}]$ is with $M$ digits. We use $\bm{z}$ to
learn attention weights $\bm{w}_{\bm{x}}=[w_{1},w_{2},\cdots,w_{M}]$ in order
to determine which digit of $\bm{x}$ is more important in prediction.
Formally, PriMeD learns $\bm{w}_{\bm{x}}$ with a simple attention module as
follows:

(5) $\bm{w}_{\bm{x}}=softmax(\bm{W}_{x}^{att}\bm{z}^{\intercal})$

where $\bm{W}_{\bm{x}}^{att}$ is the $M\times K$ parameter matrix and
$softmax(\cdot)$ is the softmax function. Similarly, we also have
$\bm{w}_{\bm{s}}=softmax(\bm{W}_{\bm{s}}^{att}\bm{z}^{\intercal})$ to learn
attention weights for sensitive attributes $\bm{s}$. Finally, the processed
features are input into a feed-forward neural network to derive the final
decision:

(6)
$\hat{y}=MLP([\bm{x}\odot\bm{w}_{x}^{\intercal};\bm{s}\odot\bm{w}_{s}^{\intercal};\bm{z}]),$

where $[;]$ denotes the concatenation operation, $\odot$ denotes the element-
wise multiplication operation, and $MLP(\cdot)$ is a two-layer MLP. In this
stage, by incorporating the latent factor $\bm{z}$ in the prediction, the
model can calibrate the bias from observational data and derive a higher-
quality prediction:

Table 1. Information used in our experiments. Datasets | MIMIC | Hip & Knee
---|---|---
| Sensitive
---
Attributes
| Insurance, Ethnicity
---
Gender
| Gender, Race, Age
---
| Clinical
---
Features
| Demographics
---
Clinical events, Lab events
Procedures, Diagnoses
| Height, Weight
---
Position #
20-37, 51-63
Outcomes | Mortality |  | Re-operation
---
Re-admission
DOptoDis

##  3\. Experiments

In this section, we conduct experiments to evaluate the performance of our
proposed PriMeD.

###  3.1. Datasets

We use three real-world EHR datasets to measure PriMeD’s performance.

  * •

MIMIC-III 111https://physionet.org/content/mimiciii/1.4/. MIMIC-III(Johnson et
al., 2016) is a deidentified, publicly-available dataset comprising
comprehensive clinical data of patients admitted to the Beth Israel Deaconess
Medical Center. MIMIC-III contains EHRs associated with 46,520 patients,
including over 20 tables such as medical events, diagnoses, prescriptions,
etc.

  * •

Hip & Knee. Hip & Knee are two subsets of the National Surgical Quality
Improvement Program (NSQIP) project222https://www.facs.org/quality-
programs/data-and-registries/acs-nsqip/ (Cohen et al., 2016). NSQIP provides
clinical observations and observations for patients taking surgical operations
to track surgical complications after operations. They are associated with
patients receiving Hip Arthroplasty
(CPT333https://medicaid.ncdhhs.gov/blog/2021/12/30/cpt-code-update-2022 27130)
and Knee Arthroplasty (CPT 27447) surgical operations. There are 96,441 and
156,292 pieces of data in them respectively, and each data point contains
demographics, pre-operative features, and the outcome of the surgery for a
patient.

#### data preprocessing

According to the documentation of MIMIC-
III444https://mimic.mit.edu/docs/iii/tables/ and
NSQIP555https://www.facs.org/quality-programs/data-and-registries/acs-
nsqip/participant-use-data-file/, we determine the columns/tables used in our
experiments (shown in Table 1). In the MIMIC-III dataset, we regard the
patient’s insurance status, ethnicity and gender as sensitive attributes, and
use clinical events to predict mortality. In the Hip & Knee dataset, we take
gender, race and age as sensitive attributes and use BMI (inferred from height
and weight) and multiple clinical observations to predict the readmission,
reoperation, and whether there is a prolonged hospitalization after the
operation (DOptoDis ¿ 5).

###  3.2. Evaluation Metrics & Implementation Details

Since all tasks are binary classifications, we use Area Under the Receiver
Operating Characteristic curve (AUROC) to measure its accuracy in
classification. We implement the model using Pytorch 1.10 and adopt the
$10^{-4}$ as the learning rate and $5\times 10^{-4}$ as the weight decay. The
ratio of training, validating, and testing set is 7:2:1.

Table 2. Accuracy of PriMeD and other baselines in prediction. Dataset | MIMIC-III | Knee | Hip
---|---|---|---
Task | Mortality | Reoperation | Readmission |  | Prolonged
---
Hospitalization
Reoperation | Readmission |  | Prolonged
---
Hospitalization
DNN | 0.6984 | 0.6752 | 0.6342 | 0.7061 | 0.7044 | 0.6448 | 0.7403
Re-weighting | 0.6553 | 0.6457 | 0.5973 | 0.6590 | 0.6793 | 0.6125 | 0.6983
CE odds | 0.6810 | 0.6579 | 0.6192 | 0.6773 | 0.6889 | 0.6313 | 0.7045
WFC | 0.6741 | 0.6481 | 0.6008 | 0.6717 | 0.6865 | 0.6217 | 0.7102
RFC | 0.6889 | 0.6630 | 0.6289 | 0.6928 | 0.6956 | 0.6375 | 0.7363
FuCS | 0.6944 | 0.6774 | 0.6375 | 0.7024 | 0.7029 | 0.6426 | 0.7345
PriMeD | 0.7013 | 0.6821 | 0.6413 | 0.7146 | 0.7059 | 0.6453 | 0.7457

###  3.3. Comparison Experiments

In this subsection, we compare our model with several baseline models listed
below.

  * •

DNN. We adopt a deep neural network to make accurate predictions on
observational data. This method is served as a baseline to show the original
disparity without interference.

  * •

Re-weighting(Kamiran and Calders, 2012). Kamiran et al. propose a pre-
processing method based on the dataset re-weighting to remove bias from the
dataset. Re-weighting can reduce the discrimination while maintaining the
overall positive class probability for the training set. We use this method to
assign weights for the data, while using the same model as DNN to make the
prediction.

  * •

CE Odds(Pleiss et al., 2017). Pleiss et al. propose a post-processing method
based on calibration constraints to minimize error disparity while maintaining
calibrated probability estimates.

  * •

WFC(Jiang et al., 2020). Wasserstein Fair Classification is a post-processing
method that enforces independence between the classifier outputs and sensitive
information by minimizing Wasserstein-1 distances.

  * •

RFC(Agarwal et al., 2018). Agarwal et al. propose an in-processing method,
focusing on reducing fair classification to a sequence of cost-sensitive
classification problems to achieve the lowest error subject to the desired
constraints.

  * •

FuCS(Rezaei et al., 2021). Rezaei et al. propose a pre-processing method to
guarantee fairness under covariate shift in which the covariates change while
the conditional label distribution remains the same.

###  3.4. Performance of Prediction

Table 2 shows the performance of all baselines together with PriMeD on all
three tasks. From the table, we can observe that PriMeD outperforms all other
baselines significantly in AUROC. It is because the architecture of PriMeD is
specially designed for EHRs, while other baselines are universal classifiers.
PriMeD also outperforms the vanilla classifier, DNN, on all tasks, showing its
superiority in modeling EHR. Apart from PriMeD, FuCS achieves the second-best
performance, which may due to its ability to deal with non-iid data. When the
confounder is unobserved, EHR data may exhibit multiple distributions and is
not iid. FuCS also outperforms DNN on some tasks, indicating its ability to
learn knowledge from non-iid data. Then, RFC and CE odds achieve lower
performance at the price of guaranteeing equalized odds in prediction. WFC and
Re-weighting method have the worst performances. This fact indicates their
means of controlling fairness clearly prevent them from learning from
observational data.

If we compare different datasets/tasks, we will find the difficulty of tasks
is different as well. As for the reoperation task, the differences between the
performances of methods are relatively small, indicating the confounding
effect on this predictor is relatively weaker. The difference between methods
on the prolonged hospitalization task is quite large, indicating this
predictor is strongly influenced by the confounding effect.

Table 3. The extent of disparity of PriMeD and other baselines in prediction. Dataset |  | MIMIC-III
---
Mortality
| Knee
---
Prolonged Hospitalization
| Hip
---
Prolonged Hospitalization
Sensitive Attribute | Insurance | Ethnicity | Race | Age | Race | Age
DNN | $1.570\times 10^{-2}$ | $1.802\times 10^{-2}$ | $9.618\times 10^{-3}$ | $10.859\times 10^{-3}$ | $8.702\times 10^{-3}$ | $9.859\times 10^{-3}$
Re-weighting | $1.378\times 10^{-2}$ | $1.638\times 10^{-2}$ | $8.637\times 10^{-3}$ | $10.178\times 10^{-3}$ | $8.418\times 10^{-3}$ | $9.163\times 10^{-3}$
CE odds | $1.384\times 10^{-2}$ | $1.667\times 10^{-2}$ | $8.825\times 10^{-3}$ | $9.074\times 10^{-3}$ | $8.258\times 10^{-3}$ | $8.741\times 10^{-3}$
WFC | $1.463\times 10^{-2}$ | $1.612\times 10^{-2}$ | $8.363\times 10^{-3}$ | $8.523\times 10^{-3}$ | $8.357\times 10^{-3}$ | $8.275\times 10^{-3}$
RFC | $1.249\times 10^{-2}$ | $1.563\times 10^{-2}$ | $7.356\times 10^{-3}$ | $7.804\times 10^{-3}$ | $7.304\times 10^{-3}$ | $8.002\times 10^{-3}$
FuCS | $1.120\times 10^{-2}$ | $1.496\times 10^{-2}$ | $7.576\times 10^{-3}$ | $7.063\times 10^{-3}$ | $7.776\times 10^{-3}$ | $8.024\times 10^{-3}$
PriMeD | $1.123\times 10^{-2}$ | $1.470\times 10^{-2}$ | $6.982\times 10^{-3}$ | $9.451\times 10^{-3}$ | $7.227\times 10^{-3}$ | $7.769\times 10^{-3}$

###  3.5. Mitigate Health Disparity

In this subsection, we demonstrate the disparity of prediction between
different subgroups. Table 3 demonstrates the extent of disparities in
prediction with respect to different datasets, predictors and baseline models.
In this table, the Insurance column shows the difference in AUROC between
patients with public and private insurances, the Ethnicity and Race columns
show the difference between White and non-White patients, and the Age column
shows the difference between patients with ages above and below 65.

From Table 3, we can observe that PriMeD can effectively reduce the difference
between subgroups in prediction. It achieves the fairest performance across
all datasets and all metrics. Apart from PriMeD, we can observe that FuCS and
RFC achieve the second-best performance, which means they can balance the
performance on different demographic subgroups. From the table, we also
observe that the disparity in age is more stubborn than the disparity of race,
which means age is a stronger confounder that can cause bias.

Table 4. The performance of different variants of PreMeD. |  | Knee
---
Reoperation
| Knee
---
Readmission
| AUROC | Diff_Race | AUROC | Diff_Race
PriMeD | 0.6821 | 0.005463 | 0.6413 | 0.005731
PriMeD-Stage 1 | 0.6410 | 0.005585 | 0.5903 | 0.006121
PriMeD-Stage 2 | 0.6748 | 0.008463 | 0.6370 | 0.007951

###  3.6. Ablation Study

In order to show the effectiveness of each stage of PriMeD, Table 4
demonstrates the performance of each module of PriMeD. In this module, PriMeD-
Stage 1 means the performance of prediction while only using the stage 1 model
(the CVAE). In PriMeD-Stage 1 we use the latent vector $z$ the CVAE learned to
make classifications. Similarly, PriMeD-Stage 2 means simply using a deep
neural network to make the prediction. In PriMeD-Stage 2, we use the input
vector $b$ and $x$ to substitute the position of $z$ in the stage 2 model. In
Table 4, we use AUROC to describe the performance in prediction and use the
difference between White and non-White patients (denoted by Diff_Race) to
measure the health disparity. We can observe that the CVAE can learn
relatively fair predictions while the accuracy of classification is low. The
attention neural network can achieve quite an accurate prediction which is
similar to the DNN baseline, while the disparity issue is unaddressed.

##  4\. Related Works

The related works of this paper consist of three parts: the observation of
healthcare disparity, the review of fairness in machine learning, and a brief
introduction of the deconfounder theory.

###  4.1. Healthcare Disparity

With the increase in the number of machine learning applications in the
healthcare domain, more concerns have been raised regarding the potential
ethical issue of these models (Pfohl et al., 2021; Chen et al., 2020b, 2021).
Recently, many studies have shown that these models worsen existing health
disparities. Health disparity means the inequity for different patient
subgroups when providing healthcare services (Carter-Pokras and Baquet, 2002;
Fikree and Pasha, 2004). Although this inequity naturally exists in
observational data (Krasanakis et al., 2018; Obermeyer et al., 2019), it is
amplified by the algorithmic bias in the model (Mhasawade et al., 2021, 2021).
In response, considerable attention has been devoted to dissecting the
disparities in the machine learning models (Zhou et al., 2021; Seyyed-
Kalantari et al., 2020). According to the recent analysis, this disparity can
result from the inequity in care access and delivery history (Chen et al.,
2020a; Gaskin et al., 2012), the existence of underrepresented groups in
clinical datasets (Larrazabal et al., 2020), the misuse of biased features
during model development (Kallus and Zhou, 2018; Jiang and Nachum, 2020), and
difference of distribution between training and teat sets (Veinot et al.,
2018).

###  4.2. Fairness & Algorithmic Debiasing

Traditionally, fairness avoids any prejudice or favoritism towards any
individual or group in the decision-making process. Based on this principle,
many fairness metrics are proposed and can be categorized mainly into three
classes. One standard fairness metric is group fairness, such as demographic
parity, which requires the probability of a positive prediction to be the same
for each group (Zemel et al., 2013). Another fairness metric is individual
fairness, such as counterfactual fairness. It requires the probability of a
positive prediction to be the same for both factual and counterfactual
datapoints (Kusner et al., 2017). In this paper, our goal is to minimize the
difference in utility between patient subgroups, which is similar to (Rajkomar
et al., 2018a; Heidari et al., 2019; Pfohl et al., 2019) and is different from
either group or individual fairness.

Based on the above definitions of fairness, there are three types of
algorithms to address the fairness of machine learning models: the pre-
processing, in-processing, and post-processing methods. Pre-processing methods
either change the label of training data or assign weights for data before the
training process (Kamiran and Calders, 2012, 2010). Post-processing methods
conduct calibration of prediction results after training. Some of them
calibrate the prediction based on a holdout set, and others are based on
certain constraints to ensure fairness (d’Alessandro et al., 2017; Pleiss et
al., 2017). In-processing techniques develop model architectures to remove
discrimination during the model training process (d’Alessandro et al., 2017;
Louizos et al., 2015; Sattigeri et al., 2019). Our model adopts the in-
processing technique, because it is more flexible in dealing with complex
latent data structures.

###  4.3. Causal Inference with Unobserved Confounders

In this part, we mainly discuss the deconfounder theory. Deconfounder (Wang
and Blei, 2019, 2020) is a theory to estimate unbiased treatment effects for
observational data with the setting of multiple causes. Due to unobserved
confounders of many tasks, traditional methods can hardly learn unbiased
knowledge from the observational data. However, suppose we can observe
multiple causes of the outcome. In this case, the dependencies between causes
can be used to infer latent variables. The latent variables can be used as
substitutes for the hidden confounders. Therefore, Wang & Blei propose a two-
stage architecture to deal with the unobserved confounders. Recently,
deconfounder has been applied to many areas such as recommender systems (Wang
et al., 2020) and medical treatment estimation (Zhang et al., 2019). This
paper demonstrates that deconfounder can also be used in the fairness domain
to reduce inequity in prediction.

##  5\. Conclusion & Future Directions

This paper analyzes the disparity between different demographic groups and
proposes a solution to address it. Our goal is to minimize the disparity of
different demographics while maintaining a high utility for machine learning
healthcare applications. To achieve this goal, we propose our PriMeD to derive
unbiased predictions. By incorporating CVAE as a module to infer latent
factors for patients, PriMeD can naturally correct the disparity in the first
stage of the model, while preserving high prediction accuracy. Experiments
conducted on three real-world datasets have shown the superiority of PriMeD
over other baselines, and the visualization of the learned latent factor
further demonstrates its ability to learn fair representation without the
influence of sensitive attributes.

##  6\. Acknowledgement

This work is supported in part by NSF under grants III-1763325, III-1909323,
III-2106758, and SaTC-1930941.

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