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# The Compton Scattering Total Cross Section at Next-to-Leading Order

Roman N. Lee<EMAIL_ADDRESS>Budker Institute of Nuclear Physics, 630090,
Novosibirsk, Russia  Matthew D. Schwartz<EMAIL_ADDRESS>Department of
Physics, Harvard University, Cambridge, MA 02138  Xiaoyuan Zhang
<EMAIL_ADDRESS>Department of Physics, Harvard University,
Cambridge, MA 02138

###### Abstract

An analytic formula is given for the total scattering cross section of an
electron and a photon at order $\alpha^{3}$. This includes both the double-
Compton scattering real-emission contribution as well as the virtual Compton
scattering part. When combined with the recent analytic result for the pair-
production cross section, the complete $\alpha^{3}$ cross section is now
known. Both the next-to-leading order calculation as well as the pair-
production cross section are computed using modern multiloop calculation
techniques, where cut diagrams are decomposed into a set of master integrals
that are then computed using differential equations.

The scattering of photons off of electrons is perhaps the most important
physical processes in nature: essentially all observable phenomena involve
photon-electron interactions. The shift in the wavelength of the scattered
photon, $\Delta\lambda=\frac{h}{mc}(1-\cos\theta)$, as first observed by
Compton in 1923 convincingly demonstrated that light comprises particles with
energy and momentum Compton (1923). Compton’s paper introduced the Compton
wavelength of the electron, $\lambda_{e}=\frac{h}{mc}$, which governs the
effective cross sectional area of the electron as seen by the photon
$\sigma\sim\pi\lambda_{e}^{2}$. The calculation of the Compton scattering
cross section by Dirac Dirac (1926) and Gordon Gordon (1926), and later with
full spin and relativistic corrections by Klein and Nishina Klein and Nishina
(1929), provided a convincing case of the correctness of the Dirac equation.
Photon scattering off of electrons is critical to a wide variety of scientific
enterprise, from X-ray crystallography to astrophysics.

Total cross sections at high energy are of interest for both experimental
reasons and theoretical ones. On the experimental side, they are relevant for
not just applications like cosmic rays, but also for estimating luminosity and
measuring coupling constants. On theoretical side, total cross sections
necessarily involve the singular forward-scattering region, where outgoing
particles are collinear to incoming ones. In this region, off-shell
Glauber/Coulomb modes are essential. Studying these modes has led to insights
such as Regge physics and the BFKL equation Lipatov (1976); Kuraev  _et al._
(1976); Balitsky and Lipatov (1978). For Compton scattering, the singularities
are in a region where the outgoing electron is collinear to the incoming
photon and dominated by $t$-channel fermion exchange. Despite their
importance, very few analytic results are known for total cross sections
beyond the leading order.

Bethe and Heitler Bethe and Heitler (1934), as well as Racah Racah (1934),
considered a related process of pair production in a background
electromagnetic field by the high-energy photon, $\gamma Z\to Ze^{+}e^{-}$.
This process has a total cross section which scales asymptotically like
$m^{-2}\ln s$, rather than $s^{-1}$, an indication of the relevance of the
Glauber region. Upon formal substitution $Z=-1$, the leading high-energy
asymptotics of this cross section also determines the high-energy limit of
$e^{-}\gamma\to e^{-}e^{+}e^{-}$ process (cf. Ref. Baier  _et al._ (1966)).
The complete analytical result for the total cross section of the latter was
first computed only recently Lee  _et al._ (2021) (using the same technology
developed for this paper), confirming the leading high-energy asymptotics of
Bethe and Heitler. The analytic $e^{+}e^{-}\to\gamma\gamma$ cross section at
NLO has also been completed within the last year by the same methods Lee
(2020a). The cross section of $\gamma\gamma\to e^{+}e^{-}$ can, in principle,
be extracted as the Abelian limit of the $gg\to t\bar{t}$ total cross section
Czakon and Mitov (2010). Compton scattering is thus the last pure QED total
cross section not known analytically at order $\alpha^{3}$ in QED. Even the
leading high-energy asymptotics of the cross section is not known.

The importance of Compton scattering led numerous investigators to explore
corrections beyond the leading order. The next-to-leading order (NLO) Compton
scattering process includes both loop contributions to $e^{-}\gamma\to
e^{-}\gamma$ and real double-Compton scattering $e^{-}\gamma\to
e^{-}\gamma\gamma$. These two contributions are separately infrared divergent,
but the divergence cancels when they are added, as guaranteed by the Bloch-
Nordsieck theorem Bloch and Nordsieck (1937). The virtual graphs were computed
by Feynman and Brown in 1951 Brown and Feynman (1952), and the cancellation of
the infrared divergence in the differential cross section was shown using a
photon mass cutoff. The double Compton process was studied by Mandl and Skyrme
in 1952 Mandl and Skyrme (1952). Recently, the total cross section for double
Compton scattering has been calculated in Ref. Lee  _et al._ (2021). The
asymptotic behavior of Compton scattering at high energy at the amplitude
level has been examined by numerous authors (e.g. Gell-Mann  _et al._ (1964);
McCoy and Wu (1976); Sen (1983)). Polarized differential Compton scattering at
NLO was studied by Swartz Swartz (1998) and by Denner and Dittmaier Denner and
Dittmaier (1999). Although numerical results for the total NLO cross section
can be obtained by integrating these differential cross sections over the
scattering angle, no analytic formula has yet been produced. The result of
this paper is that final missing analytic form.

To compute the total cross section, one approach is to use the optical theorem
to extract it from the imaginary part of the $e^{-}\gamma\to e^{-}\gamma$
forward scattering amplitude. However, rather than compute the full 2-loop
forward amplitude and then take its imaginary part, it is simpler to compute
the cut diagrams directly. These diagrams are shown in Fig. 1. The cuts that
put $e^{-}\gamma$ on-shell are the virtual corrections, while those putting
$e^{-}\gamma\gamma$ on-shell correspond to real emission. There are also
contributions to the total $e^{-}\gamma$ cross section at order $\alpha^{3}$
from final states with 3 charged particles. These were computed in Lee  _et
al._ (2021) so we do not consider them here.

To compute the cuts, we apply integration by parts and differential equations
to 2 or 3 particle cuts separately. For example, one of 2-particle cut master
integrals of interest looks like

$\mathcal{I}_{1}=\begin{gathered}\includegraphics{LittleD.pdf}\end{gathered}\\\
=\frac{(2\pi)^{2}}{2i\pi^{d}}\int
d^{d}p_{2}d^{d}k_{2}d^{d}k_{3}\delta(p_{2}^{2}-m^{2})\theta(p_{2}^{0})\delta(k_{2}^{2})\theta(k_{2}^{0})\\\
\times\frac{1}{k_{3}^{2}[(p_{1}-k_{2}-k_{3})^{2}-m^{2}]}\delta^{d}(p_{1}+k_{1}-p_{2}-k_{2})$
(1)

Applying loop computation technology to cut graphs significantly simplifies
the problem: the extra $\delta$-functions reduce the number of master
integrals and, therefore, the size of the differential system. Even more
important, the cuts relevant for the NLO correction to the Compton scattering
cross section prevent the appearance of the non-polylogarithmic master
integrals — the massive sunrise graphs.

The main tool we use for the IBP (integration-by-parts) reduction is LiteRed
Lee (2014, ) which allows for the account of the individual cuts. For the
reduction of the differential system to $\epsilon$-form we use Libra package
Lee (2020b). It is helpful to rewrite the integrals in terms of threshold
variables like

$x=\frac{s-m^{2}}{m^{2}},\quad y=\sqrt{\frac{x}{x+4}}\,.$ (2)

A variable like $y$ can be used to rationalize the weights of the appearing
iterated integrals. Both $x$ and $y$ vanish at threshold $s\to m^{2}$. We find
that using the threshold limit to set the boundary conditions of the master
integrals provides a dramatic simplification: in this limit the loop and phase
space integrals factorize, making them much easier to evaluate. For example,
expanding around $y=0$, Eq. (1) has the boundary condition

$\mathcal{I}_{1}=-\frac{2^{2d-6}\pi^{2}\csc\left(\frac{\pi
d}{2}\right)}{(d-3)^{2}\Gamma(d-3)}y^{2d-6}\\\
-\frac{2^{2d-7}\pi^{2}\csc\left(\frac{\pi
d}{2}\right)\Gamma\left(\frac{3-d}{2}\right)}{\Gamma\left(\frac{d-1}{2}\right)}y^{4d-12}+\cdots$
(3)

The differential equations couple many different master integrals. Once they
are solved using the boundary conditions, the individual masters may be
extracted. For example, we find

$e^{2\epsilon\gamma_{E}}\mathcal{I}_{1}=\frac{\pi
x}{(x+1)\epsilon}+\pi\bigg{(}\frac{(x-1)\ln(1+x)-3x\ln x+5x}{x+1}\\\
+\frac{\ln^{2}(1+x)+\text{Li}_{2}(1-x^{2})-\pi^{2}/6}{2x}\bigg{)}+\mathcal{O}(\epsilon)$
(4)

We check the results for the master integrals by constructing their sums which
determine the imaginary parts of the corresponding uncut diagrams and
evaluating the latter numerically using Fiesta Smirnov (2016). More details of
related calculations using the same technology can be found in Lee  _et al._
(2021); Lee (2020a).

Figure 1: Cut Feynman diagrams contributing to the NLO cross section. Cuts
drawn in blue indicate double-Compton contributions and red cuts indicate loop
contributions. The last eight diagrams involve an insertion of the mass
counterterm and require separate integrals.

We use dimensional regularization $d=4-2\epsilon$ to regulate the infrared and
ultraviolet divergences. The renormalization in the on-shell scheme amounts to
the following relation between the renormalized and bare virtual corrections
$\sigma_{\text{virt}}=\left(\sigma_{\text{virt}}\right)_{\text{bare}}+2\delta
Z_{\psi}\sigma_{\text{LO}}+\delta\sigma_{m}$, where $\delta
Z_{\psi}=-\frac{(4\pi\alpha)(3-2\epsilon)\Gamma(\epsilon)}{(4\pi)^{2-\epsilon}(1-2\epsilon)}\left(\frac{e^{\gamma_{E}}}{4\pi}\right)^{\epsilon}$
is the one-loop contribution to the wave function renormalization constant,
$\sigma_{\text{LO}}$ is the Born cross section defined below in Eq. (5), and
$\delta\sigma_{m}$ is the sum of the last eight diagrams in Fig. 1, where
cross denotes the one-loop mass counterterm $i\delta
m=im\frac{(4\pi\alpha)(3-2\epsilon)\Gamma(\epsilon)}{(4\pi)^{2-\epsilon}(1-2\epsilon)}\left(\frac{e^{\gamma_{E}}}{4\pi}\right)^{\epsilon}$
vertex (cf. Ref. Lee (2020a)). After the renormalization we can put
$\epsilon=0$ in the sum $\sigma_{\text{virt}}+\sigma_{\text{real}}$ as the
infrared divergences cancel between virtual and real contributions.

Let us write the cross section for $e^{-}\gamma\to e^{-}\gamma(+\gamma)$ to
order $\alpha^{3}$ as
$\sigma_{\text{tot}}=\sigma_{\text{LO}}+\sigma_{\text{NLO}}$ where

$\sigma_{\text{LO}}=\frac{\pi\,\alpha^{2}}{m^{2}(x+1)}\Big{[}\frac{x^{3}+18x^{2}+32x+16}{x^{2}(x+1)}\\\
+\frac{\left(2x^{3}-6x^{2}-24x-16\right)}{x^{3}}\ln(x+1)\Big{]}$ (5)

and

$\sigma_{\text{NLO}}=\frac{\alpha^{3}}{m^{2}x^{3}}\Big{\\{}-\frac{x\left(273x^{3}-982x^{2}-2960x-1744\right)}{24(x+1)^{2}}+\frac{37x^{4}-54x^{3}-339x^{2}-428x-184}{4(x+1)^{2}}\ln(x+1)\\\
+\frac{x^{2}\left(14x^{4}+17x^{3}-17x^{2}-22x-8\right)}{2(1-x)(1+x)^{3}}\ln
x-\frac{4x^{6}+35x^{5}-31x^{4}-755x^{3}-1765x^{2}-1506x-440}{2(x+1)^{2}(x+4)}\ln^{2}(x+1)\\\
-\frac{x^{6}+7x^{5}-28x^{4}-239x^{3}-449x^{2}-338x-88}{(x+1)^{2}(x+4)}\text{Li}_{2}(-x)-\frac{1}{3}\left(x^{2}-16x-23\right)\ln^{3}(x+1)+\left(x^{2}-x+2\right)\left[\text{Li}_{2}(1-x)-\frac{\pi^{2}}{6}\right]\\\
+\frac{x^{4}+7x^{3}+x^{2}-3x-2}{(x+1)^{2}}\ln(x+1)\ln
x+\left(x^{2}+2x-6\right)\Big{[}\text{Li}_{3}(x^{2})-\text{Li}_{2}(x^{2})\ln
x\Big{]}\\\
-\frac{4\left(x^{5}+26x^{4}+146x^{3}+316x^{2}+288x+96\right)}{(x+1)^{2}(x+4)}G(-2,-1;x)+8\left(x^{2}-4x-6\right)G(-1,-2,-1;x)\\\
+4(2x^{2}-x-6)G(-1,-1,0;x)+2\left(2x^{2}-7x-12\right)G(-1,0,-1;x)-(5x^{2}+32x-8)G(0,-1,-1;x)\\\
-3(x-2)(x+4)y\Big{[}G(0,y,-1;x)+2G(y,-1,0;x)\Big{]}-\frac{8y\left(x^{4}+3x^{3}-18x^{2}-68x-24\right)}{(x+4)x}G(y,0,-1;x)\\\
+\frac{3y\left(5x^{4}+14x^{3}-96x^{2}-352x-128\right)}{(x+4)x}G(y,-1,-1;x)-\frac{16y\left(x^{4}+2x^{3}-24x^{2}-80x-48\right)}{(x+4)x}G(y,-2,-1;x)\\\
+\frac{3x^{4}+18x^{3}+44x^{2}-8x-64}{x}yG(y,-1;x)-\frac{6y\left(x^{3}-12x-8\right)}{x}G(-4,y,-1;x)+3\left(3x^{2}-8\right)G(y,y,-1;x)\Big{\\}}$
(6)

The $G$ functions are defined iteratively via

$G(a,a_{1},\ldots,a_{n};x)=\int_{0}^{x}dw_{a}(x^{\prime})G(a_{1},\ldots,a_{n};x^{\prime}),$
(7)

where weights are

$dw_{y}(x)=\frac{ydx}{x},\quad dw_{a}(x)=\frac{dx}{x-a}\quad(a=-4,-2,-1,0)$
(8)

and $G(0;x)=\ln x$. When $y$ is not among the letters, these functions are
conventional Goncharov polylogarithms as the notation suggests. The weight
$dw_{y}$ can be rationalized by changing variables from $x$ to $y$ in Eq. (2).
Then one can express the $G$ functions as linear combinations of
$G(b_{1},\ldots,b_{n};y)$ with $b_{k}\in\left\\{0,\pm 1,\pm
i,\pm{i}/{\sqrt{3}}\right\\}$, or, alternatively, as linear combinations of
$G(\ldots;z)$, with $z=\frac{1-y}{1+y}$ and indices being the 4th or 6th roots
of unity.

Some of the integrals can be evaluated into relatively simple forms with an
additional judicious variable change. For example, using
$z=\frac{1-y}{1+y}=\frac{2+x-\sqrt{x(x+4)}}{2}$, we can compute

$G(y,-1;x)=\int_{0}^{x}\frac{dx^{\prime}}{\sqrt{x(x+4)}}\int_{0}^{x^{\prime}}\frac{dx^{\prime\prime}}{x^{\prime\prime}+1}\\\
=-\int_{1}^{z}d\ln z^{\prime}\int_{1}^{z^{\prime}}d\ln\frac{z^{\prime\prime
3}+1}{z^{\prime\prime}(z^{\prime\prime}+1)}\\\
=\frac{1}{3}\text{Li}_{2}(-z^{3})-\text{Li}_{2}(-z)+\frac{1}{2}\ln^{2}z-\frac{\pi^{2}}{18}$
(9)

In the Appendix we present the expressions for all the $G$ functions entering
Eq. (6) in terms of the classical polylogarithms $\text{Li}_{n}$. The same
expressions, as well as the total cross section, are given in a Mathematica
notebook included with the arXiv submission of this paper. As a check, we
compare to the numerical values given in Table 3 of Denner and Dittmaier
(1999), and find perfect agreement.

Figure 2 plots the size of the NLO correction relative to the leading order as
a function of center-of-mass energy $E_{\text{CM}}=\sqrt{s}$, using
$\alpha=1/137.036$. At threshold $s\to m^{2}$, the NLO correction vanishes.
This is guaranteed by Thirring’s theorem Thirring (1950); Dittmaier (1997);
Denner and Dittmaier (1999), and is a non-trivial check on our computation.
More precisely, using PolyLogTools Duhr and Dulat (2019), we find that near
threshold $(s\sim m^{2})$, the cross section behaves as

$\sigma_{\text{tot}}=\frac{\pi\alpha^{2}}{m^{2}}\left[\frac{8}{3}-\frac{8}{3}x+\cdots\right]\\\
+\frac{\alpha^{3}}{m^{2}}x^{2}\left[-\frac{16}{9}\ln
x+\frac{7}{15}+\cdots\right]$ (10)

with the $\alpha^{3}$ term vanishing like $(s-m^{2})^{2}$.

We see from Figure 2 that the NLO correction to the cross section grows with
energy, providing a 30% correction already at 1 TeV (pair production and
electroweak corrections are also increasingly important at high energy
Dittmaier (1994)). At high energy, the asymptotic behavior is

$\displaystyle\sigma_{\text{tot}}=\frac{\pi\alpha^{2}}{s}\left[2\ln\frac{s}{m^{2}}+1+\cdots\right]$
(11)
$\displaystyle\\!+\\!\frac{\alpha^{3}}{s}\\!\left[\frac{1}{3}\ln^{3}\frac{s}{m^{2}}\\!-\\!\frac{1}{2}\ln^{2}\frac{s}{m^{2}}\\!+\\!\frac{17}{4}\ln\frac{s}{m^{2}}\\!-\\!\frac{75}{8}\\!-\\!\frac{\pi^{2}}{2}+4\zeta_{3}\right]$

This is a new result. The $\frac{1}{s}\ln^{3}\frac{s}{m^{2}}$ behavior of the
NLO result dominates over the $\frac{1}{s}\ln\frac{s}{m^{2}}$ behavior at LO;
this explains the growth of their ratio at high energy in Figure 2.

To summarize, we have calculated the Compton scattering total cross section
with accuracy $O(\alpha^{3})$. The result can be represented in terms of the
Goncharov’s polylogarithms of the argument
$z=\frac{\sqrt{s+3m^{2}}-\sqrt{s-m^{2}}}{\sqrt{s+3m^{2}}+\sqrt{s-m^{2}}}$ with
letters involving $6$th and $4$th root of unity 111If we want to avoid also
the classical polylogarithms $\text{Li}_{n}(x^{2})$ and $\text{Li}_{n}(1-x)$,
we should include also the letters $\varphi^{2}$ and $\varphi^{-2}$, where
$\varphi=\frac{1+\sqrt{5}}{2}$ is the golden ratio., or in temrs of classical
polylogarithms. The analytic form of the total cross section may be of use in
numerous applications, from cosmic ray astrophysics to particle colliders as
well as to theoretical investigations into factorization and forward/backward
scattering.

Our analytic results reveal some remarkable features. For one, when continued
to a multi-valued analytic function, the cross section manifests a branch
point at $s=2m^{2}$, due to terms like $\text{Li}_{3}(x^{2})$. Although the
singularity is on an unphysical sheet and the cross section is smooth around
$s=2m^{2}$, the presence of this branch point affects the threshold and high-
energy expansions: both diverge from the exact result near $s=2m^{2}$ no
matter how many terms one retains. Moreover, these divergences cannot be
eliminated by any rational variable change. This is in sharp contrast to the
cross section of $e^{-}\gamma\to e^{-}e^{+}e^{-}$ Lee  _et al._ (2021) which
is nicely approximated after only a few terms of high-energy asymptotics, or a
few terms of threshold asymptotics when expanding in
$r=\frac{s-9m^{2}}{s-m^{2}}$. One might hope to give a physical interpretation
to the branch point by connecting sequential monodromies to multiple cuts, as
was done at the amplitude level in Bourjaily  _et al._ (2021).

Another intriguing feature is the appearance of a double log
$\ln\frac{s^{2}}{m^{2}}$ in the NLO cross section over the LO cross section.
Normally, double logarithms are due to overlapping soft and collinear Sudakov
singularities. However, the soft singularities are expected to cancel in QED
cross sections due to the Block-Nordsieck theorem Bloch and Nordsieck (1937)
leaving at most single logarithms at each order in $\alpha$. It would be
interesting to understand the origin of this double logarithm, and its
connection to DGLAP or BFKL evolution both of which are single logarithmic.

Figure 2: NLO corrections to the Compton scattering cross section, as compared
to the leading order. At $E_{\text{CM}}=1\text{GeV}$ the correction is 8.2%
growing to a 30% correction at 1 TeV.

The authors would like to thank C. Vergu, H-X Zhu and Y. Zhu for discussions.
MDS is supported in part by the U.S. Department of Energy under grant DE-
SC0013607. RNL is supported by Russian Science Foundation under grant
20-12-00205.

##  Appendix A Appendix: Explicit expressions in terms of classical
polylogarithms

$\displaystyle G$
$\displaystyle(-2,-1;x)=\text{Li}_{2}(-x-1)+\ln(x+1)\ln(x+2)+\frac{\pi^{2}}{12},$
(12) $\displaystyle G$
$\displaystyle(y,-1;x)=\frac{\text{Li}_{2}\left(-z^{3}\right)}{3}-\text{Li}_{2}(-z)+\frac{1}{2}\ln^{2}{z}-\frac{\pi^{2}}{18},$
(13) $\displaystyle G$
$\displaystyle(-1,-2,-1;x)=2\text{Li}_{3}(-x-1)-\text{Li}_{2}(-x-1)\ln(x+1)+\frac{\pi^{2}}{12}\ln(x+1)+\frac{3\zeta_{3}}{2},$
(14) $\displaystyle G$
$\displaystyle(-1,-1,0;x)=\text{Li}_{3}(-x)+\text{Li}_{3}\left(\frac{x}{x+1}\right)+\frac{1}{2}\ln^{2}(x+1)\ln{x}-\frac{1}{6}\ln^{3}(x+1),$
(15) $\displaystyle G$
$\displaystyle(-1,0,-1;x)=-2\text{Li}_{3}(-x)-2\text{Li}_{3}\left(\frac{x}{x+1}\right)+\text{Li}_{2}(-x)\ln(x+1)+\frac{1}{3}\ln^{3}(x+1),$
(16) $\displaystyle G$
$\displaystyle(0,-1,-1;x)=\text{Li}_{3}(-x)+\text{Li}_{3}\left(\frac{x}{x+1}\right)-\text{Li}_{2}(-x)\ln(x+1)-\frac{1}{6}\ln^{3}(x+1),$
(17) $\displaystyle G$
$\displaystyle(0,y,-1;x)=2\operatorname{Re}\left[2\text{Li}_{3}\left(\frac{1-z}{1-r\,z}\right)-2\text{Li}_{3}\left(\frac{z-1}{z-r}\right)+\text{Li}_{3}\left(1-r/z\right)-\text{Li}_{3}\left(1-z/r\right)\right]+2\text{Li}_{3}\left(1-z^{-1}\right)-2\text{Li}_{3}(1-z)$
$\displaystyle+\left[\text{Li}_{2}(1-z)-\text{Li}_{2}\left(1-z^{-1}\right)\right]\ln\left(x+1\right)+\arctan\left(\sqrt{3}\tfrac{1-z}{1+z}\right)\left[\frac{2\pi}{3}\ln\left(x+1\right)+\frac{2\left(\psi^{\prime}\left(\tfrac{1}{6}\right)-2\pi^{2}\right)}{5\sqrt{3}}\right]-\frac{1}{3}\ln^{3}{z}+\frac{\pi^{2}}{18}\ln{z},$
(18) $\displaystyle G$
$\displaystyle(y,0,-1;x)=2\operatorname{Re}\left[2\text{Li}_{3}\left(r-r^{2}/z\right)-2\text{Li}_{3}\left(r-r^{2}z\right)+\text{Li}_{3}\left(1-z/r\right)-\text{Li}_{3}\left(1-r/z\right)\right]+\big{[}\text{Li}_{2}\left(1-z^{-1}\right)-\text{Li}_{2}(1-z)$
$\displaystyle-\ln{x}\ln{z}\big{]}\ln\left(x+1\right)-\arctan\left(\sqrt{3}\tfrac{1-z}{1+z}\right)\left[2\pi\ln{z}+\frac{2}{5\sqrt{3}}\left(\psi^{\prime}\left(\tfrac{1}{6}\right)-2\pi^{2}\right)\right]+\frac{11\pi^{2}}{18}\ln{z},$
(19) $\displaystyle G$
$\displaystyle(y,-1,0;x)=\frac{4}{9}\text{Li}_{3}\left(-z^{3}\right)+\frac{1}{3}\text{Li}_{2}\left(-z^{3}\right)\ln{x}-2\text{Li}_{3}\left(z-z^{2}\right)-2\text{Li}_{3}(z-1)+\frac{2}{9}\text{Li}_{3}\left((z-1)^{3}\right)-4\text{Li}_{3}(-z)$
$\displaystyle-2\text{Li}_{3}(z)-\text{Li}_{2}(-z)\ln{x}-\frac{1}{6}\ln^{3}{z}+\frac{1}{6}\pi^{2}\ln{z}-\frac{1}{9}\pi^{2}\ln(1-z)-\frac{2}{3}\zeta_{3},$
(20) $\displaystyle G$
$\displaystyle(-4,y,-1;x)=2\operatorname{Re}\bigg{[}\text{Li}_{3}\left(1+z^{-1}\right)-\text{Li}_{3}(1+z)+\text{Li}_{3}\left(1-r/z\right)-\text{Li}_{3}\left(1-z/r\right)+2\text{Li}_{3}\left(\frac{1-z/r}{1+z}\right)-2\text{Li}_{3}\left(\frac{z-r}{z+1}\right)\bigg{]}$
$\displaystyle-\left[\text{Li}_{2}(-z)-\text{Li}_{2}\left(-z^{-1}\right)+\ln\left(x+4\right)\ln{z}\right]\ln\left(x+1\right)-\frac{2}{3}\arctan\left(\sqrt{3}\tfrac{1-z}{1+z}\right)\left[\frac{2}{5\sqrt{3}}\left(\psi^{\prime}\left(\tfrac{1}{6}\right)-2\pi^{2}\right)-\pi\ln\left(x+4\right)\right]$
$\displaystyle-\frac{1}{3}\ln^{3}{z}+\frac{5}{9}\pi^{2}\ln{z},$ (21)
$\displaystyle G$
$\displaystyle(y,-2,-1;x)=2\operatorname{Re}\bigg{[}\text{Li}_{3}\left(\frac{1-r/z}{1-i\,r}\right)+\text{Li}_{3}\left(\frac{1-r/z}{1+i\,r}\right)-\text{Li}_{3}\left(\frac{1-z/r}{1+i/r}\right)-\text{Li}_{3}\left(\frac{1-z/r}{1-i/r}\right)-\text{Li}_{3}\left(1-r/z\right)$
$\displaystyle+\text{Li}_{3}\left(1-z/r\right)+\frac{1}{2}\ln\left[(1-z/r)(1-r/z)\right]\Big{[}\text{Li}_{2}\left(\frac{1-z/r}{1+i/r}\right)+\text{Li}_{2}\left(\frac{1-z/r}{1-i/r}\right)-\text{Li}_{2}\left(\frac{1-r/z}{1-i\,r}\right)$
$\displaystyle-\text{Li}_{2}\left(\frac{1-r/z}{1+i\,r}\right)+\text{Li}_{2}\left(1-r/z\right)-\text{Li}_{2}\left(1-z/r\right)\Big{]}\bigg{]}-\frac{1}{36}\left(5\pi^{2}+18\ln^{2}(2-\sqrt{3})\right)\ln{z},$
(22) $\displaystyle G$
$\displaystyle(y,-1,-1;x)=2\operatorname{Re}\bigg{[}\text{Li}_{3}\left(1-z/r\right)-\text{Li}_{3}\left(1-{r}/{z}\right)+\text{Li}_{3}\left(\frac{1-z/r}{\sqrt{3}iz}\right)-\text{Li}_{3}\left(\frac{r-z}{\sqrt{3}i}\right)$
$\displaystyle-\frac{1}{2}\ln\left(x+1\right)\left[\text{Li}_{2}\left({r}/{z}\right)-\text{Li}_{2}\left(r\,z\right)\right]\bigg{]}+\frac{1}{12}\ln{z}\left(\ln^{2}{z}+3\ln^{2}\left[3\left(x+1\right)\right]-6\ln^{2}{3}+\pi^{2}\right)$
$\displaystyle-\left[\arctan\left(\sqrt{3}\tfrac{1-z}{1+z}\right)\right]^{2}\ln{z}-\frac{1}{15}\arctan\left(\sqrt{3}\tfrac{1-z}{1+z}\right)\bigg{[}5\pi\ln\left[3\left(x+1\right)\right]+\frac{2}{\sqrt{3}}\left(\psi^{\prime}\left(\tfrac{1}{6}\right)-2\pi^{2}\right)\bigg{]},$
(23) $\displaystyle G$
$\displaystyle(y,y,-1;x)=\text{Li}_{3}(-z)-\frac{1}{9}\text{Li}_{3}\left(-z^{3}\right)-\frac{1}{6}\ln^{3}{z}+\frac{\pi^{2}}{18}\ln{z}+\frac{2}{3}\zeta_{3}.$
(24)

Here $\psi^{\prime}(x)$ is the derivative of the digamma function
$\psi(x)=\Gamma^{\prime}(x)/\Gamma(x)$ and $r=e^{i\pi/3}$ is the primitive
sixth root of unity.

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# WCLD: Curated Large Dataset of Criminal Cases from Wisconsin Circuit Courts

Elliott Ash1 Naman Goel2∗ Nianyun Li1∗ Claudia Marangon1∗ Peiyao Sun1∗
1Center for Law & Economics, ETH Zurich
2Department of Computer Science, University of Oxford
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Authors are ordered alphabetically.

###### Abstract

Machine learning based decision-support tools in criminal justice systems are
subjects of intense discussions and academic research. There are important
open questions about the utility and fairness of such tools. Academic
researchers often rely on a few small datasets that are not sufficient to
empirically study various real-world aspects of these questions. In this
paper, we contribute WCLD, a curated large dataset of 1.5 million criminal
cases from circuit courts in the U.S. state of Wisconsin. We used reliable
public data from 1970 to 2020 to curate attributes like prior criminal counts
and recidivism outcomes. The dataset contains large number of samples from
five racial groups, in addition to information like sex and age (at judgment
and first offense). Other attributes in this dataset include neighborhood
characteristics obtained from census data, detailed types of offense, charge
severity, case decisions, sentence lengths, year of filing etc. We also
provide pseudo-identifiers for judge, county and zipcode. The dataset will not
only enable researchers to more rigorously study algorithmic fairness in the
context of criminal justice, but also relate algorithmic challenges with
various systemic issues. We also discuss in detail the process of constructing
the dataset and provide a datasheet. The WCLD dataset is available at
https://clezdata.github.io/wcld/.

##  1 Introduction

Machine learning algorithms are increasingly being used or proposed to be used
in various sensitive use-cases, including criminal justice. In such scenarios,
a major concern is that the algorithms can perpetuate and amplify different
forms of social biases. There is plenty of empirical evidence supporting this
concern in the literature [1]. For example, the seminal work by ProPublica [2]
found COMPAS, a recidivism risk estimation tool, to be unfair against black
defendants: making different errors for different demographic groups and at
different rates. While any component in the machine learning pipeline can
introduce bias and there are also fundamental limitations of algorithms [3,
4], it is also widely acknowledged that training and evaluation datasets are
major contributors [5, 6] to unfairness in algorithms. Therefore, it is
important for academic researchers to have access to large datasets to study
the problem rigorously. Further, the datasets should also contain rich
information to allow understanding the interaction of data and algorithmic
issues with various systemic issues. A standard research dataset used in this
context is the COMPAS dataset [7]. While this dataset has been extremely
useful so far in bringing attention to some of the issues with machine
learning algorithms, it is often limiting in improving our understanding of
the algorithms’ behavior because of its small size and other limitations
(discussed in Section 2). Further, researchers often use synthetic datasets
and simulations [8, 9, 10, 11, 12] to complement their analysis. Availability
of large scale real-world datasets will bring more grounding to this
complementary analysis approach [13, 9].

In this paper, we contribute a large scale curated dataset of criminal cases
from circuit courts in the U.S. state of Wisconsin. The dataset has been
constructed from publicly available raw case records, to make that information
easy to use in machine learning research. The dataset has a defined set of
attributes and prediction variable for research. The attributes include type
of offense, sex, race, prior criminal count (for each offense type), age (at
judgment and at first offense) and neighbourhood characteristics like
population density, proportion who attended college, proportion eligible for
food stamp, African American population share, Hispanic population share,
proportion of male, proportions who live in rural and urban area and median
household income. The prediction variable for research is the observed
recidivism within a follow-up period of 2 years, measured in different ways.
We also provide more information for each row in the dataset that can be used
for further analysis. This information includes charge severity, case
decisions, year of filing, sentence length, charge severity for previous
cases, sentence lengths for previous cases, pseudo-identifiers for judges,
county and zipcode. Mappings for county and zipcode can be made available to
the researchers on request. Information that we have excluded from the dataset
are name of the defendant, their address, date of birth, dates of case events
etc. There are about 1.5 millions rows in the dataset. We will also strive to
add more features to this dataset in the future.

We hope that the academic research community will find this dataset useful not
only for benchmarking machine learning and fairness algorithms, but also for
critically understanding the strengths and weaknesses of various algorithms
and how these issues interact with biases due to geography, time, judges,
sentencing etc.

In the rest of the paper, we describe the process of constructing the dataset
from raw case records accessed through the Wisconsin Circuit Courts Access
API. We discuss various choices made while curating the dataset and their
relevance in using this dataset for research. We provide summary statistics of
various features in the dataset. For reference, we also report the performance
of two basic machine learning classifiers (logistic regression and XGBoost) on
the dataset. Appendix F contains datasheet [14] for this dataset.

##  2 Related Work

The COMPAS dataset [7] is one of the most frequently used dataset in machine
learning fairness literature. It has around 7000 observations from a single
court (Broward County, Florida) over two years (2013 and 2014). It has very
few samples for racial groups other than Caucasian and African-American. The
dataset is limited to the set of defendants assessed with COMPAS at the pre-
trial stage, and it does not include information on judges or other officials.
The validity of this dataset as a reliable fairness benchmark is often
disputed [15].

[16] contribute datasheets for 15 publicly available datasets in criminal
justice from the US: such as survey data from victims, policing data, court
records, data about enforcement agencies and officers, data about extremists
etc. The datsets listed Section 4.4 of their paper are the most relevant in
the context of our contribution. However, those are mostly raw case records,
not datasets with curated attributes and recidivism outcomes. [13] survey a
few other datasets in criminal justice domain that have similar limitations.
[17] contribute a dataset that contains information on the personnel,
activities, use of force, and complaints in the Chicago Police Department
(CPD).

In the field of machine learning fairness more broadly, there are several
notable works that introduce new larger datasets or make larger datasets more
usable. For example, [18] construct five datasets related to income,
employment, health coverage, commute time, and housing. They observe that many
results vary with the data highlighting the importance of more data-centric
empirical research. [15] release a toolkit to make the Home Mortgage
Disclosure Act datasets more easily usable. They also conclude that the
variance in small datasets like COMPAS can muddle the reliability of
conclusions about algorithmic fairness interventions. We refer the readers to
[19] for a detailed survey on datasets used in algorithmic fairness research.

In a recent paper [13], we reported results of a study that was made possible
on subsets of the WCLD dataset. In that study, we used simulations to create
several semi-synthetic datasets from the WCLD dataset to understand the impact
of a few data-centric factors on algorithmic fairness. However, this is only
one example and we believe that researchers will use this dataset for
understanding various other issues as well. Further, the study in [13] used
only some of the information that we are releasing with this dataset. For
example, the additional information on judge, county, detailed offense types
(wcisclass), charge severity, probation, detention, sentencing etc can be used
for understanding various systemic biases and how they interact with
algorithmic fairness interventions. We have now also curated a variable on
violent recidivism. With this submission, we also provide a detailed datasheet
for the dataset, lists its various limitations and describe in detail the
origin and the process of curating the dataset. This datasheet and all the
other details will be useful for researchers while considering conducting
further research with this dataset, and even to inform decisions about further
possible improvements in the dataset.

##  3 Dataset Construction

###  3.1 Wisconsin Circuit Courts Access (WCCA)

WCCA was created in April 1999, and it contains public case records and docket
information from the 72 county courts of the U.S. state of Wisconsin. The WCCA
website allows the public to search and read the records. The court record
summaries provided by WCCA are public records under Wisconsin open records law
sections 19.31-19.39 of the Wisconsin Statutes.

The REST interface of WCCA is a subscription based service provided by WCCA to
allow automated processes to download and monitor public circuit courts
information. We purchased a subscription to this service in May 2020 for the
purpose of academic research. We paid $12,500$ USD to the ‘Wisconsin Supreme
Court’ from our research budget, for 12 months of electronic access to WCCA
information through the REST interface.

###  3.2 Case Records

Using the REST interface subscription, we were able to download dockets from
1970 to 2020. We queried all case numbers in each county during the period and
subsequently, using these case numbers, we queried individual cases for all
available information. There are around 11 million records, out of which
around 2.5 million are criminal. The case records include the charges in
current offense, the outcomes and sentences of cases and the defendants’
demographic information (e.g. sex, race, address, and date of birth). There is
additional information available about various case events (e.g. hearings,
bail decisions, bail amounts etc), information on the associated attorneys and
government officials involved, including prosecutors and judges.

###  3.3 Attributes

From these raw case records, we set out to create a curated dataset with
numerical and categorical attributes and prediction variable for research.
Attributes like type of offense (felony, misdemeanor and criminal traffic),
wcisclass (more specific categorization of offense), year of filing, defendant
sex, and race do not require much effort to extract as these are directly
available in the case records. As per WCCA declaration (please see the
datasheet in appendix F): “In criminal cases, any designation in any race
field contains subjective information generally provided by the agency that
filed the case.” We found that for each case, the defendants are assigned
exactly one race. For a small number of defendants, different cases assign a
different race to the same defendant. This doesn’t necessarily imply mixed-
race but could be (and perhaps more likely to be) a data quality issue. More
specifically, for $\approx 0.3\%$ of the defendants, there were more than two
races reported (across different cases). For another $\approx 0.3\%$ of the
defendants, there were two races reported (across different cases) and none of
the two races was caucasian. There was perhaps no obvious way to determine the
race in these records. These records are not included in the dataset. For
$\approx 5\%$ of defendants, there were two races reported (across different
cases) and one of the two races was caucasian. We observed that for most of
these defendants, non-caucasian race was the most frequently reported race in
cases associated with the respective defendant. For reporting model
(un)fairness in Section 4.2, we used the non-caucasian race, but a separate
column called ‘all_cases_races’ in the dataset indicates all the races
reported for the defendants across cases.

We construct the rest of the attributes and the variable for prediction from
the information that is indirectly available in the case records. We use a
combination of first name, last name, and date of birth as a unique identifier
for a defendant. This identifier allows us to conduct a search in the database
of case records to match the defendant across multiple cases and construct the
additional attributes 111It may be noted that this identifier is not robust to
name change by some of the defendants (which we can not rule out). We provide
this identifier in the dataset as a separate column.. The detailed process of
constructing the additional attributes is described as follows.

####  3.3.1 Prior Criminal Counts

Using the database search, we obtain the prior count of each of the three
crime types - felony, misdemeanor and criminal traffic - of the defendant for
each of the cases. We were able to collect cases from as early as 1970. We use
all these case records for constructing the prior criminal count.

####  3.3.2 Age

We also infer age at judgment and age at first offense for each case. Age at
judgment is calculated based on the date of birth of the defendant and
judgment disposition date of the case. Age at first offense for each case is
the age when the defendant committed the first crime found in the database.

####  3.3.3 Neighborhood Characteristics

We link 9 local demographics variables to our data from a zipcode level
dataset processed from 2010 census data [20]. These are population density,
proportion who attended college, proportion eligible for food stamp, African
American population share, Hispanic population share, proportion of male,
proportions who live in rural and urban areas and median household income. For
around 3% of the cases in our dataset, the records didn’t contain zipcode. For
such cases, we have used average county characteristics. Note that the
neighborhood characteristics are from the year 2010, not necessarily from the
respective years of individual cases. However, we include the year of filing
for each case in the dataset. Researchers can use the year variable to link
with external datasets for respective years (not only neighborhood
characteristics but other information that they may consider interesting and
relevant). Zipcode for the cases can also be made available to the researchers
on request for this purpose, for example.

####  3.3.4 Charge Severity

Felonies and misdemeanors are sub-classified by severity level in Wisconsin.
Charge severity in raw case records accessed through the REST interface can be
Felony A, B, BC, C, D, E, F, G, H, I, U and Misdemeanor A, B, C, U. We map
each charge severity to a unique numerical value. The numerical values reflect
the increasing charge severity. Mapping is provided in Table 5 in Appendix G.
There can be multiple charges in a case; we only keep the highest charge
severity for each case.

Using the defendant identifier, we also created the prior counts of charge
severity for each prior charge of the defendants. These columns (for each of
the 15 charge severity types) are also available in the dataset.

###  3.4 Sentence Length

We calculate the sentence length using the available information about the
number of assigned days of jail. In case of multiple charges in the same date,
we take the maximum of jail sentence for all charges. If this value is zero,
we use the available information about the number of assigned days of house
correction (maximum value in case of multiple charges).

For each row, we also provide summary statistics for the previous sentence
lengths in four additional columns (max, min, mean, and median sentence
length).

####  3.4.1 Other Columns

The dataset contains two other binary variables: not$\\_$detained and
probation. not$\\_$detained = 0 implies that the defendant was detained in the
case, and not$\\_$detained = 1 implies released, probation or sentence. The
other variable, probation = 1 implies that the defendant got probation.

###  3.5 Recidivism

The choice of the prediction variable in machine learning is a difficult and
value-laden decision [21]. In this dataset, we provide a commonly used
variable in machine learning research, which is whether the defendant
recidivates or not. There are well-founded ethical concerns about recidivism
prediction task. Please see Section 5.3 for important discussion on ethics. We
provide this variable in the dataset to facilitate empirical research on
(fundamental) limitations and/or opportunities for improvement. Please see
Section 5.2 for discussion on intended uses and limitations. We emphasise that
the researchers can use this dataset for other research questions as well and
do not have to necessary use the recidivism variable (as a prediction variable
or otherwise).

To construct this variable, we had to decide on a follow-up period within
which committing a new offense is deemed as recidivism. We pick the usual
choice [2, 22] of 2 year follow-up period. Specifically, we measure whether
the defendant would commit a crime within two years from the date of judgment.

In any dataset, whether we observe recidivism or not for a defendant is
fundamentally biased by the decisions of judges and sentence. This is a common
problem [23, 24]. Further, defendants serve different sentence lengths and
depending on the sentence, it may not be possible to observe recidivism within
2 year follow-up period. Assigning a missing outcome to every case with a
sentence throws away a lot of useful data. Yet extending the follow up period
for two years after the assigned sentence period instead of the judgment date
is also problematic because defendants often serve more or less than the
assigned sentence. Since there is not a comparable data source that has the
exact jail record of every defendant in Wisconsin, we don’t observe the actual
(served) sentence length. There is no consensus in the literature about how to
deal with this problem. We use a cutoff for sentence length, of 180 days, such
that we don’t have to throw away a lot of useful data and still leave enough
time in the follow-up period for the defendant to reveal crime potential.
Above this cutoff, we treat the defendants’ outcome as missing, even if they
do not re-offend within the follow-up period. As an alternate, we can also
define the recidivism variable by varying the sentence length cut-off from 180
days to 2 years, and extending the follow-up period of 2 years by adding the
sentence length. Note that the second way takes into account the sentence
length given by the judge, which creates even more bias in recidivism
variable. Both these recidivism variables are available in our dataset.

###  3.6 Violent Recidivism

In machine learning fairness research, violent recidivism (as opposed to
recidivism without any specific criminal context) may also be considered [7,
25]. In the FBI’s Uniform Crime Reporting (UCR) Program, violent crime is
composed of four offenses: murder and nonnegligent manslaughter, forcible
rape, robbery, and aggravated assault. In our setting, the challenge in
constructing a violent recidivism variable is annotating the large number of
records as violent/non-violent. In the raw case records, we have textual
descriptions for various charges in a case. There were about $\approx 45,000$
charge descriptions across records, making manual annotation difficult. To the
best of our knowledge, there are no publicly available violent/non-violent
classifications of these charge descriptions. Note that charge descriptions
are different from type of offense (felony, misdemeanor and criminal traffic)
and wcisclass columns. We used GPT-4 [26], a pre-trained large language model,
for categorising charge descriptions as referring to violent or non-violent
crimes. We provided GPT-4 with the FBI’s UCR definition of violent crime in
the prompt and asked it to classify a given charge description as violent/non-
violent while spelling out the thought process or explanation for the
classification. Appendix H provides further details about the process. Due to
lack of any ground truth, we do not have a quantitative assessment of the
accuracy of the violent/non-violent labels thus obtained. But we did observe
several limitations of this approach. We found that GPT often struggled to
label charge descriptions that were ambiguous, had insufficient context or
contained abbreviations. We also noticed that if prompted multiple times, GPT
could classify some charge descriptions differently due to ambiguity and
provided plausible explanations for both classifications. We manually chose
classifications for a very small fraction of these instances but there can be
more such inconsistencies that were not caught. In some other cases, it
explicitly stated that due to insufficient information, it could not provide a
classification. Further, charge descriptions may have standard meanings in
Wisconsin courts which GPT may not have access to in its training data or may
not have taken into consideration due to our prompt’s focus on FBI’s UCR
definition. Thus, the process is not perfect and certainly not a replacement
for expert annotations.

Using the classifications provided by GPT, we identified whether at least one
of the charge descriptions in the case referred to a violent crime. Similar to
the recidivism variables in the previous section, the violent recidivism
variable is also binary (1 meaning a violent recidivism was observed within a
two year follow-up period, with 180 days sentence length cutoff) and contains
missing values. Due to space constraints, we do not discuss violent recidivism
variable in the rest of this paper, unless explicitly stated. In addition to
violent recidivism, another column in the dataset marks whether the current
case had a violent charge. We also make available the violent/non-violent
classifications for charge descriptions and the explanations provided by GPT
as a separate file.

###  3.7 Data Filtering

Through the REST interface, we were able to download case records from as
early as 1970, but the records in earlier years tend to be incomplete and the
number of cases much smaller. Therefore, we only keep the cases from 2000 in
the dataset. It only means that the rows in the curated dataset are only those
cases that appear in the courts from 2000. The pre-2000 information for such
cases is still included in the form of prior criminal count of defendants. For
around 0.005% cases in the dataset, WCCA filing date of older cases is during
or after 2000. We have not removed these cases from the dataset. Given the 2
year follow-up period, we exclude cases that are disposed after 2018 since
there is not enough time to observe recidivism (we had case records until 2020
only). Note that the post 2018 information is used for observing recidivism
outcomes in the dataset. We also point out that during pre-processing, we
excluded dismissed cases that do not result in conviction. We also deleted
records of defendants that do not have sex and/or race data available or if
race could not be determined from the available conflicting race information
across cases (see Section 3.3 for details.). We also excluded cases that only
have forfeiture (non-crime) charge. Information excluded in pre-processing
step is not used in the dataset.

###  3.8 Redacted Information

While the case records on the WCCA website contain personal information of the
defendants, the website also displays notice to the employers informing about
the violation of state law to discriminate against a job applicant because of
an arrest or conviction record. As an ethics consideration on our end, we have
removed personal information from the curated dataset that we contribute in
this paper. In particular, we have excluded the names of the defendants, their
addresses, dates of birth, dates of case events, case numbers etc. We have
also replaced information about judge, zipcode and county with pseudo-
identifiers, but mappings for zipcode and county can be made available to
researchers on request.

##  4 Exploratory Analysis

###  4.1 Summary Statistics

Table 1 presents summary statistics of the dataset thus constructed with
around 1.5 million cases from 2000 to 2018. There are five race groups in the
dataset, with around 65% Caucasian, 23% African American and 7% Hispanic. The
‘Asian or Pacific Islander’ and ‘American Indian or Alaskan Native’ groups are
abbreviated as ‘Asian’ and ‘Native American’ respectively. The proportion of
male defendants in the dataset (80%) is significantly higher than female, and
most crimes in the dataset are committed at a younger age (below 60).
Misdemeanors are the most frequent crime type except for Hispanic with
criminal traffic (45%) being the most common crime type.

Table 1: Summary of the Dataset | Full sample | Caucasian |  African American | Hispanic |  Native American |  Asian
---|---|---|---|---|---|---
_Sample size_ | 1,476,967 | 964,922 | 333,036 | 101,607 | 63,862 | 13,540
_Sample share_ |  | 65.33% | 22.55% | 6.88% | 4.32% | 0.92%
 _Sex_ |  |  |  |  |  |
Male | 80.40% | 79.05% | 83.47% | 88.88% | 69.65% | 87.57%
 _Age_ |  |  |  |  |  |
Below 30 | 51.38% | 49.45% | 54.13% | 56.91% | 53.71% | 68.60%
30 to 60 | 47.44% | 49.09% | 45.17% | 42.61% | 45.58% | 30.85%
_Case type_ |  |  |  |  |  |
Felony | 32.18% | 30.76% | 39.98% | 21.09% | 29.80% | 36.39%
Misdemeanor | 43.04% | 43.67% | 43.14% | 34.12% | 47.55% | 40.89%
Criminal Traffic | 24.78% | 25.57% | 16.88% | 44.79% | 22.66% | 22.73%

Table 2 shows the summary statistics for the two recidivism variables, when
the sentence length cut-off is 180 days and 2 years as discussed in Section
3.5. While the rate of missing prediction variable in the dataset varies
significantly in the two variables, the recidivism rate (excluding ‘missing’)
doesn’t show the same level of significant difference. In the rest of the
paper, we use the recidivism variable with 180 days cut-off length, unless
otherwise stated. The observed recidivism rate is the highest ($\sim$56%) with
Native American.

Table 2: Statistics of Two Recidivism Variables Missing Variable Rate
---
Sentence Cut-off | Overall | Caucasian |  African American | Hispanic |  Native American |  Asian
180 Days | 0.0807 | 0.0667 | 0.1286 | 0.0653 | 0.0686 | 0.0767
2 Years | 0.0471 | 0.0367 | 0.0824 | 0.0365 | 0.0374 | 0.0513
Recidivism Rate (excluding ‘missing’)
Sentence Cut-off | Full sample | Caucasian |  African American | Hispanic |  Native American |  Asian
180 Days | 0.4221 | 0.4034 | 0.4643 | 0.3876 | 0.5647 | 0.3780
2 Years | 0.4168 | 0.3996 | 0.4534 | 0.3825 | 0.5593 | 0.3724

Figure 1 shows the recidivism rate for different races over the years. Overall
recidivism rate and the recidivism rate difference between groups is
decreasing over time. We also observed marginal shifts in the distribution of
offense types, group proportions, sex, and age. We discuss the limitations of
this interpretation in Section 5.2.

Figure 1: Change in Data Distribution with Time

For each type of offense in the dataset, there are more specific
categorization of offense in the dataset under the wcisclass variable. For
misdemeanor, there are 55 distinct wcisclass values (and 0.33% missing). For
felony, there are 60 distinct wcisclass values (and 0.34% missing), and for
criminal traffic, there are 39 distinct wcisclass values (and 0.49% missing).
We list the 10 most common wcisclasss for each offense type in Table 3. We
provide summary statistics about highest_charge_severity in Table 6 in
appendix G. Figure 2 in appendix G also compares the distributions of
different offense types in the two recidivism variables, when the recidivism
value is missing and when it is not not-missing.

Table 3: 10 Most Frequent wcisclass for Each Type of Offense Misdemeanor | Felony | Criminal Traffic
---|---|---
wcisclass | Percentage | wcisclass | Percentage | wcisclass | Percentage
Disorderly Conduct | 15.51 | Drug Possession | 15.98 | OAR/OAS | 42.74
Battery | 14.44 | Bail Jumping | 10.43 |  Operating While Intoxicated | 42.25
Resisting Officer | 13.62 | Burglary | 7.49 |  Operate Without License | 8.37
Drug Possession | 8.76 |  Drug Manufacture/ Deliver | 6.94 | Hit and Run | 1.98
Bail Jumping | 8.75 | Theft | 6.28 |  Unidentified Misdemeanor Traffic | 1.16
Theft | 6.31 |  Operating while intoxicated | 5.83 | Bail Jumping | 0.9
Retail Theft (Shoplifting) | 5.62 | Other Felony | 5.03 |  Other Misdemeanor | 0.78
Criminal Damage | 5.61 | Forgery | 3.76 | nan | 0.49
Drug Paraphernalia | 2.96 | Battery | 3.25 | Resisting Officer | 0.41
Worthless Checks | 2.69 |  Substantial/ Aggravated Battery | 3.06 | Drug Possession | 0.21

###  4.2 Machine Learning Predictions

The purpose of this section is not to indicate that the following is the
“right" way or the only way to use this dataset in research. But for reference
only, we report performance metrics of two machine learning classifiers on
this dataset. The predictors included in the models were sex, type of offense,
prior criminal count (for each type), and age (at judgment and at first
offense). Logistic Regression and XGBoost classifiers were from Python’s
scikit-learn [27] and XGBoost [28] packages respectively. The packages are
public and freely available. LR learns a linear model while XGBoost learns a
decision tree ensemble. We split the entire data into 70% train and 30% test,
and the results are reported on the test data. For LR classifier, we include
L2 regularization and select the regularization parameter via 10-fold cross-
validation. For XGBoost, we include both L1 and L2 regularization and tune the
hyperparameters via grid search and 5-fold cross-validation. 95% confidence
intervals are constructed by multiple train and test splits. Details of
compute resources can be found at
https://scicomp.ethz.ch/wiki/Euler#Introduction.

Table 4 shows the performance of the LR classifier and the XGBoost classifier
on this dataset across all five race groups. FPR stands for False Positive
Rate, FNR for False Negative Rate, PR for the rate of positive classification,
AUC stands for Area under the ROC Curve. We observe from the table that Native
American is the most disadvantaged group in the sense that it has the highest
FPR and PR, and the lowest FNR. The largest FPR difference is around 13%, the
largest FNR difference is around 18%, and the largest PR difference is around
20%, all between Native American and Hispanic with XGBoost classifier.
Hispanic and Caucasian groups receive the most favorable decisions (1-PR) in
both models, followed by Asian and African American groups. The overall
accuracy of XGBoost and LR is not very different but the FNR, FPR and PR
differences between groups is higher for XGBoost. These results are for the
recidivism variable with 180 days sentence cut-off. In Table 7 in appendix G,
we also provide results for the recidivism variable with 2 years sentence cut-
off. We did not observe a notable difference in performance metrics for the
two recidivism variables but it may be worth investigating the differences at
a more granular level.

In addition to the discussion in Section 2 on the limitations of the COMPAS
dataset, it is useful to consider the following as well for comparison with
our dataset. [29] report about COMPAS/ProPublica dataset: “The (unconstrained)
logistic regression classifier leads to an accuracy of 0.668. However, the
classifier yields false positive rates of 0.35 and 0.17, respectively, for
blacks and whites (i.e., DFPR = 0.18), and false negative rates of 0.31 and
0.61 (i.e., DFNR = 0.30).” The COMPAS dataset has the following number of
samples from different racial groups. African-American: 3175, Asian: 31,
Caucasian: 2103, Hispanic: 509, Native American: 11. Due to very small
numbers, racial groups other than African-American and Caucasian are generally
not analysed in the literature using the COMPAS. Our dataset significantly
addresses that limitation.

Table 4: Recidivism Prediction Performance on the Dataset, with 95% Confidence
Intervals

| Overall | Caucasian |  African American
---|---|---|---
_XGBoost_ |  |  |
Accuracy | 0.6588 $\pm$ 0.0018 | 0.6648 $\pm$ 0.0020 | 0.6459 $\pm$ 0.0034
AUC | 0.7039$\pm$ 0.0018 | 0.7044$\pm$ 0.0023 | 0.7033$\pm$ 0.0035
FPR | 0.2244 $\pm$ 0.0041 | 0.2159 $\pm$ 0.0042 | 0.2454 $\pm$ 0.0048
FNR | 0.5008 $\pm$ 0.0040 | 0.5113 $\pm$ 0.0045 | 0.4792 $\pm$ 0.0067
PR | 0.3405 $\pm$ 0.0035 | 0.3261 $\pm$ 0.0038 | 0.3734 $\pm$ 0.0041
_Logistic Regression_ |  |  |
Accuracy | 0.6461 $\pm$ 0.0013 | 0.6560 $\pm$ 0.0020 | 0.6206 $\pm$ 0.0027
AUC | 0.6822 $\pm$ 0.0013 | 0.6825 $\pm$ 0.0022 | 0.6806 $\pm$ 0.0030
FPR | 0.1532 $\pm$ 0.0022 | 0.1479 $\pm$ 0.0026 | 0.1667 $\pm$ 0.0031
FNR | 0.6282 $\pm$ 0.0029 | 0.6334 $\pm$ 0.0035 | 0.6244 $\pm$ 0.0039
PR | 0.2456 $\pm$ 0.0021 | 0.2363 $\pm$ 0.0024 | 0.2638 $\pm$ 0.0023
| Hispanic |  Native American |  Asian
---|---|---|---
_XGBoost_ |  |  |
Accuracy | 0.6567 $\pm$ 0.0055 | 0.6303 $\pm$ 0.0076 | 0.6760 $\pm$ 0.0108
AUC | 0.6719$\pm$ 0.0068 | 0.6878$\pm$ 0.0090 | 0.7079 $\pm$ 0.0122
FPR | 0.2016 $\pm$ 0.0073 | 0.3272 $\pm$ 0.0145 | 0.2203 $\pm$ 0.0147
FNR | 0.5674 $\pm$ 0.0096 | 0.4025 $\pm$ 0.0106 | 0.4944 $\pm$ 0.0225
PR | 0.2910 $\pm$ 0.0060 | 0.4800 $\pm$ 0.0100 | 0.3283 $\pm$ 0.0128
_Logistic Regression_ |  |  |
Accuracy | 0.6535 $\pm$ 0.0048 | 0.6026 $\pm$ 0.0057 | 0.6735 $\pm$ 0.0116
AUC | 0.6558 $\pm$ 0.0056 | 0.6746 $\pm$ 0.0068 | 0.6946 $\pm$ 0.0136
FPR | 0.1292 $\pm$ 0.0051 | 0.2395 $\pm$ 0.0081 | 0.1386 $\pm$ 0.0118
FNR | 0.6903 $\pm$ 0.0085 | 0.5189 $\pm$ 0.0080 | 0.6350 $\pm$ 0.0199
PR | 0.1991 $\pm$ 0.0049 | 0.3761 $\pm$ 0.0065 | 0.2243 $\pm$ 0.0106

##  5 Discussion

###  5.1 License

The dataset is distributed under the Creative Commons 4.0 BY-NC-SA
(Attribution-NonCommercial-ShareAlike) license. The choice of this license is
motivated by the intended uses of the dataset and other considerations that we
discuss next.

###  5.2 Intended Uses and Limitations

The dataset is intended to be used for academic research only. We focus in
this paper on algorithmic fairness analysis as a specific use-case, but it
doesn’t have to restricted to this. We have provided pseudo-identifiers for
judge, county and zipcode and as stated earlier, mappings for county and
zipcode can be provided to researchers on request to explore different
research use-cases or to explore the interaction of algorithmic or data
centric factors with systemic issues such as biases of judges, county or
location factors etc. There is more information available in the raw case
records that is more difficult to parse and thus, not available at present in
the curated dataset. As we discuss in Section 5.4, we will strive to update
the dataset in the near future if we find this additional information to be of
sufficiently good quality. This will further extend the scope of research uses
of the dataset. These may include, for e.g., studying the process of
sentencing itself, or measuring the impact of defendant attributes on the
severeness of sentencing or bail.

The dataset addresses several limitations of prior datasets (larger size,
large number of samples from different racial groups, different courts,
counties, more attributes, less variance, observations over longer time etc).
At the same time, there are several fundamental limitations [11] that are
difficult or even impossible to address in any dataset. Examples include
missing values and biases encoded in the recidivism variable, as discussed in
Section 3.5, and in the attributes. Such biases in the data can not be
avoided. Depending on the context, they may be handled in other ways, for
e.g., through appropriate algorithmic intervention or even critically
rethinking the use of algorithms in the first place. We argue that since these
limitations are fundamental and can not be addressed easily through data
collection or curation processes, it is important that the datasets are
available to researchers for critical research. The researchers must be aware
of the limitations of the datasets however. For example, the unavailability of
prison/jail records implies that the reported recidivism is not guaranteed to
be accurate, and we are not able to measure the extent of this inaccuracy. In
addition, since we only have cases from Wisconsin Circuit Courts, we can not
capture recidivism elsewhere, also, an error that is hard to quantify the
impact of. Moreover, the fact that convictions do not faithfully represent
criminality is another source of bias and a fundamental issue with the
prediction task definition itself. We further bring attention to the fact that
the WCLD dataset is built using case records for a long period of time. There
are several changes in the society and external variables over such a long
time period that can not be captured in a dataset. This temporal factor must
be kept in mind while using the dataset for research. However, it is also a
strength of the dataset since it can be used to study the algorithms’ behavior
under distributions shifts. We note that not all distribution shifts are
reflective of the real-world societal changes but are nevertheless realities
of the data collection and curation processes, and therefore, availability of
such datasets for research is important. In case of WCCA, case records from
earlier years like late 1900s tend to be less in number (there can be multiple
possible reasons for this), and that may have caused underestimation of prior
criminal counts for older cases in the WCLD dataset.

As part of the subscription information, WCCA also informed of a few other
limitations of the data available through the REST interface. We list those
limitations in the datasheet (Section F.3 in appendix F).

###  5.3 Ethical Considerations

The court record summaries provided by WCCA are public records under Wisconsin
open records law sections 19.31-19.39 of the Wisconsin Statutes. Court records
not open to public inspection by law are not available. WCCA information does
not include information that may be confidential, sealed, or redacted in
accordance with all applicable statutes, court orders, and rules related to
confidentiality, sealing, and redaction. We refer the reader to the WCCA FAQs
[30] for further information. We took additional steps to redact sensitive
information as discussed in Section 3.8

Further, readers should not interpret summary statistics of this dataset (for
example, those listed in Tables 1 and 2) as ‘ground truth’ but rather as
characteristics of the dataset only.

Researchers using this dataset are expected to carefully take into account the
intended uses and limitations listed in Section 5.2 and follow practices for
responsible use of criminal justice data in research [11]. Beyond limitations
of our dataset and data more broadly in this context, fundamental limitations
of algorithms (including but not limited to mathematical limitations [3, 4])
pose challenge to addressing the ethical concerns [11, 31, 32] related to
recidivism prediction task itself [33]. Such limitations and ethical concerns
about the task should also be taken into account while drawing conclusions
from the research using this dataset.

###  5.4 Future Work

In the case records that we downloaded using the REST interface, there is
additional information available about various case events (e.g. hearings,
bail decisions, bail amounts, sentences), information on the associated
attorneys and government officials involved, including prosecutors. It may be
useful to parse this information and update the dataset. We will strive to do
these and update the dataset depending on the quality and usefulness of the
additional information available. Readers interested in contributing to the
dataset are encouraged to reach out to us.

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##  F Datasheet

###  F.1 Motivation

For what purpose was the dataset created? The raw case records were created
presumably for public inspection and for record keeping in the courts.

The external census dataset used in curation was created for census.

The final curated dataset, contributed in this paper, was created for academic
research on algorithmic fairness.

Who created the dataset, and on behalf of which entity? According to Wisconsin
Circuit Courts Access website, the raw data records that we accessed through
their website are an exact copy of the case information entered into the
circuit court case management system by court staff in the counties where the
case files are located.

Nianyun Li, Claudia Marangon and Peiyao Sun curated the dataset in its current
form with the help of Elliott Ash and Naman Goel.

Who funded the creation of the dataset? The funding of creation of original
data records is unclear, but presumably the state funded the court staff in
the counties where the case files are located.

The creation of the curated dataset was funded by ETH Zurich.

###  F.2 Composition

What do the instances that comprise the dataset represent? The instances
represent case and defendant information.

How many instances are there in total (of each type)? There are around 1.5
million instances.

Does the dataset contain all possible instances or is it a sample of instances
from a larger set? The dataset is a sample of instances from a larger set.

What data does each instance consist of? Each instance contains defendant’s
new$\\_$id, age, sex, type of offense, wcisclass, year of filing, race, age at
judgment, age at first offense, 9 neighborhood characteristics (population
density, proportion who attended college, proportion eligible for food stamp,
African American population share, Hispanic population share, proportion who
live in rural and urban area, median household income, highest charge
severity, not$\\_$detained, probation, recid (180 days sentence length
cutoff), recid (2 years sentence length cutoff), violent_recid (180 days
sentence length cutoff), jail, county, violent_crime. The details of each of
these variables are provided in the accompanying paper and the metadata file
in the data directory. Each instance also contains prior criminal count for
each type of offense, prior sentence length statistics, prior charge severity
counts. The counts were created from the available raw case records by
performing database search, and are therefore possibly underestimated.

Is there a label or target associated with each instance? For research, recid
(180 days sentence length cutoff) and recid (2 years sentence length cutoff)
are two variables associated with each instance. There is also a violent_recid
(180 days sentence length cutoff) variable available for research. These are
not ground truth labels in a traditional sense but only variables defined by
the authors. Details in Section 3.5and ethics discussion in Section 5.3 of the
accompanying paper.The first variable is recidivism as observed in the case
records by performing database search, within a 2 year follow-up period since
judgment disposition date, using a 180 days cut-off for sentence. The second
is obtained by using a 2 year cut-off sentence and extending the follow-up
period of 2 years by adding the sentence length.

Is any information missing from individual instances? Yes, the recidivism
variables can not be observed for defendants depending on their sentence.
Thus, it is missing for some defendants. Details in Section 3.5.

Are relationships between individual instances made explicit? The dataset is
anonymized and therefore some relationship between individual instances may be
lost. We have included defendant pseudo-identifier in the dataset constructed
based on first name, last name and date of birth.

Are there recommended data splits? No. But we have included two possible
random splits in the dataset (one is completely random thus only ensuring
different cases in train and test splits, the other also ensures different
defendants in train and test splits).

Are there any errors, sources of noise, or redundancies in the dataset? The
errors are possible in the raw case information entered by the court staff.
Known errors in the curated dataset construction are discussed in Section 5.2.

Is the dataset self-contained, or does it rely on external resources? The
dataset has been curated using case records from WCCA and census data from
2010. The curated dataset is self-contained.

Does the dataset contain data that might be considered confidential? No

Does the dataset contain data that, if viewed directly, might be offensive,
insulting, or threatening? No

Does the dataset relate to people? Yes, the raw case records relate to the
defendants. However, directly identifiable information such as names,
addresses, date of birth, case numbers etc has been removed in the curated
dataset.

Does the dataset identify any subpopulations? Yes, the dataset contains data
from five racial groups as marked by WCCA, sex and age groups.

Is it possible to identify individuals? The raw case records available on WCCA
are public information and it is possible to identify individuals there. For
the curated dataset that we release, we have removed directly identifiable
information such as names, addresses, date of birth, case numbers etc.

Does the dataset contain data that might be considered sensitive in any way?
The raw case records available on WCCA are public information and some of the
information such as defendant’s personal information may be considered
sensitive. For the curated dataset that we release, we have removed directly
identifiable information such as names, addresses, date of birth, case numbers
etc.

###  F.3 Collection Process

How was the data associated with each instance acquired? Through the REST
interface of WCCA.

What mechanisms or procedures were used to collect the data? Through the REST
interface of WCCA. We queried all case numbers in each county during the
period and subsequently, using these case numbers, we queried individual cases
for all available information. Different attributes were then derived using
the process described in the accompanying paper.

If the data are a sample from a larger set, what was the sampling strategy?
The dataset is composed primarily of new cases filed between 2000-2018. The
dataset excludes dismissed cases that do not result in conviction, records of
defendants that do not have sex and/or race data and cases that only have
forfeiture (non-crime) charge. https://wcca.wicourts.gov/faq.html provides
more information on records that might have been deleted. WCCA also informed
us of a few limitations of the data as part of the subscription. These are
listed as follows:

  1. 1.

WCCA Information includes only court records open to public view under
Wisconsin’s Open Records Law, Wis. Stat. 19.31-19.39. Court records not open
to public inspection by law are not available.

  2. 2.

WCCA Information does not include information that may be confidential,
sealed, or redacted in accordance with all applicable statutes, court orders,
and rules related to confidentiality, sealing, and redaction.

  3. 3.

WCCA Information consists of information entered into the CCAP 222WCCA was
formerly CCAP. case management system by the Clerk of Circuit Court or
Register in Probate in each county. CCAP is not responsible for the accuracy
or timeliness of WCCA information.

  4. 4.

WCCA Information does not comprise the complete court record. Copies of
documents must be obtained from the Clerk of Circuit Court or Register in
Probate.

  5. 5.

WCCA Information is only a snapshot of the information accessible in the CCAP
case management system on the date the information is downloaded by the
Subscriber.

  6. 6.

WCCA Information is not the Judgment and Lien Docket under Wis. Stat. 806.10.
The Judgment and Lien Docket is available from the Clerk of Circuit Court.

  7. 7.

Court records which predate the implementation of the CCAP case management
system in the county in which the records were created are not accessible
under this Agreement, except to the extent such records have been back loaded.

  8. 8.

In criminal cases, any designation in any race field contains subjective
information generally provided by the agency that filed the case.

  9. 9.

Searching WCCA Information by a particular field or code may not return all
cases in which a particular event occurred unless at the time the record was
created the case management system required the field or code to be completed
in order to proceed to make the rest of the record

Who was involved in the data collection process and how were they compensated?
The case records were created by court staff in respective county courts and
were presumably, compensated by state.

The authors of this paper collected the case records from WCCA and were
employees of ETH Zurich during the data curation process. They were
compensated by ETH Zurich in the form of fixed monthly salaries. Nianyun Li,
Naman Goel, Peiyao Sun, Claudia Marangon were/are on fixed-term contracts with
ETH Zurich.

Over what time frame was the data collected? Authors had access to the raw
case records through WCCA REST interface during the period July 2020 - July
2021. However, data collection was finished by Feb 2021.

Were any ethical review processes conducted? The authors are not aware of the
ethical review process followed in WCCA or county courts for creation of the
case records. As part of the curation of the dataset that we contribute, no
formal/institutional ethical review process was conducted.

Does the dataset relate to people? Yes, the raw case records relate to
defendants. However, directly identifiable information such as names,
addresses, date of birth, case numbers etc has been removed in the curated
dataset.

Did you collect the data from the individuals directly, or obtain it via third
parties? We obtained the raw case records from a third party, WCCA
(https://wcca.wicourts.gov).

Were the individuals notified about the data collection? The authors are not
aware of it. However, presumably, the defendants were aware that the
information about their cases (and hence the related information about them)
is kept in court records and is public information under Wisconsin state laws,
when exceptions do not apply.

Did the individuals in question consent to the collection and use of their
data? The authors are not aware of it. The information is available publicly
under Wisconsin state laws.

If consent was obtained, were the consenting individuals provided with a
mechanism to revoke their consent in the future or for certain uses? WCCA
provides option in certain limited cases to petition to have case-records
removed. Please see https://wcca.wicourts.gov/faq.html.

For the curated dataset, directly identifiable information such as names,
addresses, date of birth, case numbers etc has already been removed.

Has analysis of the potential impact of the dataset and its use on data
subjects been conducted? Authors are not aware if Wisconsin state or WCCA had
done any such analysis.

For the curated dataset, we have thought carefully about it and redacted all
the information that, according to us, could potentially affect subjects.

###  F.4 Pre-processing and Cleaning

Was any preprocessing of the data done? The accompanying paper describes the
details of curating the dataset from the raw case records.

Was the “raw” data saved in addition to the cleaned data? Yes, we saved the
data on our institute’s secure servers (until our research requires). We do
not plan to make this data available to others.

Is the software used to clean the data available? We didn’t use any ‘data
cleaning software’. We have described the steps taken in curating the dataset
in the accompanying paper. If useful, we can also provide SQL commands for
these steps.

###  F.5 Uses

Has the dataset been used for any tasks already? The curated dataset that we
release has only been used for academic research. We are not aware who else
has used the raw case records from WCCA and for which tasks.

Is there a repository that links to any or all papers that use the dataset?
Not to our knowledge.

What (other) tasks could the dataset be used for? The dataset is for academic
research.

Is there anything about the composition of the dataset or the way it was
collected and cleaned that might impact future uses? We have listed
limitations of the data in Section 5.2 and in earlier parts of this datasheet
(for example, see Section F.3).

Are there tasks for which the dataset should not be used? The dataset should
not be used for purposes other than academic research.

###  F.6 Distribution

Will the dataset be distributed to third parties outside of the entity on
behalf of which the dataset was created? Yes, public data.

How will the dataset be distributed? The dataset is hosted at
http://clezdata.github.io/wcld/. Downloads are subject to research only use
acknowledgement. In case of any difficulties in accessing the data in the
future, interested readers can contact the authors.

When will the dataset be distributed? The dataset is hosted at
http://clezdata.github.io/wcld/.

Will the dataset be distributed under a copyright, other IP license, or terms
of use? The dataset is distributed under the Creative Commons 4.0 BY-NC-SA
license.

Have any third parties imposed IP-based or other restrictions on the data
associated with the instances? No.

Do any export controls or other regulatory restrictions apply to the data? No.

###  F.7 Maintenance

Who is supporting/hosting/maintaining the dataset? Elliott Ash.

How can the data owner/curator be contacted? Through email:
<EMAIL_ADDRESS>
Is there an erratum? Not at the time of publishing this paper.

Will the dataset be updated? The existing entries in the dataset are unlikely
to be modified. New information may be added.

If the dataset relates to people, are there applicable limits on the retention
of data associated with the instances? Public information under applicable
law.

Will older versions of the dataset continue to be supported/hosted/maintained?
In the unlikely event that the entries in the dataset are to be modified,
older version will also be made available, for example, using a version
control system.

If others want to extend/augment/build on/contribute to the dataset, is there
a mechanism for them to do so? Individuals interested in contributing are
encouraged to contact Elliott Ash at<EMAIL_ADDRESS>
##  G Additional Information

Table 5 shows the mappings from charge severity categories on WCCA and the
numerical ranking that we assigned to these categories. Values 1-6 were
assigned for forfeiture charges and hence, not shown in the table. Table 6
provides summary statistics for the column charge severity.

Table 5: Mapping from Charge Severity to Numerical Ranking Charge | highest_charge_severity
---|---
Felony A | 21
Felony B | 20
Felony BC | 19
Felony C | 18
Felony D | 17
Felony E | 16
Felony F | 15
Felony G | 14
Felony H | 13
Felony I | 12
Felony U | 11
Misdemeanor A | 10
Misdemeanor B | 9
Misdemeanor C | 8
Misdemeanor U | 7
Table 6: Highest Charge Severity in the Dataset highest_charge_severity | Count | Percentage
---|---|---
7 | 516004 | 34.94
10 | 460898 | 31.21
9 | 145750 | 9.87
13 | 116008 | 7.85
12 | 74062 | 5.01
15 | 37580 | 2.54
14 | 30561 | 2.07
18 | 26678 | 1.81
11 | 21893 | 1.48
16 | 21217 | 1.44
17 | 17088 | 1.16
20 | 6187 | 0.42
19 | 1581 | 0.11
21 | 845 | 0.06
8 | 615 | 0.04

(a) Missing Recidivism Outcome

(b) Not-Missing Recidivism Outcome

Figure 2: Differences in the distribution of type of offense depending on the
sentence cutoff length in recidivism variables.

Table 7 shows results of the machine learning classifiers for the recidivism
variable with 2 years sentence cut-off.

Table 7: Recidivism (2 year Sentence Cut-Off) Prediction with 95% Confidence
Intervals

| Overall | Caucasian |  African American
---|---|---|---
_XGBoost_ |  |  |
Accuracy | 0.6589 $\pm$ 0.0016 | 0.6652 $\pm$ 0.0019 | 0.6460 $\pm$ 0.0038
AUC | 0.7018 $\pm$ 0.0018 | 0.7026 $\pm$ 0.0021 | 0.7005 $\pm$ 0.0034
FPR | 0.2177 $\pm$ 0.0029 | 0.2094 $\pm$ 0.0029 | 0.2353 $\pm$ 0.0046
FNR | 0.5136 $\pm$ 0.0043 | 0.5232 $\pm$ 0.0049 | 0.4970 $\pm$ 0.0064
PR | 0.3298 $\pm$ 0.0030 | 0.3163 $\pm$ 0.0032 | 0.3567 $\pm$ 0.0039
_Logistic Regression_ |  |  |
Accuracy | 0.6457 $\pm$ 0.0014 | 0.6555 $\pm$ 0.0016 | 0.6215 $\pm$ 0.0029
AUC | 0.6784 $\pm$ 0.0015 | 0.6794 $\pm$ 0.0019 | 0.6760 $\pm$ 0.0034
FPR | 0.1478 $\pm$ 0.0022 | 0.1430 $\pm$ 0.0023 | 0.1571 $\pm$ 0.0026
FNR | 0.6429 $\pm$ 0.0024 | 0.6469 $\pm$ 0.0030 | 0.6453 $\pm$ 0.0047
PR | 0.2351 $\pm$ 0.0018 | 0.2270 $\pm$ 0.0020 | 0.2467 $\pm$ 0.0028
| Hispanic |  Native American |  Asian
---|---|---|---
_XGBoost_ |  |  |
Accuracy | 0.6576 $\pm$ 0.0054 | 0.6273 $\pm$ 0.0064 | 0.6733 $\pm$ 0.0149
AUC | 0.6724 $\pm$ 0.0056 | 0.6852$\pm$ 0.0066 | 0.7016 $\pm$ 0.0182
FPR | 0.1997 $\pm$ 0.0079 | 0.3223 $\pm$ 0.0123 | 0.2153 $\pm$ 0.0130
FNR | 0.5720 $\pm$ 0.0097 | 0.4125 $\pm$ 0.0105 | 0.5140 $\pm$ 0.0245
PR | 0.2872 $\pm$ 0.0076 | 0.4706 $\pm$ 0.0091 | 0.3163 $\pm$ 0.0116
_Logistic Regression_ |  |  |
Accuracy | 0.6540 $\pm$ 0.0054 | 0.5988 $\pm$ 0.0050 | 0.6728 $\pm$ 0.0152
AUC | 0.6533 $\pm$ 0.0061 | 0.6699 $\pm$ 0.0070 | 0.6878 $\pm$ 0.0176
FPR | 0.1293 $\pm$ 0.0058 | 0.2358 $\pm$ 0.0090 | 0.1339 $\pm$ 0.0113
FNR | 0.6948 $\pm$ 0.0087 | 0.5316 $\pm$ 0.0082 | 0.6521 $\pm$ 0.0222
PR | 0.1967 $\pm$ 0.0055 | 0.3658 $\pm$ 0.0069 | 0.2138 $\pm$ 0.0096

##  H Violent/Non-Violent Labels for Charge Descriptions using GPT-4

The prompt used for GPT was as follows:

“In the FBI’s Uniform Crime Reporting (UCR) Program, violent crime is composed
of four offenses: murder and nonnegligent manslaughter, forcible rape,
robbery, and aggravated assault. Violent crimes are defined in the UCR Program
as those offenses which involve force or threat of force.
I will next provide you a list of charge descriptions. For each charge
description in the list, I want you to provide me a single word answer
(Violent/Non-Violent), depending on whether that charge description refers to
a violent crime or not. Before providing the answer, I want you to provide an
explanation or thought process of how you go from charge description to the
answer. The format of your response should be Charge Description;;Thought
Process;;Violent/Non-Violent. Do not include any other text in your response.
The charge description in your response should be exactly same as the charge
description in my list (do not correct the formatting or spellings etc in
charge descriptions) and in the same order as my list.
Here is the list: ”

We appended charge descriptions (50 at a time) to the prompt and repeatedly
prompted the model until we obtained labels for all the charge descriptions.
The model used was ‘gpt-4’ as on 20-Sep-2023. We set the system message as
‘You are a helpful assistant.’ The cost of inference was approximately $150,
including the costs of trial and error with different prompting and models
etc.