###### Abstract

We show that the natural scaling of measurement for a particular problem
defines the most likely probability distribution of observations taken from
that measurement scale. Our approach extends the method of maximum entropy to
use measurement scale as a type of information constraint. We argue that a
very common measurement scale is linear at small magnitudes grading into
logarithmic at large magnitudes, leading to observations that often follow
Student’s probability distribution which has a Gaussian shape for small
fluctuations from the mean and a power law shape for large fluctuations from
the mean. An inverse scaling often arises in which measures naturally grade
from logarithmic to linear as one moves from small to large magnitudes,
leading to observations that often follow a gamma probability distribution. A
gamma distribution has a power law shape for small magnitudes and an
exponential shape for large magnitudes. The two measurement scales are natural
inverses connected by the Laplace integral transform. This inversion connects
the two major scaling patterns commonly found in nature. We also show that
superstatistics is a special case of an integral transform, and thus can be
understood as a particular way in which to change the scale of measurement.
Incorporating information about measurement scale into maximum entropy
provides a general approach to the relations between measurement, information
and probability.

###### keywords:

maximum entropy; information theory; superstatistics; power law; Student’s
distribution; gamma distribution

10.3390/e12030289 12 Received: 18 January 2010; in revised form: 16 February
2010 / Accepted: 23 February 2010 / Published: 26 February 2010 Measurement
Invariance, Entropy, and Probability Steven A. Frank 1,2,⋆ and D. Eric Smith 2
E-mail<EMAIL_ADDRESS>
##  1 Introduction

Suppose you have a ruler that is about the length of your hand. With that
ruler, you can measure the size of all the visible objects in your office.
That scaling of objects in your office with the length of the ruler means that
those objects have a natural linear scaling in relation to your ruler.

Now consider the distances from your office to various galaxies. Your ruler is
of no use, because you cannot distinguish whether a particular galaxy moves
farther away by one ruler unit. Instead, for two galaxies, you can measure the
ratio of distances from your office to each galaxy. You might, for example,
find that one galaxy is twice as far as another, or, in general, that a galaxy
is some percentage farther away than another.

Percentage changes define a ratio scale of measure, which has natural units in
logarithmic measure (Hand04Measurement, ). For example, a doubling of distance
always adds $\log(2)$ to the logarithm of the distance, no matter what the
initial distance.

Measurement naturally grades from linear at local magnitudes to logarithmic at
distant magnitudes when compared to some local reference scale. The transition
between linear and logarithmic varies between problems. Measures from some
phenomena remain primarily in the linear domain, such as measures of height
and weight in humans. Measures for other phenomena remain primarily in the
logarithmic domain, such as cosmological distances. Other phenomena scale
between the linear and logarithmic domains, such as fluctuations in the price
of financial assets (Aparicio01Empirical, ) or the distribution of income and
wealth (dragulescu01exponential, ).

The second section of this article shows how the characteristic scaling of
measurement constrains the most likely probability distribution of
observations. We use the standard method of maximum entropy to find the most
likely probability distribution (Jaynes03Science, ). But, rather than follow
the traditional approach of starting with the information in the summary
statistics of observations, such as the mean or variance, we begin with the
information in the characteristic scale of measurement. We argue that
measurement sets the fundamental nature of information and shapes the
probability distribution of observations. We present a novel extension of the
method of maximum entropy to incorporate information about the scale of
measurement.

The third section emphasizes the naturalness of the measurement scale that
grades from linear at small magnitudes to logarithmic at large magnitudes.
This linear to logarithmic scaling leads to observations that often follow a
linear-log exponential or Student’s probability distribution. A linear-log
exponential distribution is an exponential shape for small magnitudes and a
power law shape for large magnitudes. Student’s distribution is a Gaussian
shape for small fluctuations from the mean and a power law for large
fluctuations from the mean. The shapes correspond to linear scaling at small
magnitudes and logarithmic scaling at large magnitudes. Many naturally
observed patterns follow these distributions. The particular form depends on
whether the measurement scale for a problem is primarily linear, primarily
logarithmic, or grades from linear to logarithmic.

The fourth section inverts the natural linear to logarithmic scaling for
magnitudes. Because magnitudes often scale from linear to logarithmic as one
moves from small to large magnitudes, inverse measures often scale from
logarithmic to linear as one moves from small to large magnitudes. This
logarithmic to linear scaling leads to observations that often follow a gamma
probability distribution. A gamma distribution is a power law shape for small
magnitudes and an exponential shape for large magnitudes, corresponding to
logarithmic scaling at small values and linear scaling at large values. The
gamma distribution includes as special cases the exponential distribution, the
power law distribution, and the chi-square distribution, subsuming many
commonly observed patterns.

The fifth section demonstrates that the Laplace integral transform provides
the formal connection between the inverse measurement scales. The Laplace
transform, like its analytic continuation the Fourier transform, changes a
magnitude with dimension $d$ on one scale into an inverse magnitude with
dimension $1/d$ on the other scale. This inversion explains the close
association between the linear to logarithmic scaling as magnitudes increase
and the inverse scale that grades from logarithmic to linear as magnitudes
increase. We discuss the general role of integral transforms in changing the
scale of measurement. Superstatistics is the averaging of a probability
distribution with a variable parameter over a probability distribution for the
variable parameter (Beck03Superstatistics, ). We show that superstatistics is
a special case of an integral transform, and thus can be understood as a
particular way in which to change the scale of measurement.

In the sixth section, we relate our study of measurement invariance for
continuous variables to previous methods of maximum entropy for discrete
variables. We also distinguish the general definition of measurement scale by
information invariance from our particular argument about the commonness of
linear-log scales.

In the discussion, we contrast our emphasis on the primacy of measurement with
alternative approaches to understanding measurement, randomness, and
probability. One common approach changes the definition of randomness and
entropy to incorporate a change in measurement scale (Tsallis09Introduction,
). We argue that our method makes more sense, because we directly incorporate
the change in measurement scale as a kind of information, rather than alter
the definition of randomness and entropy to match each change in measurement
scale. It is measurement that changes empirically between problems rather than
the abstract meaning of randomness and information. Although we focus on the
duality between linear to logarithmic scaling and its inverse logarithmic to
linear scaling, our general approach applies to any type of measure invariance
and measurement scale.

##  2 Measurement, Information Invariance, and Probability

We derive most likely probability distributions. Our method follows the
maximum entropy approach (Jaynes57Information, ; Jaynes57Informationb, ;
Jaynes03Science, ). That approach assumes that the most likely distribution
has the maximum amount of randomness, or entropy, subject to the constraint
that the distribution must capture all of the information available to us. For
example, if we know the average value of a sample of observations, and we know
that all values from the underlying probability distribution are positive,
then all candidate probability distributions must have only positive values
and have a mean value that agrees with the average of the empirically observed
values. By maximum entropy, the most random distribution constrained to have
positive values and a fixed mean is the exponential distribution.

We express the available information by constraints. Typical constraints
include the average or variance of observations. But we must use all available
information, which may include information about the scale of measurement
itself. Previous studies have discussed how the scale of measurement provides
information. However, that aspect of maximum entropy has not been fully
developed (Jaynes03Science, ; Frank09The-common, ). Our goal is to develop the
central role of measurement scaling in shaping the commonly observed
probability distributions.

In the following sections, we show how to use information about measurement
invariances and associated measurement scales to find most likely probability
distributions.

###  2.1 Maximum entropy

The method of maximum entropy defines the most likely probability distribution
as the distribution that maximizes a measure of entropy (randomness) subject
to various information constraints. We write the quantity to be maximized as

$\Lambda={\cal{E}}-\alpha C_{0}-\sum_{i=1}^{n}\lambda_{i}C_{i}$ (1)

where ${\cal{E}}$ measures entropy, the $C_{i}$ are the constraints to be
satisfied, and $\alpha$ and the $\lambda_{i}$ are the Lagrange multipliers to
be found by satisfying the constraints. Let $C_{0}=\int p_{y}{\hbox{\rm
d}}y-1$ be the constraint that the probabilities must total one, where $p_{y}$
is the probability distribution function of $y$. The other constraints are
usually written as $C_{i}=\int p_{y}f_{i}(y){\hbox{\rm d}}y-\left\langle
f_{i}(y)\right\rangle$, where the $f_{i}(y)$ are various transformed
measurements of $y$. Angle brackets denote mean values. A mean value is either
the average of some function applied to each of a sample of observed values,
or an a priori assumption about the average value of some function with
respect to a candidate set of probability laws. If $f_{i}(y)=y^{i}$, then
$\left\langle y^{i}\right\rangle$ are the moments of the distribution—either
the moments estimated from observations or a priori values of the moments set
by assumption. The moments are often regarded as “normal” constraints,
although from a mathematical point of view, any properly formed constraint can
be used.

Here, we confine ourselves to a single constraint of measurement. We express
that constraint with a more general notation, $C_{1}=\int
p_{y}\textrm{T}[f(y)]{\hbox{\rm
d}}y-\left\langle\textrm{T}[f(y)]\right\rangle$, where $\textrm{T}()$ is a
transformation. We could, of course, express the constraining function for $y$
directly through $f(y)$. However, we wish to distinguish between an initial
function $f(y)$ that can be regarded as a normal measurement, in any sense in
which one chooses to interpret the meaning of normal, and a transformation of
normal measurements denoted by $\textrm{T}()$ that arises from information
about the measurement scale.

The maximum entropy distribution is obtained by solving the set of equations

${{\partial\Lambda}\over{\partial p_{y}}}={{\partial{\cal{E}}}\over{\partial
p_{y}}}-\alpha-\lambda\textrm{T}[f(y)]=0$ (2)

where one checks the candidate solution for a maximum and obtains $\alpha$ and
$\lambda$ by satisfying the constraint on total probability and the constraint
on $\left\langle\textrm{T}[f(y)]\right\rangle$. We assume that we can treat
entropy measures as the continuous limit of the discrete case.

In the standard approach, we define entropy by Shannon information

${\cal{E}}=-\int p_{y}\log(p_{y}){\hbox{\rm d}}y$ (3)

which yields the solution of Equation (2) as

$p_{y}=ke^{-\lambda\textrm{T}[f(y)]}$ (4)

where $k$ and $\lambda$ satisfy the two constraints.

###  2.2 Measurement and transformation

Maximum entropy, in order to be a useful method, must capture all of the
available information about a particular problem. One form of information
concerns transformations to the measurement scale that leave the most likely
probability distribution unchanged. Suppose, for example, that we obtain the
same information from measurements of $x$ and transformed measurements,
$\textrm{G}(x)$. Put another way, if one has access only to measurements on
the $\textrm{G}(x)$ scale, one has the same information that would be obtained
if the measurements were reported on the $x$ scale. We say that the
measurements $x$ and $\textrm{G}(x)$ are equivalent with respect to
information, or that the transformation $x\rightarrow\textrm{G}(x)$ is an
invariance (Hand04Measurement, ; luce08measurement, ; narens08meaningfulness,
).

To capture this information invariance in maximum entropy, we must express our
measurements on a transformed scale. In particular, we must choose the
transformation, $\textrm{T}()$, for expressing measurements so that

$\textrm{T}(x)=\gamma+\delta\textrm{T}[\textrm{G}(x)]$ (5)

for some arbitrary constants $\gamma$ and $\delta$. Putting this definition of
$\textrm{T}(x)$ into Equation (4) shows that we get the same maximum entropy
solution whether we use the direct scale $x$ or the alternative measurement
scale, $\textrm{G}(x)$, because the $k$ and $\lambda$ constants will adjust to
the constants $\gamma$ and $\delta$ so that the distribution remains
unchanged.

Given the transformation $\textrm{T}(x)$, the derivative of that
transformation expresses the information invariance in terms of measurement
invariance. In particular, we have the following invariance of the measurement
scale under a change ${\hbox{\rm d}}x$

${\hbox{\rm d}}\textrm{T}(x)\propto{\hbox{\rm d}}\textrm{T}[\textrm{G}(x)]$
(6)

We may also examine $m_{x}=\textrm{T}^{\prime}(x)={\hbox{\rm
d}}\textrm{T}(x)/{\hbox{\rm d}}x$ to obtain the change in measurement scale
required to preserve the information invariance between $x$ and
$\textrm{G}(x)$.

If we know the measurement invariance, $\textrm{G}(x)$, we can find the
correct transformation from Equation (5). If we know the transformation
$\textrm{T}(x)$, we can find $\textrm{G}(x)$ by inverting Equation (5) to
obtain

$\textrm{G}(x)=\textrm{T}^{-1}\left[{{\textrm{T}(x)-\gamma}\over{\delta}}\right]$
(7)

Alternatively, we may deduce the transformation $\textrm{T}(x)$ by examining
the form of a given probability distribution and using Equation (4) to find
the associated transformation.

In summary, $x$ and $\textrm{G}(x)$ provide invariant information, and the
transformation of measurements $\textrm{T}(x)$ captures that information
invariance in terms of measurement invariance.

###  2.3 Example: ratio and scale invariance

Suppose the information we obtain from positive-valued measurements depends
only on the ratio of measurements, $y_{2}/y_{1}$. In this particular case, all
measurements with the same ratio map to the same value, so we say that the
measurement scale has ratio invariance. Pure ratio measurements also have
scale invariance, because ratios do not depend on the magnitude or scale of
the observations.

We express the invariances that characterize a measurement scale by the
transformations that leave the information in the measurements unchanged
(Hand04Measurement, ; luce08measurement, ; narens08meaningfulness, ). If we
obtain values $x$ and use the measurement scale from the transformation
$\textrm{T}(x)=\log(x)$, the information in $x$ is the same as in
$\textrm{G}(x)=x^{c}$, because $\textrm{T}(x)=\log(x)$ and
$\textrm{T}[\textrm{G}(x)]=c\log(x)$, so in general
$\textrm{T}(x)\propto\textrm{T}[\textrm{G}(x)]$, which means that the
information in the measurement scale given by $\textrm{T}(x)$ is invariant
under the transformation $\textrm{G}(x)$.

We can express the invariance in a way that captures how measurement relates
to information and probability. The transformation $\textrm{T}(x)=\log(x)$
shrinks increments on the uniform scaling of $x$ so that each equally spaced
increment on the original uniform scale shrinks to length $1/x$ on the
transformed scale. We can in general quantify the deformation in incremental
scaling by the derivative of the transformation $\textrm{T}(x)$ with respect
to $x$. In the case of the logarithmic measurement scale with ratio
invariance, the measure invariance in Equation (6) is

${\hbox{\rm d}}\log(x)\propto{\hbox{\rm
d}}\log[\textrm{G}(x)]\Rightarrow{{1}\over{x}}\propto{{c}\over{x}}$

showing in another way that the logarithmic measure $\textrm{T}(x)$ is
invariant under the transformation $\textrm{G}(x)$. With regard to probability
or information, we can think of the logarithmic scale with ratio invariance as
having an expected density of probability per increment in proportion to
$1/x$, so that the expected density of observations at scale $x$ decreases in
proportion to $1/x$. Roughly, we may also say that the information value of an
increment decreases in proportion to $1/x$. For example, the increment length
of our hand is an informative measure for the visible objects near us, but
provides essentially no information on a cosmological scale.

If we have measurements $f(y)=y$, and we transform those measurements in a way
consistent with a ratio and scale invariance of information, then we have the
transformed measures $\textrm{T}[f(y)]=\log(y)$. The constraint for maximum
entropy corresponds to $\left\langle\log(y)\right\rangle$, which is logarithm
of the geometric mean of the observations on the direct scale $y$. Given that
constraint, the maximum entropy distribution is a power law

$p_{y}=ke^{-\lambda\textrm{T}[f(y)]}=ke^{-\lambda\log(y)}=ky^{-\lambda}$

For $y\geq 1$, we can solve for the constants $k$ and $\lambda$, yielding
$p_{y}=\delta y^{-(1+\delta)}$, with
$\delta=1/\left\langle\log(y)\right\rangle$.

##  3 The linear to Logarithmic Measurement Scale

###  3.1 Measurement

In the previous section, we obtained ratio and scale invariance with a measure
$m_{x}=\textrm{T}^{\prime}(x)\propto 1/x$. In this section, we consider the
more general measure

$m_{x}\propto{{1}\over{1+bx}}$

At small values of $x$, the measure becomes linear, $m_{x}\propto 1$, and at
large values of $x$, the measure becomes ratio invariant (logarithmic),
$m_{x}\propto 1/x$. This measure has scale dependence with ratio invariance at
large scales, because the measure changes with the magnitude (scale) of $x$,
becoming ratio invariant at large values of $x$. The parameter $b$ controls
the scale at which the measure grades between linear and logarithmic.

Given $m_{x}=\textrm{T}^{\prime}(x)$, we can integrate this deformation of
measurement to obtain the associated scale of measurement as

$\textrm{T}(x)={{1}\over{a}}\log(1+bx)=\log(1+bx)^{{1}\over{a}}\propto\log(1+bx)$
(8)

where we have expressed the proportionality constant as $1/a$ and we have
dropped the constant of integration. The expression $\log(1+bx)$ is just a
logarithmic measurement scale for positive values in relation to a fixed
origin at $x=0$, because $\log(1)=0$. The standard logarithmic expression,
$\log(x)$, has an implicit origin for positive values at $x=1$, which is only
appropriate for purely ratio invariant problems with no notion of an origin to
set the scale of magnitudes. In most empirical problems, there is some
information about the scaling of magnitudes. Thus, $\log(1+bx)$ is more often
the natural measurement scale.

Next, we seek an expression $\textrm{G}(x)$ to describe the information
invariance in the measurement scale, such that the information in $x$ and in
$\textrm{G}(x)$ is the same. The expression in Equation (6), ${\hbox{\rm
d}}\textrm{T}(x)\propto{\hbox{\rm d}}\textrm{G}[\textrm{T}(x)]$, sets the
condition for information invariance, leading to

$\textrm{G}(x)={{(1+bx)^{{1}\over{a}}-1}\over{b}}$ (9)

On the measurement scale $\textrm{T}(x)$, the information in $x$ is the same
as in $\textrm{G}(x)$, because

${\hbox{\rm d}}\textrm{T}(x)\propto{\hbox{\rm
d}}\textrm{T}[\textrm{G}(x)]\Rightarrow{{b/a}\over{1+bx}}\propto{{b/a^{2}}\over{1+bx}}$

We now use $x=f(y)$ to account for initial normal measures that may be taken
in any way we choose. Typically, we use direct values, $f(y)=y$, or squared
values, $f(y)=y^{2}$, corresponding to initial measures related to the first
and second moments—the average and variance. For now, we use $f(y)$ to hold
the place of whatever direct values we will use. Later, we consider the
interpretations of the first and second moments.

###  3.2 Probability

The constraint for maximum entropy corresponds to
$\left\langle\textrm{T}[f(y)]\right\rangle=\left\langle\log[1+bf(y)]^{{1}\over{a}}\right\rangle$,
a value that approximately corresponds to an interpolation between the linear
mean and the geometric mean of $f(y)$. Given that constraint, the maximum
entropy distribution from Equation (4) is

$p_{y}\propto[1+bf(y)]^{-\alpha}$ (10)

where $\alpha=\lambda/a$ acts as a single parameter chosen to satisfy the
constraint, and $b$ is a parameter derived from the measurement invariance
that expresses the natural scale of measurement for a particular problem.

From Equation (10), we can express simple results when in either the purely
linear or purely logarithmic regime. For small values of $bf(y)$ we can write
$p_{y}\propto e^{-\alpha bf(y)}$. For large values of $bf(y)$ we can write
$p_{y}\propto f(y)^{-\alpha}$, where we absorb $b^{-\alpha}$ into the
proportionality constant. Thus, the probability distribution grades from
exponential in $f(y)$ at small magnitudes to a power law in $f(y)$ at large
magnitudes, corresponding to the grading of the linear to logarithmic
measurement scale.

###  3.3 Transition between linear and logarithmic scales

We mentioned that one can obtain the parameter $\alpha$ in Equation (10)
directly from the constraint $\left\langle\textrm{T}[f(y)]\right\rangle$,
which can be calculated directly from observed values of the process or set by
assumption. What about the parameter $b$ that sets the grading between the
linear and logarithmic regimes?

When we are in the logarithmic regime at large values of $bf(y)$,
probabilities scale as $p_{y}\propto f(y)^{-\alpha}$ independently of $b$.
Thus, with respect to $b$, we only need to know the magnitude of observations
above which ratio invariance and logarithmic scaling become reasonable
descriptions of the measurement scale.

In the linear regime, $p_{y}\propto e^{-\alpha bf(y)}$, thus $b$ only arises
as a constant multiplier of $\alpha$ and so can be subsumed into a single
combined parameter $\beta=\alpha b$ estimated from the single constraint.
However, it is useful to consider the meaning of $b$ in the linear regime to
provide guidance for how to interpret $b$ in the mixed regime in which we need
the full expression in Equation (10).

When $f(y)=y$, the linear regime yields an exponential distribution
$p_{y}\propto e^{-\alpha by}$. In this case, $b$ weights the intensity or rate
of the process $\alpha$ that sets the scaling of the distribution.

When $f(y)=y^{2}$, the linear regime yields a Gaussian distribution
$p_{y}\propto e^{-\alpha by^{2}}$, where $2\alpha b$ is the reciprocal of the
variance that defines the precision of measurements—the amount of information
a measurement provides about the location of the average value. In this case,
$b$ weights the precision of measurement. The greater the value of $b$, the
more information per increment on the measurement scale.

###  3.4 Linear-log exponential distribution

When $f(y)=y$, we obtain from Equation (10) what we will call the linear-log
exponential distribution

$p_{y}\propto[1+by]^{-\alpha}$ (11)

for $y>0$. This distribution is often called the generalized type II Pareto
distribution or the Lomax distribution (Johnson94Distributions, ). Small
values of $by$ lead to an exponential shape, $p_{y}\propto e^{-\alpha by}$.
Large values of $by$ lead to power law tails, $p_{y}\propto y^{-\alpha}$. The
parameter $b$ determines the grading from the exponential to the power law.
Small values of $b$ extend the exponential to higher values of $y$, whereas
large values of $b$ move the extent of the power law shape toward smaller
values of $y$. Many natural phenomena follow a linear-log exponential
distribution (Tsallis09Introduction, ).

###  3.5 Student’s distribution

When $f(y)=y^{2}$, we obtain from Equation (10) Student’s distribution

$p_{y}\propto[1+by^{2}]^{-\alpha}$ (12)

Here, we assume that $y$ expresses deviations from the average. Small
deviations lead to a Gaussian shape around the mean, $p_{y}\propto e^{-\alpha
by^{2}}$. Large deviations lead to power law tails, $p_{y}\propto
f(y)^{-\alpha}$. The parameter $b$ determines the grading from the Gaussian to
the power law. Small values of $b$ expand the Gaussian shape far from the
mean, whereas large values of $b$ move the extent of the power law shape
closer to the central value at the average. Many natural phenomena expressed
as deviations from a central value follow Student’s distribution
(Tsallis09Introduction, ).

The ubiquity of both Student’s distribution and the linear-log exponential
distribution arises from the fact that the grading between linear measurement
scaling at small magnitudes and logarithmic measurement scaling at large
magnitudes is inevitably widespread. Many cases will be primarily in the
linear regime and so be mostly exponential or Gaussian except in the extreme
tails. Many other cases will be primarily in the logarithmic regime and so be
mostly power law except in the regime of small deviations near the origin or
the central location. Other cases will produce measurements across both scales
and their transition.

##  4 The Inverse Logarithmic to Linear Measurement Scale

We have argued that the linear to logarithmic measurement scale is likely to
be common. Magnitudes such as time or distance naturally grade from linear at
small scales to logarithmic at large scales.

Many problems measure inverse dimensions, such as the reciprocals of time or
distance. If magnitudes of time or space naturally grade from linear to
logarithmic as scale increases from small to large, then how do the
reciprocals scale? In this section, we argue that the inverse scale naturally
grades from logarithmic to linear as scale increases from small to large.

We first describe the logarithmic to linear measurement scale and its
consequences for probability. We then show the sense in which the logarithmic
to linear scale is the natural inverse of the linear to logarithmic scale.

###  4.1 Measurement

The transformation

$\textrm{T}(x)=x+b\log(x)$

corresponds to the change in measurement scale $m_{x}=1+b/x$. As $x$ becomes
small, the measurement scaling $m_{x}\rightarrow 1/x$ becomes the ratio-
invariant logarithmic scale. As $x$ increases, the measurement scaling
$m_{x}\rightarrow 1$ becomes the uniform measure associated with the standard
linear scale. Thus, the scaling $m_{x}=1+b/x$ interpolates between logarithmic
and linear measurements, with the weighting of the two scales shifting from
logarithmic to linear as $x$ increases from small to large values.

###  4.2 Probability

The constraint for maximum entropy corresponds to
$\left\langle\textrm{T}[f(y)]\right\rangle=\left\langle
f(y)+b\log[f(y)]\right\rangle$, a value that interpolates between the linear
mean and the geometric mean of $f(y)$. Given that constraint, the maximum
entropy distribution is $p_{y}\propto f(y)^{-\lambda b}e^{-\lambda f(y)}$,
with $\lambda$ chosen to satisfy the constraint.

The direct measure $f(y)=y$ for positive values is the gamma distribution

$p_{y}\propto y^{-\lambda b}e^{-\lambda y}$ (13)

As $y$ becomes small, the distribution approaches a power law form,
$p_{y}\propto y^{-\lambda b}$. As $y$ becomes large, the distribution
approaches an exponential form in the tails, $p_{y}\propto e^{-\lambda y}$.
Thus, the distribution grades from power law at small scales to exponential at
large scales, corresponding to the measurement scale that grades from
logarithmic to linear as magnitude increases. Larger values of $b$ extend the
power law to higher magnitudes by pushing the logarithmic to linear change in
measure to higher magnitudes. The combination of power law and exponential
shapes in the gamma distribution is the direct inverse of the linear-log
exponential distribution given in Equation (11).

The squared values $f(y)=y^{2}$, which we interpret as squared deviations from
the average value, lead to

$p_{y}\propto y^{-\lambda b}e^{-\lambda y^{2}/2}$ (14)

where the exponent of two on the first power law component is subsumed in the
other parameters. This distribution is a power law at small scales with
Gaussian tails at large scales, providing the inverse of Student’s
distribution in Equation (12). This distribution is a form of the generalized
gamma distribution (Johnson94Distributions, ), which we call the gamma-Gauss
distribution. This distribution may, for example, arise as the sum of
truncated power laws or Lévy flights (Frank09The-common, ).

##  5 Integral Transforms and Superstatistics

The previous sections showed that linear to logarithmic scaling has a simple
relation to its inverse of logarithmic to linear scaling. That simple relation
suggests that the two inverse scales can be connected by some sort of
transformation of measure. We will now show the connection.

Suppose we start with a particular measurement scale given by $\textrm{T}(x)$
and its associated probability distribution given by

$p_{x}\propto e^{-\alpha\textrm{T}(x)}$

Consider a second measurement scale $\widetilde{\textrm{T}}(\sigma)$ with
associated probability distribution

$p_{\sigma}\propto e^{-\lambda\widetilde{\textrm{T}}(\sigma)}$

What sort of transformation relates the two measurement scales?

The integral transforms often provide a way to connect two measurement scales.
For example, we could write

$p_{x}\propto\int_{\sigma^{-}}^{\sigma^{+}}p_{\sigma}g_{x|\sigma}{\hbox{\rm
d}}\sigma$ (15)

This expression is called an integral transform of $p_{\sigma}$ with respect
to the transformation kernel $g_{x|\sigma}$. If we interpret $g_{x|\sigma}$ as
a probability distribution of $x$ given a parameter $\sigma$, and $p_{\sigma}$
as a probability distribution over the variable parameter $\sigma$, then the
expression for $p_{x}$ is called a superstatistic: the probability
distribution, $p_{x}$, that arises when one starts with a different
distribution, $g_{x|\sigma}$, and averages that distribution over a variable
parameter with distribution $p_{\sigma}$ (Beck03Superstatistics, ).

It is often useful to think of a superstatistic as an integral transform that
transforms the measurement scale. In particular, we can expand Equation (15)
as

$e^{-\alpha\textrm{T}(x)}\propto\int_{\sigma^{-}}^{\sigma^{+}}e^{-\lambda\widetilde{\textrm{T}}(\sigma)}g_{x|\sigma}{\hbox{\rm
d}}\sigma$ (16)

which shows that the transformation kernel $g_{x|\sigma}$ changes the
measurement scale from $\widetilde{\textrm{T}}(\sigma)$ to $\textrm{T}(x)$. It
is not necessary to think of $g_{x|\sigma}$ as a probability distribution—the
essential role of $g_{x|\sigma}$ concerns a change in measurement scale.

The Laplace transform provides the connection between our inverse linear-
logarithmic measurement scales. To begin, expand the right side of Equation
(16) using the Laplace transform kernel $g_{x|\sigma}=e^{-\sigma x}$, and use
the inverse logarithmic to linear measurement scale,
$\widetilde{\textrm{T}}(\sigma)=\sigma+b\log(\sigma)$. Integrating from zero
to infinity yields

$e^{-\alpha\textrm{T}(x)}\propto(1+x/\lambda)^{b\lambda-1}$

with the requirement that $b\lambda<1$. From this, we have
$\textrm{T}(x)\propto\log(1+x/\lambda)$, which is the linear to logarithmic
scale. Thus, the Laplace transform inverts the logarithmic to linear scale
into the linear to logarithmic scale. The inverse Laplace transform converts
in the other direction.

If we use $x=y$, then the transform relates the linear-log exponential
distribution of Equation (11) to the gamma distribution of Equation (13). If
we use $x=y^{2}$, then the transform relates Student’s distribution of
Equation (12) to the gamma-Gauss distribution of Equation (14).

The Laplace transform inverts the measurement scales. This inversion is
consistent with a common property of Laplace transforms, in which the
transform inverts a measure with dimension $d$ to a measure with dimension
$1/d$. One sometimes interprets the inversion as a change from a direct
measure to a rate or frequency. Here, it is only the inversion of dimension
that is significant. The inversion arises because, in the transformation
kernel $g_{x|\sigma}=e^{-\sigma x}$, the exponent $\sigma x$ is typically non-
dimensional, so that the dimensions of $\sigma$ and $x$ are reciprocals of
each other. The transformation takes a distribution in $\sigma$ given by
$p_{\sigma}$ and returns a distribution in $x$ given by $p_{x}$. Thus, the
transformation typically inverts the dimension.

##  6 Connections and Caveats

###  6.1 Discrete versus continuous variables

We used measure invariance to analyze maximum entropy for continuous
variables. We did not discuss discrete variables, because measure invariance
applied to discrete variables has been widely and correctly used in maximum
entropy (Jaynes03Science, ; Sivia06Tutorial, ; Frank09The-common, ). In the
Discussion, we describe why previous attempts to apply invariance to
continuous variables did not work in general. That failure motivated our
current study.

Here, we briefly review measure invariance in the discrete case for comparison
with our analysis of continuous variables. We use the particular example of
$N$ Bernoulli trials with a sample measure of the number of successes
$y=0,1,\ldots,N$. Frank (Frank09The-common, ) describes the measure invariance
for this case: “How many different ways can we can obtain $y=0$ successes in
$N$ trials? Just one: a series of failures on every trial. How many different
ways can we obtain $y=1$ success? There are $N$ different ways: a success on
the first trial and failures on the others; a success on the second trial, and
failures on the others; and so on. The uniform solution by maximum entropy
tells us that each different combination is equally likely. Because each value
of $y$ maps to a different number of combinations, we must make a correction
for the fact that measurements on $y$ are distinct from measurements on the
equally likely combinations. In particular, we must formulate a measure…that
accounts for how the uniformly distributed basis of combinations translates
into variable values of the number of successes, $y$. Put another way, $y$ is
invariant to changes in the order of outcomes given a fixed number of
successes. That invariance captures a lack of information that must be
included in our analysis.”

In this particular discrete case, transformations of order do not change our
information about the total number of successes. Our measurement scale
expresses that invariance, and that invariance is in turn captured in the
maximum entropy distribution.

The nature of invariance is easy to see in the discrete case by combinatorics.
The difficulty in past work has been in figuring out exactly how to capture
the same notion of invariance in the continuous case. We showed that the
answer is perhaps as simple as it could be: use the transformations that do
not change information in the context of a particular problem. Jaynes
(Jaynes03Science, ) hinted at this approach, but did not develop and apply the
idea in a general way.

###  6.2 General measure invariance versus particular linear-log scales

Our analysis followed two distinct lines of argument. First, we presented the
general expression for invariance as a form of information in maximum entropy.
We developed that expression particularly for the case of continuous
variables. The general expression sets the conditions that define measurement
scales and the relation between measurement and probability. But the general
expression does not tell us what particular measurement scale will arise in
any problem.

Our second line of argument claimed that various types of grading between
linear and logarithmic measures arise very commonly in natural problems. Our
argument for commonness is primarily inductive and partially subjective. On
the inductive side, the associated probability distributions seem to be those
most commonly observed in nature. On the subjective side, the apparently
simplest assumptions about invariance lead to what we called the common
gradings between linear and logarithmic scales. We do not know of any way to
prove commonness or naturalness. For now, we are content that the general
mathematical arguments lead in a simple way to those probability distributions
that appear to arise commonly in nature.

A different view of what is common or what is simple would of course lead to
different information invariances, measurement scales, and probability
distributions. In that case, our general mathematical methods would provide
the tools by which to analyze the alternative view.

##  7 Discussion

We developed four topics. First, we provided a new extension to the method of
maximum entropy in which we use the measurement scale as a primary type of
information constraint. Second, we argued that a measurement scale that grades
from linear to logarithmic as magnitude increases is likely to be very common.
The linear-log exponential and Student’s distributions follow immediately from
this measurement scale. Third, we showed that the inverse measure that grades
from logarithmic at small scales to linear at large scales leads to the gamma
and gamma-Gauss distributions. Fourth, we demonstrated that the two
measurement scales are natural inverses related by the Laplace integral
transform. Superstatistics are a special case of integral transforms and can
be understood as changes in measurement scale.

In this discussion, we focus on measurement invariance, alternative
definitions of entropy, and maximum entropy methods.

Jaynes (Jaynes03Science, ) summarized the problem of incorporating measurement
invariance as a form of information in maximum entropy. The standard
conclusion is that one should use relative entropy to account for measurement
invariance. In our notation, for a measurement scale $\textrm{T}(y)$ with
measure deformation $m_{y}=\textrm{T}^{\prime}(y)$, the form of relative
entropy is the Kullback-Leibler divergence

${\cal{E}}=-\int p_{y}\log\left({{p_{y}}\over{m_{y}}}\right){\hbox{\rm d}}y$

in which the $m_{y}$ is proportional to a prior probability distribution that
incorporates the information from the measurement scale and leads to the
analysis of maximum relative entropy. This approach works in cases where the
measure change, $m_{y}$, is directly related to a change in the probability
measure. Such changes in probability measure typically arise in combinatorial
problems, such as a type of measurement that cannot distinguish between the
order of elements in sets.

For continuous deformations of the measurement scale, using $m_{y}$ as a
relative scaling for probability does not always give the correct answer. In
particular, if one uses the constraint $\left\langle f(y)\right\rangle$ and
the measure $m_{y}$ in the above definition of relative entropy, the maximum
relative entropy gives the probability distribution

$p_{y}\propto m_{y}e^{-\lambda f(y)}$

which is often not the correct result. Instead, the correct result follows
from the method we gave, in which the information from measurement invariance
is incorporated by transforming the constraint as
$\left\langle\textrm{T}[f(y)]\right\rangle$, yielding the maximum entropy
solution

$p_{y}\propto e^{-\lambda\textrm{T}[f(y)]}$

It is possible to change the definition of entropy, such that maximum entropy
applied to the transformed measure of entropy plus the direct constraint
$\left\langle f(y)\right\rangle$ gives the correct answer
(Tsallis09Introduction, ). The resulting probability distributions are of
course the same when transforming the constraint or using an appropriate
matching transformation of the entropy measure. We discuss the mathematical
relation between the alternative transformations in a later paper.

We prefer the transformation of the constraint, because that approach directly
shows how information about measurement alters scale and is incorporated into
maximum entropy. By contrast, changing the definition of entropy requires each
measurement scale to have its own particular definition of entropy and
information. Measurement is inherently an empirical phenomenon that is
particular to each type of problem and so naturally should be changed as the
nature of the problem changes. The abstract notions of entropy and information
are not inherently empirical factors that change among problems, so it seems
perverse to change the definition of randomness with each transformation of
measurement.

The Tsallis and Rényi entropy measures are transformations of Shannon entropy
that incorporate the scaling of measurement from linear to logarithmic as
magnitude increases (Tsallis09Introduction, ). Those forms of entropy
therefore could be used as a common alternative to Shannon entropy whenever
measurements scale linearly to logarithmically. Although mathematically
correct, such entropies change the definition of randomness to hide the
particular underlying transformation of measurement. That approach makes it
very difficult to understand how alternative measurement scales alter the
expected types of probability distributions.

##  8 Conclusions

Linear and logarithmic measurements seem to be the most common natural scales.
However, as magnitudes change, measurement often grades between the linear and
logarithmic scales. That transition between scales is often overlooked. We
showed how a measurement scale that grades from linear to logarithmic as
magnitude increases leads to some of the most common patterns of nature
expressed as the linear-log exponential distribution and Student’s
distribution. Those distributions include the exponential, power law, and
Gaussian distributions as special cases, and also include hybrids between
those distribution that must commonly arise when measurements span the linear
and logarithmic regimes.

We showed that a measure grading from logarithmic to linear as magnitude
increases is a natural inverse scale. That measurement scale leads to the
gamma and gamma-Gauss distributions. Those distributions are also composed of
exponential, power law, and Gaussian components. However, those distributions
have the power law forms at small magnitudes corresponding to the logarithmic
measure at small magnitudes, whereas the inverse scale has the power law
components at large magnitudes corresponding to the logarithmic measure at
large magnitudes.

The two measurement scales are natural inverses connected by the Laplace
transform. That transform inverts the dimension, so that a magnitude of
dimension $d$ on one scale becomes an inverse magnitude of dimension $1/d$ on
the other scale. Inversion connects the two major scaling patterns commonly
found in nature. Our methods of incorporating information about measurement
scale into maximum entropy also apply to other forms of measurement scaling
and invariance, providing a general method to study the relations between
measurement, information, and probability.

## Acknowledgements

SAF is supported by National Science Foundation grant EF-0822399, National
Institute of General Medical Sciences MIDAS Program grant U01-GM-76499, and a
grant from the James S. McDonnell Foundation. DES thanks Insight Venture
Partners for support.

## References

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# Information-Theoretic Bias Reduction via Causal View of Spurious Correlation

Seonguk Seo1 Joon-Young Lee2 Bohyung Han1

###### Abstract

We propose an information-theoretic bias measurement technique through a
causal interpretation of spurious correlation, which is effective to identify
the feature-level algorithmic bias by taking advantage of conditional mutual
information. Although several bias measurement methods have been proposed and
widely investigated to achieve algorithmic fairness in various tasks such as
face recognition, their accuracy- or logit-based metrics are susceptible to
leading to trivial prediction score adjustment rather than fundamental bias
reduction. Hence, we design a novel debiasing framework against the
algorithmic bias, which incorporates a bias regularization loss derived by the
proposed information-theoretic bias measurement approach. In addition, we
present a simple yet effective unsupervised debiasing technique based on
stochastic label noise, which does not require the explicit supervision of
bias information. The proposed bias measurement and debiasing approaches are
validated in diverse realistic scenarios through extensive experiments on
multiple standard benchmarks.

## Introduction

Various recognition algorithms based on deep neural networks have achieved
remarkable performance improvement by learning useful patterns from a large
number of training examples, and have started to be deployed in many real-
world applications. However, the objective of the optimization problem is
mainly concerned about the final accuracy, which makes trained models
vulnerable to unexpected decision rules affected by spurious correlation.
Although such decision rules work well on most of training examples, they
often lead to poor worst-case generalization performance and make learned
models unfair to the examples in under-represented groups. For instance, a few
existing works (Kim 2016; Buolamwini and Gebru 2018) have discovered the
performance gap across demographic subgroups in real-world face recognition
tasks; the accuracy of a model on _darker-skinned women_ is often much lower
than that on _lighter-skinned men_. This tendency becomes a major weakness in
achieving algorithmic fairness and generalizing on unseen test environments
with domain or distribution shifts.

Mitigating spurious correlation has recently emerged as an important issue in
learning debiased model (Hardt, Price, and Srebro 2016; Zhao et al. 2017;
Zhang, Lemoine, and Mitchell 2018; Li and Vasconcelos 2019; Wang et al. 2019;
Gong, Liu, and Jain 2020; Sagawa et al. 2020b, a). Given the target and bias
variables, they typically suppress spurious correlation by making the
algorithm independent of the bias variables while maintaining the predictions
to the target variables. To this end, various measurements about algorithmic
fairness or bias have been introduced, e.g., demographic parity (Calders,
Kamiran, and Pechenizkiy 2009), equality of odds (Hardt, Price, and Srebro
2016), and group-fairness accuracy (Sagawa et al. 2020a; Zhang et al. 2020).
Although they have been widely used to maintain fairness, their accuracy- or
logit-based bias measurement schemes may lead to inaccurate algorithmic bias
measurement. For example, we observe that fine-tuning the last linear
classification layer of a model is sufficient to achieve high unbiased
accuracy and worst-group accuracy, even without updating the biased
representations.

To better quantify algorithmic bias, we first formulate the spurious
correlation from a causal point of view. Following the strict definition of
fairness, we derive the condition of independence between a bias variable and
a predicted target variable, which is evaluated by their mutual information.
However, in our causal view, it gives a biased estimation of spurious
correlation. To handle the issue, we propose a new bias measurement technique
based on conditional mutual information using feature representations, which
is claimed to quantify the algorithmic bias more accurately while maintaining
the original objective, fairness.

Based on the new bias measurement, we propose a debiasing framework to
mitigate the algorithmic bias by augmenting a loss term for bias
regularization, which is derived by the conditional mutual information. We
also introduce a simple yet effective unsupervised debiasing technique by
exploiting stochastic label noise, which also prevents a model from capturing
spurious correlation even without the prior knowledge of bias information. Our
experiments verify that both approaches are helpful for alleviating the
algorithmic bias while preserving model accuracy.

The main contributions of our work are summarized as follows.

  * $\bullet$

We propose an information-theoretic measurement technique for the amount of
algorithmic bias via conditional mutual information on learned feature
representations, which is derived from the causal view of spurious
correlation.

  * $\bullet$

Based on the new bias measurement approach, we propose two novel debiasing
frameworks; one employs a information-theoretic bias regularization loss and
the other is with stochastic label noise even without the supervision of bias
information.

  * $\bullet$

We evaluate our bias measurement scheme and debiasing techniques in various
realistic scenarios and achieve promising results on multiple standard
benchmarks.

## Setup and Preliminaries

Let $(X,Y,Z)\in\mathbb{R}^{m}\times\mathbb{R}\times\mathbb{R}$ be a triplet
representing a joint distribution over the space
$(\mathcal{X},\mathcal{Y},\mathcal{Z})$, where $X$ is an input variable, $Y$
is a target variable, and $Z$ is a bias variable. We denote a learned model by
a function $c\in\mathcal{C}$ mapping $\mathcal{X}$ to $\mathcal{Y}$. The
classification function $c:X\rightarrow\hat{Y}$ consists of the feature
extractor $g:X\rightarrow F$ and the linear classifier
$h:F\rightarrow\hat{Y}$, i.e., $c(\cdot)=h(g(\cdot))$, where
$F\in\mathbb{R}^{d}$ is a feature representation and $\hat{Y}\in\mathcal{Y}$
is the predicted target variable. Following the conventional notations in
probability theory, we will use the comma (,) to denote the joint distribution
and the semicolon (;) to separate the input arguments of mutual information.
For example, $I(X;Y,Z)$ indicates the mutual information between $X$ and the
joint distribution of $Y$ and $Z$.

### Group-Fairness Accuracy

In classification tasks, unbiased accuracy, worst-group accuracy, or accuracy
disparity are employed to address non-uniform accuracy and evaluate algorithm
fairness and robustness in the presence of dataset bias (Sagawa et al. 2020a;
Zhang et al. 2020; Koh et al. 2021; Zhang and Sang 2020). Formally, if
$Y_{i}\in\\{1,...,A\\}$ and $Z_{i}\in\\{1,...,B\\}$ denote the target and bias
values of $X_{i}$, respectively, then the unbiased accuracy is given by

$\displaystyle\frac{1}{AB}\sum_{a,b}\frac{\sum_{i}{\mathbbm{1}(c(X_{i})=Y_{i}=a,Z_{i}=b)}}{\sum_{i}{\mathbbm{1}(Y_{i}=a,Z_{i}=b)}},$
(1)

which indicates the average accuracy over all groups defined by a pair of
target and bias values. Other metrics, the worst-group accuracy and the
accuracy disparity, are obtained by the worst accuracy of all groups and the
discrepancy between the best and the worst group, respectively.

However, the unbiased accuracy may not be a good metric for evaluating the
amount of bias in data by itself. Given a baseline model converged
sufficiently with hair color classification on the CelebA dataset, we fix its
feature extractor $f(\cdot)$ and fine-tune its classification layer $h(\cdot)$
with a simple resampling technique to reduce class imbalance problem. We
select gender as a bias variable for evaluation. Although the features in this
model remain unchanged even after the fine-tuning phase, it attains a
significant performance gain, 8% points in the unbiased accuracy and 19%
points in the worst-group accuracy. This result implies that the accuracy- or
logit-based bias measurement methods may not be able to capture the innate
bias residing in the features, and that high fairness scores can be achieved
by adjusting prediction scores in linear classification layers.

### Estimating Mutual Information

To identify the feature-level bias, we first introduce the mutual information,
which measures the co-dependence between two variables. For random variables
$X$ and $Y$ over the space $\mathcal{X}\times\mathcal{Y}$, the mutual
information is defined as

$\displaystyle I(X;Y)=H(X)-H(X|Y)=H(Y)-H(Y|X),$ (2)

where $H(\cdot)$ denotes the Shannon entropy. The mutual information is also
defined by the Kullback-Leibler (KL) divergence between the joint distribution
of two random variables and the products of their marginal distributions:

$\displaystyle I(X;Y)$ $\displaystyle=D_{KL}(P_{(X,Y)}||P_{X}\otimes P_{Y}),$
(3)

which implies that the larger the divergence between the joint and the product
of the marginals is, the stronger the dependence between $X$ and $Y$ is.

However, its exact computation is tractable only for discrete variables or
continuous variables under some constraints. Recently, Belghazi et al.
(Belghazi et al. 2018) propose a neural estimator of mutual information (MINE)
between continuous high-dimensional random variables, by offering a lower
bound of the mutual information based on the Donsker-Varadhan
representation111The Donsker-Varadhan representation provides the KL-
divergence as a supremum over all functions $T$: $D_{\text{KL}}(P\parallel
Q)=\sup_{T:\Omega\rightarrow\mathbb{R}}\mathbb{E}_{P}[T]-\log(\mathbb{E}_{Q}[\exp(T)])$.
(Donsker and Varadhan 1975),

$\displaystyle I(X;Y)\geq I_{\phi}(X;Y)$
$\displaystyle=\sup_{\phi\in\Phi}\mathbb{E}_{P_{(X,Y)}}[f_{\phi}(x,y)]$ (4)
$\displaystyle-\log(\mathbb{E}_{P_{X}\otimes{P_{Y}}}[\exp(f_{\phi}(x,y))]),$

where $f_{\phi}(\cdot,\cdot)$ is a statistical neural network parameterized by
$\phi\in\Phi$. The expectations $\mathbb{E}_{P_{(X,Y)}}$ and
$\mathbb{E}_{P_{X}\otimes{P_{Y}}}$ are approximated using empirical sampling
from the joint distributions and the products of the marginal distributions,
respectively. By maximizing the right-hand side of (Estimating Mutual
Information) with respect to $\phi$, we can obtain a tighter lower bound,
which leads to a more accurate estimation of the mutual information.

## Cobias: Bias Measurement with Conditional Mutual Information

(a) The diagram of structural causal model. We aim to measure the co-
dependence between $Z$ and $F$ via $Z\rightarrow X\rightarrow F$, but it is
tricky because there exists a backdoor path $Z\leftarrow C\rightarrow
Y\rightarrow X\rightarrow F$.

(b) Venn diagram of information-theoretic measures for $F,Y,$ and $Z$. The
region in orange denotes the mutual information between $F$ and $Z$, excluding
the amount of information that explains $Y$.

Figure 1: The causal and information-theoretic diagrams of our problem
setting.

We interpret the spurious correlation in a causal view, and present the bias
measurement technique based on the conditional independence derived from the
causal view.

### Causal Model of Spurious Correlation

We formulate the causal relationship among input image $X$, target label $Y$,
bias variable $Z$, context prior $C$, and feature representation $F$ using a
structural causal model (Spirtes et al. 2000). Figure 1(a) illustrates the
relationship, where the direct link $A\rightarrow B$ indicates that $A$ is the
cause of $B$. Note that we adopt the feature vector $F$, instead of logits or
predicted target values as discussed before.

From the viewpoint of information leakage (Dwork et al. 2012), it is possible
to quantify the algorithmic bias by measuring the co-dependence between
feature and bias variables via mutual information, e.g., $I(Z;F)$. However, as
depicted in Figure 1(a), the path between $Z$ and $F$ contains not only the
direct path $Z\rightarrow X\rightarrow F$, but also a backdoor path
$Z\leftarrow C\rightarrow Y\rightarrow X\rightarrow F$. Because the model is
trained to maximize the mutual information between $Y$ and $F$ via
$Y\rightarrow X\rightarrow F$, the backdoor path is constructed naturally if
there exists the correlation between $Y$ and $Z$. To focus on the direct path,
we should block the backdoor path, which is easily done by conditioning on
$Y$. Since the path $C\rightarrow Y\rightarrow X$ is a serial connection,
conditioning on $Y$ blocks the backdoor path and we can now measure the mutual
information between $Z$ and $F$ via the direct path only.

### Cobias: Conditional Mutual Information as Bias Measurement

Generalizing the mutual information to multivariate cases, we derive the
conditional mutual information with three variables $X$, $Y$, and $Z$, which
is given by

$\displaystyle I(X;Y|Z)$
$\displaystyle=\mathbb{E}_{\mathcal{Z}}[D_{KL}(P_{(X|Z,Y|Z)}||P_{X|Z}\otimes
P_{Y|Z})].$ (5)

Note that this equation is an extension of (3) to a conditional setting. The
conditional mutual information measures the conditional independence between
the relevant variables, i.e., $I(X;Y|Z)=0\Leftrightarrow X\perp Y|Z$. Compared
to the standard setting, this is particularly important if co-dependence
between $X$ and $Y$ changes conditioned on variable $Z$.

We also consider three variables in our problem setting, which include feature
representation vector $F$, target variable $Y$, and bias variable $Z$. As
discussed earlier, we concentrate on the mutual information between $F$ and
$Z$ conditioned on $Y$ and derive a bias measurement as follows:

$\displaystyle\text{Cobias}:=I(F;Z|Y)$ $\displaystyle=I(F;Z)-I(F;Z;Y)$ (6)
$\displaystyle=I(F;Z,Y)-I(F;Y),$ (7)

where $I(F;Z;Y)$ is an interaction information among three variables and
$(Z,Y)\in\mathbb{R}^{2}$ denotes the joint variables of $Z$ and $Y$. We
estimate the conditional mutual information by computing the difference
between two mutual information $I(F;Z,Y)$ and $I(F;Y)$ as expressed in (7),
which is derived from the chain rule of mutual information.

Figure 1(b) presents a Venn diagram of the information-theoretic measures for
those variables. Our bias measurement $I(F;Z|Y)$ quantifies the mutual
information between feature and bias variable $I(F;Z)$ excluding the amount of
the information that also explains the target variable $I(F;Y;Z)$, which
corresponds to the region in orange.

## Debiasing Frameworks

This section introduces the proposed two debiasing techniques, which are given
by the bias-supervised debiasing regularization and the unsupervised
stochastic label noise.

### Cobias as Debiasing Regularizer

To mitigate the algorithmic bias, we directly incorporate the proposed bias
measurement $I(F;Z|Y)$ as an additional loss term to learn our model. Then,
given the extra supervision about bias attributes $Z$, the final objective
function is formulated as

$\displaystyle\min_{\theta,\psi}\ell(h(F;\psi),Y)+\beta I(F;Z|Y)$
$\displaystyle=\min_{\theta,\psi}\ell(h(f(X;\theta);\psi),Y)+\beta
I_{\phi}(f(X;\theta);Z|Y),$ (8)

where $\ell(\cdot,\cdot)$ is a cross-entropy loss, $\theta$ and $\psi$ denote
the parameters of the feature extractor $f(\cdot)$ and the classification
layer $h(\cdot)$, respectively, $\phi$ is the parameters of the mutual
information estimator network. The second term in the objective function plays
a role as a regularizer to prevent the feature representation from being
biased, where $\beta$ is the hyperparameter of its weight. Note that this
regularization term aims to reduce inherent bias within feature
representations and has complementary characteristics to the task-specific
loss term. Since the loss does not depend on model architecture or algorithm,
it can be easily adopted in other existing debiasing frameworks. Because the
feature representation $F=f(X;\theta)$ is updated during training, the
conditional mutual information $I(F;Z|Y)$ also needs to be re-estimated each
time, which results in the revision of the objective function as

$\displaystyle\min_{\theta,\psi}\max_{\phi}\ell(h(f(X;\theta);\psi),Y)+\beta
I_{\phi}(f(X;\theta);Z|Y).$ (9)

Because the objective function is a minimax problem, we alternate to train and
update the parameters $(\theta,\psi)$ and $\phi$ in every epoch.

To compute $I(F;Z|Y)$, we should measure the difference between two estimated
mutual information values, $I_{\phi_{1}}(F;Z,Y)$ and $I_{\phi_{2}}(F;Y)$ as
shown in (7). However, minimizing the difference of the two values obtained
from the two different estimators parameterized by $\phi_{1}$ and $\phi_{2}$
may hamper training stability. We sidestep this issue by minimizing a
surrogate mutual information, $I(F,Y;Z)$, with respect to $\theta$, instead of
$I(F;Z|Y)$. Because $I(f(X;\theta),Y;Z)=I(f(X;\theta);Z|Y)+I(Y;Z)$ and
$I(Y;Z)$ is a constant independent of $\theta$, minimizing
$I(f(X;\theta);Z|Y)$ with respect to $\theta$ is equivalent to minimizing
$I(f(X;\theta),Y;Z)$. This enables us to train the network based on (9) using
a single estimator network and consequently facilitates stable training.

### Stochastic Label Noise

We now introduce a simple yet effective unsupervised debiasing approach
exploiting synthetic label noise. Stochastic label noise perturbs the target
label from its true class to any other class with a noise rate $\rho$ on each
mini-batch independently. Let $Y$ and $\widetilde{Y}$ be the true and noisy
target labels, respectively. Then, $p(\widetilde{Y}=k|Y=y)=\frac{\rho}{K-1}$
for all $k\neq y$ and $1-\rho$ for $k=y$, where $K$ is the number of classes.
The objective function is given by

$\displaystyle\min_{\theta,\psi}\ell(h(f(X;\theta);\psi),\widetilde{Y}).$ (10)

Stochastic label noise, even without any explicit supervision of bias
information, mitigates the algorithmic bias while minimizing the performance
degradation in the classification for target variable. Intuitively, this is
because spurious correlation is more susceptible to label noise than true
causation. Label noise gives an implicit ensemble effect (Xie et al. 2016),
which helps to find invariant properties between a feature representation and
its target label. Because spurious correlation does not appear to be stable
properties (Arjovsky et al. 2019; Woodward 2005), label noise helps to absorb
spurious correlation and mitigate algorithmic bias.

From the perspective of information theory, it is natural that injecting label
noise reduces the mutual information between target variable $Y$ and bias
variable $Z$, i.e., $I(Z;\widetilde{Y})<I(Z;Y)$. In addition, by adding
stochastic label noise, the reduction of mutual information between $Y$ and
$Z$ turns out to be more significant than that of $Y$’s entropy. According to
our experiment, the ratio $R:={I(Z;\widetilde{Y})}/{I(Y;\widetilde{Y})}$ is
0.187 in the absence of label noise with $Y=$ “hair color” and $Z=$ “gender”
in the CelebA dataset. However, it decreases when we add several different
levels of label noise, and the tendency is consistent in other $(Y,Z)$ pairs
if they are correlated. This observation implies that injecting label noise is
helpful for alleviating spurious correlation without affecting classification
performance. Note that this label perturbation method is orthogonal to
existing debiasing frameworks and can be applied to them with no modification.

## Experiments

### Experimental Setup

#### Datasets

We conduct experiments on the three standard benchmarks: CelebA (Liu et al.
2015), Waterbirds (Sagawa et al. 2020a), and FairFace (Kärkkäinen and Joo
2021). CelebA is a large-scale face dataset composed of 202,599 celebrity
images with 40 attributes. This dataset is available for non-commercial
research purposes. We follow the original train-val-test split (Liu et al.
2015) throughout the experiments. Waterbirds (Sagawa et al. 2020a) is a
synthesized dataset with 4,795 training examples, which are created by
combining bird images in the CUB dataset (Wah et al. 2011) and background
images from the Places dataset (Zhou et al. 2017). Following the setup in
(Sagawa et al. 2020a), each image has two attributes; one is the type of bird,
$\\{$waterbird, landbird$\\}$ and the other is the background place,
$\\{$water, land$\\}$. FairFace (Kärkkäinen and Joo 2021) is a recently
proposed face image dataset containing 108,501 images, which are collected
from the YFCC-100M Flickr dataset (Thomee et al. 2016). The dataset has seven
race groups222White, Black, Indian, East Asian, Southeast Asian, Middle
Eastern, and Latino., nine age groups, and two gender groups.

#### Implementation details

We use the ResNet-18 (He et al. 2016) pretrained on ImageNet (Deng et al.
2009) as our backbone network for all experiments. We train our models using
the stochastic gradient descent method with the Adam optimizer for 50 epoch.
The learning rate is $1\times 10^{-4}$, and the batch size is 256. We set a
weight decay to $1\times 10^{-2}$ for Waterbirds and $1\times 10^{-4}$ for the
other two datasets. The weight of the bias regularizer $\beta$ is fixed to 5.
The noise rate $\rho$ is set to 0.1 for Waterbirds and 0.2 for the rest. For
all methods, we leverage a simple resampling technique based on the class size
to alleviate the class imbalance issue. Our algorithms are implemented in the
Pytorch (Paszke et al. 2019) framework and all experiments are conducted on a
single unit of NVIDIA Titan XP GPU.

#### Evaluation metrics

In addition to average accuracy, we evaluate all the compared algorithms with
three main metrics, Cobias, unbiased accuracy, and worst-group accuracy, to
provide a comprehensive view of the algorithmic bias. We also adopt other
fairness metrics such as bias amplification (BA) (Zhao et al. 2017), equalized
opportunity difference (EO), and disparate impact (DI) (Hardt, Price, and
Srebro 2016), where low values are preferred for the extra metrics.

### Results

Table 1: Experimental results of our debiasing frameworks on the test split of
the CelebA dataset. We set gender attribute to the bias variable.

| Target | Cobias | Unbiased Acc. | Worst-group Acc. |  BA |  EO |  DI | Average Acc.
---|---|---|---|---|---|---|---|---
ERM | Hair Color | 0.435 | 84.1 | 53.2 |  0.014 | 0.48 | 0.113 | 95.1
ERM + label noise | Hair Color | 0.286 | 87.8 | 66.9 | -0.002 | 0.35 | 0.036 | 93.8
ERM + bias regularizer | Hair Color | 0.111 | 88.1 | 67.4 |  0.001 | 0.31 | 0.039 | 94.5
Group DRO (Sagawa et al. 2020a) | Hair Color | 0.409 | 89.4 | 77.6 |  0.005 | 0.17 | 0.044 | 94.2
Group DRO + label noise | Hair Color | 0.270 | 90.6 | 84.0 | -0.004 | 0.07 | 0.030 | 93.5
Group DRO + bias regularizer | Hair Color | 0.045 | 90.7 | 85.4 |  0.002 | 0.09 | 0.018 | 94.0
ERM | Pale Skin | 0.387 | 82.7 | 59.6 |  0.012 | 0.38 | 0.273 | 95.5
ERM + label noise | Pale Skin | 0.192 | 87.6 | 73.8 |  0.002 | 0.19 | 0.049 | 94.7
ERM + bias regularizer | Pale Skin | 0.066 | 86.1 | 68.7 |  0.006 | 0.28 | 0.153 | 95.2
Group DRO (Sagawa et al. 2020a) | Pale Skin | 0.346 | 88.8 | 84.6 |  0.005 | 0.11 | 0.109 | 93.8
Group DRO + label noise | Pale Skin | 0.189 | 90.3 | 86.4 |  0.001 | 0.06 | 0.002 | 94.2
Group DRO + bias regularizer | Pale Skin | 0.114 | 89.1 | 85.5 |  0.003 | 0.10 | 0.063 | 93.9
ERM | Smiling | 0.126 | 92.1 | 88.6 |  0.002 | 0.01 | 0.008 | 92.5
ERM + label noise | Smiling | 0.017 | 92.4 | 88.7 | -0.015 | 0.10 | 0.057 | 92.8
ERM + bias regularizer | Smiling | 0.011 | 92.4 | 89.0 | -0.011 | 0.08 | 0.038 | 92.8
Group DRO (Sagawa et al. 2020a) | Smiling | 0.130 | 92.1 | 89.6 |  0.003 | 0.02 | 0.012 | 92.2
Group DRO + label noise | Smiling | 0.019 | 92.5 | 90.1 | -0.003 | 0.01 | 0.011 | 92.6
Group DRO + bias regularizer | Smiling | 0.024 | 92.7 | 89.9 | -0.005 | 0.01 | 0.019 | 92.8

#### CelebA

Table 1 presents the experiment results on the test split of the CelebA
dataset. Among all attributes, we choose blond hair, pale skin, and smiling as
target variables while gender is used as the bias variable, setting up three
different classification tasks. We employ two baseline algorithms, including
empirical risk minimization (ERM) and group distributionally robust
optimization (Group DRO) (Sagawa et al. 2020a), to which the proposed two
debiasing methods are applied. As shown in Table 1, incorporating the bias
regularization loss significantly reduces Cobias and other fairness scores,
while preserving the average classification accuracy. This implies that the
proposed bias measurement is meaningful for identifying the algorithmic bias.
We also observe that applying stochastic label noise is effective to mitigate
the algorithmic bias even without extra supervision, which validates our claim
that spurious correlation is more sensitive to label noise than true
causation. Note that Group DRO yields high bias scores in spite of its high
group-fairness accuracy, which implies that it may adjust the parameters for
bias-related features in the classification layer instead of learning debiased
representations.

Table 2: Experimental results on the test split of the Waterbirds dataset,
where bird type and background place are set to target and bias variables,
respectively.

| Cobias | Unbiased | Worst | Average
---|---|---|---|---
ERM | 0.482 | 81.4 | 52.3 | 83.6
ERM + label noise | 0.413 | 83.0 | 58.2 | 83.1
ERM + bias regularizer | 0.107 | 84.3 | 72.2 | 88.4
Group DRO | 0.376 | 86.3 | 72.4 | 85.7
Group DRO + label noise | 0.348 | 86.7 | 70.6 | 84.7
Group DRO + bias regularizer | 0.098 | 86.7 | 76.9 | 87.4

Table 3: Experimental results on the test split of the Fairface dataset, where
age and race are set to target and bias variables, respectively.

| Cobias | Unbiased | Worst | Average
---|---|---|---|---
ERM | 0.019 | 47.6 | 16.9 | 52.1
ERM + label noise | 0.017 | 48.7 | 18.6 | 53.2
ERM + bias regularizer | 0.004 | 49.6 | 18.5 | 53.4
Group DRO | 0.015 | 48.4 | 18.8 | 51.5
Group DRO + label noise | 0.004 | 49.2 | 16.8 | 52.5
Group DRO + bias regularizer | 0.002 | 49.4 | 19.3 | 53.6

#### Waterbirds

The results on the Waterbirds dataset are presented in Table 2, where the bird
type and the background place are set to target and bias variables,
respectively. According to our experiments, the proposed debiasing frameworks
consistently produce promising results in this small-scale synthesized
dataset. Compared to the CelebA dataset, injecting label noise is less
effective to reduce Cobias in this dataset, which is partly due to its small
size.

#### FairFace

We also evaluate our frameworks on the FairFace dataset and demonstrate the
results in Table 3, where the target is age and the bias is race. Our
frameworks still improves both Cobias and group-fairness accuracy when
combined with both ERM and Group DRO algorithms. Note that the Cobias scores
are particularly small because the FairFace dataset is constructed with an
emphasis of balanced race composition, as stated in (Kärkkäinen and Joo 2021).

### Analysis

#### Domain generalization scenario

Table 4: Experimental results in domain generalization setting on the CelebA
dataset, where the bias variable is fixed to gender for all settings.

| Target | Domain (train) | Domain (test) |  Cobias | Unbiased Acc. | Worst-group Acc. | Average Acc.
---|---|---|---|---|---|---|---
ERM | Hair Color | Old | Young | 0.432 | 81.5 | 43.5 | 93.8
ERM + label noise | Hair Color | Old | Young | 0.367 | 88.1 | 68.3 | 93.9
ERM + bias regularizer | Hair Color | Old | Young | 0.272 | 90.4 | 79.7 | 93.0
Group DRO (Sagawa et al. 2020a) | Hair Color | Old | Young | 0.387 | 86.2 | 68.1 | 94.2
Group DRO + label noise | Hair Color | Old | Young | 0.171 | 88.6 | 77.4 | 93.9
Group DRO + bias regularizer | Hair Color | Old | Young | 0.061 | 89.5 | 78.6 | 94.1
ERM | Hair Color | Slim | Chubby | 0.234 | 75.6 | 29.5 | 98.1
ERM + label noise | Hair Color | Slim | Chubby | 0.107 | 82.7 | 54.5 | 97.9
ERM + bias regularizer | Hair Color | Slim | Chubby | 0.073 | 84.5 | 54.2 | 97.9
Group DRO (Sagawa et al. 2020a) | Hair Color | Slim | Chubby | 0.192 | 84.5 | 67.6 | 96.5
Group DRO + label noise | Hair Color | Slim | Chubby | 0.062 | 87.2 | 64.2 | 96.8
Group DRO + bias regularizer | Hair Color | Slim | Chubby | 0.029 | 87.1 | 69.7 | 96.9

We extend our task to domain generalization scenario, where training and test
sets belong to different domains. This scenario is realistic because domain
and distribution shifts may occur simultaneously. To build this setup, we
introduce another variable, other than for target or bias attributes, which is
for domain attribute. The examples in the training and test datasets are not
supposed to have the same value for the domain variable. For example, if we
have a domain attribute, Young, then training examples may be composed of the
images with the old (Young = false) while test dataset may contain the images
of young people only (Young = true). Since the domain attribute, Young in this
case, is correlated to both the target and bias variables, it affects the
group distributions in the training and test sets significantly and they are
not consistent no longer. Table 4 presents the domain generalization results
with two domain attributes, Young and Chubby, where our frameworks outperform
the baselines consistently in the presence of domain shift. We also observe
that ERM with the bias regularizer often gives better results than the
original Group DRO method. This results validate that our frameworks are
particularly effective when domain and distribution shifts exist at the same
time.

Table 5:  Ablative results for the weight parameter $\beta$ of the bias
regularization loss on the CelebA dataset.

Bias weight |  Cobias | Unbiased Acc. | Worst-group Acc.
---|---|---|---
0 | 0.435 | 84.1 | 53.2
1 | 0.202 | 84.7 | 58.3
2 | 0.183 | 87.2 | 62.1
5 | 0.111 | 88.1 | 67.4
10 | 0.046 | 87.3 | 65.6

Table 6:  Ablative results for the noise rate $\rho$ in the stochastic label
noise on the CelebA dataset.

Noise rate |  Cobias | Unbiased Acc. | Worst-group Acc.
---|---|---|---
0.0 | 0.435 | 84.1 | 53.2
0.1 | 0.362 | 87.3 | 65.7
0.2 | 0.286 | 87.8 | 66.9
0.4 | 0.209 | 86.1 | 60.6

#### Ablation study

To validate the effectiveness of the bias regularization loss and the
stochastic label noise scheme, we perform the ablation study on the weight
parameter $\beta$ and the noise rate $\rho$. Table 5 and 6 show the ablative
results on the CelebA dataset with hair color and gender for target and bias
variables, respectively. The results show that increasing the weight of the
bias regularization loss and the noise rate improves both Cobias and group-
fairness accuracy within sufficiently wide ranges.

#### Feature visualization

(a) ERM

(b) ERM + label noise

(c) ERM + bias regularizer

Figure 2: The t-SNE plots of feature representations from the ResNet-18 models
trained with ERM and our debiasing frameworks on the CelebA dataset using the
hair color classification. We visualize the distribution of samples who have
the same target value (blond hair). Blue and orange colors denote different
gender values (female and male, respectively).

Figure 2 illustrates the t-SNE visualizations (Van der Maaten and Hinton 2008)
of the feature embeddings of the samples in the CelebA test split given by
hair color classification. For simplicity, we visualize only the examples with
blond hair. Blue and orange colors denote female and male, values of the
gender bias variable, respectively. Figure 2(a), even without using the bias
information (gender) for training, shows that the examples drawn from a bias
group are distinguishable from those from the other bias group. Compared to
vanilla ERM, as illustrated in Figure 2(b) and 2(c), ERMs with bias
regularization and stochastic label noise successfully make the examples with
different bias attributes confused on the feature embedding space, which is
desirable for learning debiased representations.

#### Comparison to logit-based bias regularizer

Table 7: Comparison to the logit-based bias regularization loss on the CelebA
dataset with gender bias.

Bias regularizer type | Target | Cobias | Unbiased Acc. | Worst-group Acc. | Average Acc.
---|---|---|---|---|---
logit-based | Hair Color | 0.219 | 84.9 | 62.7 | 94.6
feature-based (Ours) | Hair Color | 0.111 | 88.1 | 67.4 | 94.5
logit-based | Pale Skin | 0.182 | 84.0 | 60.8 | 95.7
feature-based (Ours) | Pale Skin | 0.066 | 86.1 | 68.7 | 95.2

To analyze the effectiveness of our feature-level bias measurement, we compare
the feature-based bias regularizer $I(F;Z|Y)$ in (9) with logit-based bias
regularizer $I(\hat{Y};Z|Y)$, where $\hat{Y}=h(F)\in\mathcal{Y}$ is a
predicted target variable from a classifier based on a fully connected layer,
$h(\cdot)$. Table 7 presents the results from the two regularizers, where the
feature-based approach outperforms its logit-based counterpart significantly
in terms of both Cobias and group-fairness accuracy. This means that
exploiting feature representation would be more effective to identify and
mitigate the algorithmic bias than logits.

## Related Work

Facial recognition datasets often contain inevitable biases due to their
insufficiently controlled data collection process, and consequently,
recognition algorithms tend to inherit and even amplify the dataset bias. To
address this issue, numerous debiasing frameworks have been proposed for
identifying and mitigating the potential risks posed by dataset or algorithmic
bias. These frameworks can be categorized in pre-processing (Li and
Vasconcelos 2019; Sagawa et al. 2020b; Kamiran and Calders 2012), in-
processing (Sagawa et al. 2020a; Sohoni et al. 2020; Wang et al. 2019; Zhang,
Lemoine, and Mitchell 2018; Gong, Liu, and Jain 2020; Seo, Lee, and Han 2021;
Ragonesi et al. 2021; Wang et al. 2020; Guo et al. 2020), and post-processing
(Hardt, Price, and Srebro 2016; Zhao et al. 2017) ones. Pre-processing
techniques transform the data distribution to keep the training data balanced
across groups, where dataset resampling or reweighting methods (Li and
Vasconcelos 2019; Sagawa et al. 2020b; Kamiran and Calders 2012) are usually
exploited to balance the distribution by under-sampling the majority or over-
sampling the minority classes. Debiasing through in-processing aims to build
algorithms that can learn fair representation, by taking advantage of
adversarial training (Wang et al. 2019; Zhang, Lemoine, and Mitchell 2018),
representation disentanglement (Gong, Liu, and Jain 2020; Ragonesi et al.
2021), or robust optimization (Sagawa et al. 2020a; Sohoni et al. 2020; Seo,
Lee, and Han 2021). Post-processing methods modify the predicted outputs to
meet fairness criterion, mainly by calibrating the outputs (Hardt, Price, and
Srebro 2016; Zhao et al. 2017). Our debiasing frameworks belong to in-
processing methods, which adopts a bias regularization loss or injecting label
noise during training to reduce the algorithmic bias. The bias regularization
loss is somewhat similar to (Ragonesi et al. 2021), but the major difference
is that we introduce _conditional_ mutual information from the structural
causal model. The use of unconditional mutual information as a bias
regularizer makes it difficult to achieve our goal, learning the relationship
between features and target variables.

Mutual information is an information-theoretic quantity to measure the
relationship between two variables. Because it can capture non-linear
dependencies between the variables, the mutual information is widely used in
various areas, including unsupervised representation learning (Comon 1994;
Tishby, Pereira, and Bialek 2000; Oord, Li, and Vinyals 2018; Hjelm et al.
2019; Sun et al. 2020), generative models (Chen et al. 2016; Qian and Cheung
2019), reinforcement learning (Oord, Li, and Vinyals 2018), and fair
supervised learning (Kamishima et al. 2012; Fukuchi, Kamishima, and Sakuma
2015; Ragonesi et al. 2021). However, the exact computation of mutual
information between continuous variables is basically not tractable. There
exists some non-parametric estimators based on kernel density estimation (Kwak
and Choi 2002; Suzuki et al. 2008) to deal with continuous variables, but
those are not scalable with high-dimensional or large-scale data. To overcome
this limitation, recent works on mutual information estimation focus on
training neural network to represent its variational lower bound (Belghazi et
al. 2018; Lin et al. 2019; Poole et al. 2019). Our bias measurement takes also
advantage of the variational lower bound (Donsker and Varadhan 1975; Belghazi
et al. 2018) of conditional mutual information.

## Conclusion

We proposed an information-theoretic bias measurement which can identify the
feature-level algorithmic bias from the causal view of spurious correlation.
Based on the new measurement approach, we presented two types of debiasing
frameworks with bias regularizer or label noise, each of which can be utilized
with or without explicit knowledge of bias information, respectively. We
demonstrated the effectiveness and versatility of proposed frameworks on
multiple standard benchmarks. We also conducted a detailed analysis of our
measurement and frameworks via extensive ablation studies with more realistic
scenarios.

### Acknowledgement

This work was partly supported by Samsung Advanced Institute of Technology,
the Bio & Medical Technology Development Program of the National Research
Foundation (NRF) funded by the Korea government (MSIT) [No. 2021M3A9E4080782],
and Institute of Information & communications Technology Planning & Evaluation
(IITP) grant funded by the Korean government (MSIT) [No.2021-0-01343,
Artificial Intelligence Graduate School Program (Seoul National University)].

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