# Kaon Photoproduction and the ${\boldmath\Lambda}$ Decay Parameter
$\boldmath{\mathbf{\alpha_{-}}}$

D. G. Ireland<EMAIL_ADDRESS>SUPA, School of Physics and
Astronomy, University of Glasgow G12 8QQ, United Kingdom  M. Döring
<EMAIL_ADDRESS>Institute for Nuclear Studies and Department of Physics, The
George Washington University, Washington, DC 20052, USA  D. I. Glazier
<EMAIL_ADDRESS>SUPA, School of Physics and Astronomy, University
of Glasgow G12 8QQ, United Kingdom  J. Haidenbauer  j.haidenbauer@fz-
juelich.de Institute for Advanced Simulation, Institut für Kernphysik
(Theorie) and Jülich Center for Hadron Physics, Forschungszentrum Jülich,
D-52425 Jülich, Germany  M. Mai<EMAIL_ADDRESS>Institute for Nuclear
Studies and Department of Physics, The George Washington University,
Washington, DC 20052, USA  R. Murray-Smith  Roderick.Murray-
<EMAIL_ADDRESS>School of Computing Science, University of Glasgow G12
8RZ, United Kingdom  D. Rönchen<EMAIL_ADDRESS>Helmholtz-Institut
für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical
Physics, Universität Bonn, 53115 Bonn, Germany

###### Abstract

The weak decay parameter $\alpha_{-}$ of the $\Lambda$ is an important
quantity for the extraction of polarization observables in various
experiments. Moreover, in combination with $\alpha_{+}$ from $\bar{\Lambda}$
decay it provides a measure for the matter-antimatter asymmetry. The weak
decay parameter also affects the decay parameters of the $\Xi$ and $\Omega$
baryons and, in general, any quantity in which the polarization of the
$\Lambda$ is relevant. The recently reported value by the BESIII collaboration
of $0.750(9)(4)$ is significantly larger than the current PDG value of
$0.642(13)$. In this work we make an independent estimate of $\alpha_{-}$,
using an extensive set of polarization data measured in kaon photoproduction
in the baryon resonance region and constraints set by spin algebra. The
obtained value is 0.721(6)(5). The result is corroborated by multiple
statistical tests as well as a modern phenomenological model, showing that our
new value yields the best description of the data in question. Our analysis
supports the new BESIII finding that $\mathbf{\alpha_{-}}$ is significantly
larger than the previous PDG value. Any experimental quantity relying on the
value of $\alpha_{-}$ should therefore be re-considered.

Nonleptonic hyperon decays; photonuclear reactions.

###### pacs:

14.20.Jn 13.60.Le 13.30.Eg

##  I Introduction

The decay parameter $\mathbf{\alpha_{-}}$ of the parity-violating weak decay
$\Lambda\rightarrow p\pi^{-}$ is a result of interference between $s$\- and
$p$-waves. A recent study by the BESIII collaboration Ablikim  _et al._ (2018)
reported a value of $\mathbf{\alpha_{-}}$ as $0.750\pm 0.009\pm 0.004$ for
this quantity, which is significantly different from the value quoted in the
2018 Review of Particle Physics Tanabashi  _et al._ (2018) (PDG) of $0.642\pm
0.013$.

This newly published value of $\mathbf{\alpha_{-}}$ is some 17 % higher than
the PDG value and, since both values have uncertainties at the percent level,
it represents a discrepancy of about five standard deviations. The two results
are therefore incompatible, and so an independent estimate of this quantity is
highly desirable given that $\mathbf{\alpha_{-}}$ plays an important role in
various fields of physics. For instance, comparing $\mathbf{\alpha_{-}}$ with
the parameter $\alpha_{+}$ of the decay
$\bar{\Lambda}\rightarrow\bar{p}\pi^{+}$ provides a test of $CP$ symmetry for
strange baryons and, thus can potentially shed light on the matter-antimatter
asymmetry in the universe Sakharov (1967). $\mathbf{\alpha_{-}}$ has also an
impact on several theoretical studies where its actual value enters directly.
In particular, it would affect calculations of the weak nonleptonic hyperon
decays within SU(3) chiral perturbation theory Bijnens  _et al._ (1985);
Holstein (2000); Borasoy and Marco (2003).

Over the last 40 years there have been various experiments whose results rely
on the value of $\mathbf{\alpha_{-}}$. Examples of this are the extensive
studies of the reactions $\bar{p}p\to\bar{\Lambda}\Lambda$ and
$\bar{p}p\to\bar{\Lambda}\Sigma^{0}+{\rm c.c.}$ by the PS185 Collaboration at
the LEAR facility at CERN Klempt  _et al._ (2002) that measured analyzing
powers, spin-correlation parameters and spin-transfer coefficients. Recent
results, such as the STAR measurement of heavy ion collisions to study the
vortical structure of a nearly ideal liquid The STAR Collaboration (2017), and
the ATLAS measurement of $\Lambda$ and $\bar{\Lambda}$ transverse polarization
ATLAS Collaboration  _et al._ (2015) also depend on the value of
$\mathbf{\alpha_{-}}$.

Information about other strange baryons depends on $\mathbf{\alpha_{-}}$
through chains of successive decays. For example, the decay parameter for
$\Xi$ is determined from the decays $\Xi\to\Lambda\pi\to N\pi\pi$ and deduced
from the product $\alpha_{\Xi}\mathbf{\alpha_{-}}$, which in turn affects the
measured polarization data for the reactions $K^{-}p\to
K^{+}\Xi^{-},K^{0}\Xi^{0}$ Trippe and Schlein (1967); Dauber  _et al._ (1969)
and $\gamma p\to K^{+}K^{+}\Xi^{-}$ Bono  _et al._ (2018). The decay parameter
for $\Omega^{-}$ depends likewise on the values of $\alpha_{\Xi}$, and
therefore $\mathbf{\alpha_{-}}$ Tanabashi  _et al._ (2018).

Another class of experiments that depends on $\mathbf{\alpha_{-}}$ is the
series of measurements of recoil polarization observables for kaon photo- and
electro-production in the baryon resonance region Carman  _et al._ (2003,
2009); McCracken  _et al._ (2010); Bradford  _et al._ (2007); Paterson  _et
al._ (2016). Fits to such observables by theoretical models are a crucial
element in determining the light baryon resonance spectrum Kamano  _et al._
(2016); Anisovich  _et al._ (2017); Rönchen  _et al._ (2018); Hunt and Manley
(2018), which provides a point of comparison for theoretical approaches such
as quark models, Dyson-Schwinger or Lattice QCD calculations.

Kaon photoproduction data can be also utilized to provide a new and
independent estimate for $\mathbf{\alpha_{-}}$, as will be demonstrated in the
present work. The photoproduction data set contained in the combination of
publications Bradford  _et al._ (2007); McCracken  _et al._ (2010); Paterson
_et al._ (2016) by the CLAS collaboration, is subject to strict constraints
from spin algebra (so-called Fierz identities), which can be exploited to
derive estimators for $\mathbf{\alpha_{-}}$ itself. We note that a similar
strategy has been followed once before, based on data for the reaction
$\pi^{-}p\to K^{0}\Lambda$ Astbury  _et al._ (1975). Anticipating our result,
the value for $\mathbf{\alpha_{-}}$ found in our analysis is $0.721\pm 0.006$,
i.e. close to but noticeably smaller than the number given by the BESIII
collaboration Ablikim  _et al._ (2018).

##  II Determination of $\alpha_{-}$ from Kaon Photoproduction data

Following Ref. Sandorfi  _et al._ (2011), the relative intensity distributions
of events for $\gamma+p\rightarrow K+\Lambda$ reactions in which there is no
polarization of the beam or target, but where the decay products of the
$\Lambda$ are measured, is

$1+\mathbf{\alpha_{-}}\cos\theta_{y}{{P}}.$ (1)

If the photon beam is circularly polarized we have

$1+\mathbf{\alpha_{-}}\cos\theta_{y}{{P}}+\left(\mathbf{\alpha_{-}}\cos\theta_{x}{{C_{x}}}+\mathbf{\alpha_{-}}\cos\theta_{z}{{C_{z}}}\right)P_{C}^{\gamma},$
(2)

and if the photon beam is linearly polarized the distribution is

$\displaystyle
1+\mathbf{\alpha_{-}}\cos\theta_{y}{{P}}-\left\\{{{\Sigma}}+\mathbf{\alpha_{-}}\cos\theta_{y}{{T}}\right\\}P_{L}^{\gamma}\cos
2\phi$ (3)
$\displaystyle\;-\left\\{\mathbf{\alpha_{-}}\cos\theta_{x}{{O_{x}}}+\mathbf{\alpha_{-}}\cos\theta_{z}{{O_{z}}}\right\\}P_{L}^{\gamma}\sin
2\phi.$

The $O_{j}\in\\{O_{x},\,O_{z},\,T,\,C_{x},\,C_{z},\,\Sigma,\,P\\}$ represent
the polarization observables, the direction cosines $\cos\theta_{x,y,z}$
represent the angle of the decay proton in the $\Lambda$ rest frame and $\phi$
is the angle between the reaction plane and the photon polarization axis.
Together with $\mathbf{\alpha_{-}}$, the degrees of circular and linear
polarizations, $P_{L}^{\gamma}$ and $P_{C}^{\gamma}$, enter as “calibration”
parameters. The three expressions (1), (2) and (3) represent the measurements
McCracken  _et al._ (2010), Bradford  _et al._ (2007) and Paterson  _et al._
(2016) respectively.

Assuming that the angles $\theta_{x,y,z},\phi$ are measured accurately, the
extraction of the polarization observables $O_{j}$ is possible only if the
calibration parameters $\\{\alpha_{-},\,P_{C}^{\gamma},\,P_{L}^{\gamma}\\}$
are known. The equations (1), (2) and (3), show that the extraction of
$O_{x}$, $O_{z}$ and $T$ requires the product $\alpha_{-}P_{L}^{\gamma}$,
$C_{x}$ and $C_{z}$ require $\alpha_{-}P_{C}^{\gamma}$, whilst $\Sigma$ and
$P$ require $P_{L}^{\gamma}$ and $\alpha_{-}$, respectively.

The spin algebra of pseudoscalar meson photoproduction results in several
constraints among all 15 polarization observables, known as Fierz identities
after the method used in Chiang and Tabakin (1997) to derive them. Two of
these connect the observables measured by the CLAS collaboration:

$\displaystyle O^{2}_{x}+O^{2}_{z}+C^{2}_{x}+C^{2}_{z}+\Sigma^{2}-T^{2}+P^{2}$
$\displaystyle=1$ (4) $\displaystyle\Sigma P-C_{x}O_{z}+C_{z}O_{x}-T$
$\displaystyle=0\ .$ (5)

If all observables in equations (4) and (5) are measured then these Fierz
identities can be used to estimate the calibration parameters. The published
experiments estimate the uncertainties in $P_{C}^{\gamma}$ and
$P_{L}^{\gamma}$ as systematic uncertainties, so we have some prior knowledge
of their values, giving the opportunity to estimate $\alpha_{-}$.

The CLAS data span a range of energies $W$ and scattering angles $\theta$.
Distributions of observables in $\\{W,\cos\theta\\}$ are then used to study
light baryon resonances. In the present work, we can simply treat the measured
data as an ensemble of observations, each of which are related to
$\alpha_{-}$.

There is a common region in $\\{W,\cos\theta\\}$ space among the three
measurements McCracken  _et al._ (2010), Bradford  _et al._ (2007) and
Paterson  _et al._ (2016), which is spanned by the 314 points reported in
Paterson  _et al._ (2016). Denoting by $O_{i}\equiv O(W_{i},\cos\theta_{i})$
the set of observables at kinematic point $i\equiv\\{W_{i},\cos\theta_{i}\\}$,
we have $\\{O_{x},\,O_{z},\,T,\,\Sigma,\,P\\}_{i};\,i=1,...,314$ from Paterson
_et al._ (2016). We use Gaussian process prior (GP) inference Rasmussen and
Williams (2005) with maximum a posteriori (MAP) optimisation of covariance
function hyperparameters to model the $C_{x},C_{z}$ observation uncertainties.
A second heteroscedastic GP is used, incorporating the mean of the GP
uncertainty model as observation variance, to interpolate with the data
reported in Bradford  _et al._ (2007), using the GPML package Rasmussen and
Nickisch (2010) to obtain the values of $C_{x}$ and $C_{z}$ (and their
variances) at the points $\\{W_{i},\cos\theta_{i}\\}$.

###  II.1 Statistical Analysis

With these data, the following Fierz values can be defined:

$\displaystyle{\cal F}^{(1)}_{i}=$ $\displaystyle a^{2}l^{2}\left({\cal
O}_{x,i}^{2}+{\cal O}_{z,i}^{2}-{\cal T}_{i}^{2}\right)$
$\displaystyle+a^{2}c^{2}\left({\cal C}_{x,i}^{2}+{\cal
C}_{z,i}^{2}\right)+l^{2}\mathit{\Sigma}_{i}^{2}+a^{2}{\cal P}_{i}^{2}\ ,$ (6)
$\displaystyle{\cal F}^{(2)}_{i}=$ $\displaystyle
al\left[\mathit{\Sigma}_{i}{\cal P}_{i}-{\cal T}_{i}-ac({\cal C}_{x,i}{\cal
O}_{z,i}-{\cal C}_{z,i}{\cal O}_{x,i})\right]\ ,$ (7)

where $a(=\alpha_{-}^{\rm PDG}/\alpha_{-})$, $c$ and $l$ represent relative
systematic correction factors in the calibration parameters for circular and
linear photon beam polarization. We use the convention that caligraphic
symbols denote random variables (RVs). At a given kinematic point $i$, the
observables ${\cal O}_{j,i};\,j=1,\dots,7$ are assumed independent, normally
distributed RVs, ${\cal O}_{j}\sim{\cal N}[\mu_{j},\sigma_{j}^{2}]$ that take
on values $O_{j}$. The Fierz RVs ${\cal F}^{(1,2)}$ take on values $f^{(1,2)}$
and $\mu_{j}$, $\sigma_{j}^{2}$ are the reported CLAS measurements. The use of
the Fierz identities to determine $a,\,l,\,c$ poses a series of statistical
challenges that are summarized below; the Supplemental Material Ireland  _et
al._ (2018) provides the in-depth derivation of these results, supported by
numerical tests using synthetic data.

  1. 1.

The parameters $a,l,c$ scale the $\mu_{j}$ but also the uncertainties
$\sigma_{j}$ themselves which potentially leads to a problem related to, but
not identical to, the d’Agostini bias D’Agostini (1994); Ball  _et al._
(2010). Our results are free from bias in this respect.

  2. 2.

For a typical term in the calculation of the Fierz values, the new RV ${\cal
Y}=\beta{\cal O}_{j}{\cal O}_{j^{\prime}}$, where $\beta$ is some combination
of $a,l,c$, the probability distribution function (pdf) of ${\cal Y}$ contains
a $\beta$-dependent normalization constant. This induces a bias for the pdfs
of $a,l,c$ and the normalization constant has to be removed. A similar
situation is discussed in Ref. Roe (2015).

  3. 3.

For the first Fierz identity, the naive guess based on Eq. (4), $E[{\cal
F}^{(1)}]=1$, is only correct in the limit $\sigma_{j}\to 0$. The pdf of each
summand in Eq. (6) follows a scaled, non-central $\chi^{2}$ distribution with
$E[{\cal O}_{j}^{2}]=\mu_{j}^{2}+\sigma_{j}^{2}$ and, accordingly, $E[{\cal
F}^{(1)}]=\Delta f\neq 1$. Although there exists no closed form for the
distribution of ${\cal F}^{(1)}$, namely $p^{(1)}(f^{(1)}|a,l,c)$, the
quantity $\Delta f$ can easily be calculated because expectation values add,
$E[\beta_{1}{\cal O}_{j}^{2}\pm\beta_{2}{\cal
O}_{j^{\prime}}^{2}]=\beta_{1}(\mu_{j}^{2}+\sigma_{j}^{2})\pm\beta_{2}(\mu_{j^{\prime}}^{2}+\sigma_{j^{\prime}}^{2})$
(analogously for variances). For ${\cal F}^{(2)}$, $E[{\cal O}_{j}{\cal
O}_{j^{\prime}}]=\mu_{j}\mu_{j^{\prime}}$ with $j\neq j^{\prime}$ and there is
no such shift so that the Fierz identity reads $E[{\cal F}^{(2)}]=0$.

For each kinematic point $i$, we obtain

$p_{i}({\mathfrak{O}}_{i}|a{,}l{,}c){~{}\propto~{}}p^{(1)}(f^{(1)}_{i}{=}\Delta
f_{i}|a{,}l{,}c)\,p^{(2)}(f^{(2)}_{i}{=}0|a{,}l{,}c),$ (8)

where ${\mathfrak{O}}_{i}=\cup_{j=1}^{7}{\cal O}_{j,i}$ symbolizes the data
set at point $i$. As there is no closed form for these distributions they can
be estimated by sampling: For fixed $a,l,c$, the Fierz values
$f_{i,k}^{(1,2)}$ are calculated from random sample $k$ of the ${\cal
O}_{j,i}$ and then binned around $\Delta f_{i}$ and $0$, respectively.

  4. 4.

We found that the non-linearities of the problem are small for this particular
case as discussed in the Supplemental Material Ireland  _et al._ (2018). One
can, therefore, partially linearize the problem and construct a Gaussian
likelihood for each kinematic point $i$,

$p_{i}({\mathfrak{O}}_{i}|a,l,c)\propto\exp\left[-\left(\dfrac{f^{(1)}_{i}-1}{\sigma_{{\cal
F}_{i}^{(1)}}}\right)^{2}-\left(\dfrac{f^{(2)}_{i}}{\sigma_{{\cal
F}_{i}^{(2)}}}\right)^{2}\right]$ (9)

where the $f^{(1,2)}$ are calculated according to Eqs. (6,7) with the ${\cal
O}_{j}$ replaced by their means $\mu_{j}$ (i.e. the measured values reported
in the literature) and expressions for $\sigma_{{\cal F}^{(1,2)}}$ given in
the Supplemental Material Ireland  _et al._ (2018). This probability is thus
an expression of how far away from the Fierz constraints the combination of
the ${\cal O}_{j}$ is.

As data for different energies and scattering angles are independent, the
combined likelihood can be written as the product

${\cal
P}({\mathfrak{O}}|a,l,c)=\frac{1}{Z}\prod_{i=1}^{n}p_{i}({\mathfrak{O}}_{i}|a,l,c)$
(10)

where ${\mathfrak{O}}=\cup_{i=1}^{n}{\mathfrak{O}}_{i}$ symbolizes the entire
data set and $Z$ is a normalization constant (see item 2.).

Even with the two Fierz identities as constraints, $a,l$, and $c$ are highly
correlated, and priors on $P_{C}^{\gamma}$ and $P_{L}^{\gamma}$ are required.
Systematic uncertainties in the experiments are quoted as numbers, which we
denote as $\delta_{C}$ and $\delta_{L}$, but there is no universal
prescription to code this information as a pdf. To check the robustness of the
method we used four different priors ${\cal P}(l,c)$: 1) Gaussian:
$l,c\sim{\cal N}(1,\delta_{c,l}^{2})$; 2) Uniform: $l,c\sim{\cal
U}(1-\delta_{l,c},1+\delta_{l,c})$; 3) Double Uniform: $l,c\sim{\cal
U}(1-2\delta_{l,c},1+2\delta_{l,c})$; and 4) Fixed: $l=c=1$. We take
$\delta_{l}=0.05$ and $\delta_{c}=0.02$ as representative values, according to
the systematic errors estimated in Refs. Bradford  _et al._ (2007); Paterson
_et al._ (2016). ${\cal U}$ represents a uniform pdf. The posterior density is

${\cal P}(a,l,c|{\mathfrak{O}})\propto{\cal P}({\mathfrak{O}}|a,l,c){\cal
P}(l,c)\ .$ (11)

The posteriors corresponding to the choice of priors were explored using a
Markov Chain Monte Carlo (MCMC) implementation (_emcee_ Foreman-Mackey  _et
al._ (2013)). As there were only three parameters to be determined we were
also able to scan directly across the parameters $a$, $c$ and $l$ to validate
the results of the MCMC. The results for $\alpha_{-}$ were obtained by
marginalizing over $l$ and $c$. Both methods were checked by applying them to
synthetic data that had been scaled appropriately by a “wrong” value of
$\mathbf{\alpha_{-}}$.

##  III Results

The results for the marginalized posteriors for $\alpha_{-}$ with the measured
CLAS data are depicted in Fig. 1, and the mean and standard deviation of the
marginalized pdfs are reported in Tab. 1.

Figure 1: Posterior densities for $\alpha_{-}$, given different priors for the
beam polarization calibration constants $P_{C}^{\gamma}$ and $P_{L}^{\gamma}$.
The histograms show the result of the MCMC sampling of the marginalized
posterior densities while the solid lines represent a direct scan of the
posteriors. For clarity, the results corresponding to the double width uniform
priors for $P_{C}^{\gamma}$ and $P_{L}^{\gamma}$ are omitted. Dark grey
vertical bands represent statistical uncertainty; the additional light grey
bands on the BESIII result represent systematic uncertainty.

The means of the posteriors are all consistent with each other. Whilst this is
not an exhaustive sensitivity check, the range of priors chosen reflects quite
different assumptions. This therefore suggests that the estimated value for
$\mathbf{\alpha_{-}}$ does not dependent sensitively on the choice of prior.

The Gaussian priors for $c$ and $l$ give unrealistic mean values of $c$ and
$l$ in the posterior pdf that are 3-4 standard deviations from 1.0, their
nominal values. This is possible since a normal distribution is technically
non-zero over an infinite domain. Results reported by experiments imply that
the range of values defined by the quoted systematic uncertainties should
contain the possible values of calibration parameters with high probability,
without specifying the form of a pdf. Whilst normal pdfs are often assumed for
systematic uncertainty they are perhaps not appropriate in this case.

The use of uniform pdfs as priors for $P_{C}^{\gamma}$ and $P_{L}^{\gamma}$
represents another extreme, where the implication is that the true values
_must_ lie within a given range. We take two variants: a uniform range defined
by the size of the systematic uncertainties, and a uniform distribution of
double this range. A final extreme assumption is that there is no systematic
error, and that $c=l=1$.

We make the assumption that the uniform prior for $c$ and $l$ between the
quoted systematic uncertainties represents the most realistic assumption, so
we quote the mean value of this variant (0.721) as our result, together with
the standard deviation (0.006) of the pdf of $\mathbf{\alpha_{-}}$ as the
statistical uncertainty, and a systematic uncertainty of $\pm$ half the range
of values $\frac{1}{2}(0.727-0.717)=0.005$. We denote this value by
$\alpha^{\rm CLAS}_{-}$ below.

The Supplemental Material Ireland  _et al._ (2018) provides a more detailed
representation of the results in $a,l,c$ space.

###  III.1 Refits with the Jülich-Bonn model

To cross-check the results obtained in the previous section and to estimate
the impact of a new value of $\alpha_{-}$ in calculations that employ data
such as the ones from Refs. Bradford  _et al._ (2007); McCracken  _et al._
(2010); Paterson  _et al._ (2016) as input, we use the Jülich-Bonn (JüBo)
framework. This is a dynamical coupled-channel approach that aims at the
extraction of the nucleon resonance spectrum in a combined analysis of pion-
and photon-induced hadronic scattering processes. In the JüBo approach, the
Fierz identities are fulfilled by construction. A detailed description of the
model can be found in Refs. Rönchen  _et al._ (2013) and Rönchen  _et al._
(2014); the photoproduction data of the $\eta p$ and $K^{+}\Lambda$ final
states were included recently Rönchen  _et al._ (2015, 2018), among them the
measurements of the differential cross section and several polarization
observables in $K\Lambda$ photoproduction by the CLAS Collaboration Bradford
_et al._ (2007); McCracken  _et al._ (2010); Paterson  _et al._ (2016).

Source | Value | Prior Assumption $P_{\gamma}^{L}$, $P_{\gamma}^{C}$
---|---|---
PDG | 0.642 $\pm$ 0.013 |
BES III | 0.750 $\pm$ 0.009 $\pm$ 0.004 |
CLAS | 0.719 $\pm$ 0.013 |  ${\cal N}(1.0,0.05^{2})$, ${\cal N}(1.0,0.02^{2})$
0.721 $\pm$ 0.006 ($\star$) |  ${\cal U}(0.95,1.05)$, ${\cal U}(0.98,1.02)$
0.727 $\pm$ 0.007 |  ${\cal U}(0.90,1.10)$, ${\cal U}(0.96,1.04)$
0.717 $\pm$ 0.004 | Both fixed at 1.0
| 0.721 $\pm$ 0.006 $\pm$ 0.005 | Summary of our result
Table 1: Summary of results. The result marked ($\star$) represents the most
realistic prior on $P_{C}^{\gamma}$ and $P_{L}^{\gamma}$.

In order to estimate the impact of a different value for $\alpha_{-}$ in the
JüBo model, the polarization observables $T$, $O_{x}$ and $O_{z}$ from Ref.
Paterson  _et al._ (2016), $C_{x}$ and $C_{z}$ from Ref. (Bradford  _et al._ ,
2007) and $P$ from Ref. McCracken  _et al._ (2010) are scaled by this value,
i.e. multiplied by $(\alpha^{\rm PDG}_{-}/\alpha^{\rm BESIII}_{-})$ or by
$(\alpha^{\rm PDG}_{-}/\alpha^{\rm CLAS}_{-})$ and a refit of a subspace of
free parameters of the model is performed. The data included in the refit are
limited to those that are contained in the energy range defined by the
measurement in Paterson  _et al._ (2016). Note that also the statistical data
errors entering the $\chi^{2}$ are scaled.

In addition, we also perform a refit of the unscaled data. This is necessary
because the solution JüBo2017 Rönchen  _et al._ (2018), which is the starting
point for the refits, represents the minimum of the global coupled-channels
fit including all 48,000 data points from different reactions. A refit
considering only the unscaled data listed in Tab. 2 provides a valid point of
comparison for the fit to the scaled data. We vary only parameters of the non-
pole polynomials Rönchen  _et al._ (2014) that couple to the $K\Lambda$ final
state, which amounts to 73 fit parameters. They are adjusted to the data in a
$\chi^{2}$ minimization using MINUIT on the JURECA supercomputer at the Jülich
Supercomputing Centre Jülich Supercomputing Centre (2018). In all three fits
identical fitting strategies are applied.

Observable |  $\chi^{2}/n$ (Refits)
---|---
(# data points) |  $\alpha_{-}=$ 0.642 | 0.75 | 0.721
$d\sigma/d\Omega$ (421) McCracken  _et al._ (2010) | 1.11 | 1.03 | 0.95
$\Sigma$ (314) Paterson  _et al._ (2016) | 2.55 | 2.61 | 2.56
$T$ (314) Paterson  _et al._ (2016) | 1.75 | 1.74 | 1.69
$P$ (410) McCracken  _et al._ (2010) | 1.84 | 1.66 | 1.62
$C_{x}$ (82) (Bradford  _et al._ , 2007) | 2.15 | 1.72 | 1.34
$C_{z}$ (85) (Bradford  _et al._ , 2007) | 1.58 | 1.83 | 1.62
$O_{x}$ (314) Paterson  _et al._ (2016) | 1.44 | 1.53 | 1.51
$O_{z}$ (314) Paterson  _et al._ (2016) | 1.34 | 1.58 | 1.49
all (2254) | 1.67 | 1.66 | 1.59
Table 2: $\chi^{2}/$data point of the Jülich-Bonn refits for different values
of $\alpha_{-}$. The value of $\alpha_{-}=\alpha_{-}^{\text{PDG}}=0.642$
corresponds to the refit to unscaled data, $\alpha_{-}=0.75$ correponds to the
BES-III result Ablikim  _et al._ (2018) and $\alpha_{-}=0.721$ uses the data-
driven result of this study as input for the refit.

The results are shown in Table 2. The best $\chi^{2}$ is obtained for the data
scaled by $\alpha^{\rm CLAS}_{-}$ as determined in this study, while the refit
to the data scaled by $\alpha^{\rm BESIII}_{-}$ returns a similar $\chi^{2}$
to the fit to the unscaled data ($\alpha^{\rm PDG}_{-}=0.642$). Both are
significantly worse than $\alpha^{\rm CLAS}_{-}$ which corroborates our
independent result. As a caveat, the best $\chi^{2}/n$ itself (1.59) is still
too large which suggests that for a more quantitative comparison $l$ and $c$
should also be varied as before to allow for more systematic uncertainties, or
that the model parameterization itself is not flexible enough.

##  IV Conclusions

The decay parameter $\mathbf{\alpha_{-}}$ of the $\Lambda$ is a fundamental
physical constant that is used to obtain polarization information from
reactions in which the parity-violating weak decay $\Lambda\rightarrow
p\pi^{-}$ occurs. Its value has recently been thrown into dispute by a new
measurement, thereby affecting all results that rely on it. We have made an
independent estimate of this quantity by combining an ensemble of observables
from kaon photoproduction measured at CLAS with constraints set by Fierz
identities. Our value of 0.721 $\pm$ 0.006 (statistical) $\pm$ 0.005
(systematic), clearly favours the new BESIII result of 0.750 $\pm$ 0.009 $\pm$
0.004 over the current PDG value of 0.642 $\pm$ 0.013, though it differs
manifestly from the former as well.

In view of that, it is obvious that past results which involve the $\Lambda$
decay parameter should be revisited to ensure that the derived quantities are
in line with the new and larger reference value of $\mathbf{\alpha_{-}}$,
considering, of course, the present uncertainty. This applies to data from all
experiments where the polarization of the $\Lambda$ or $\Xi$ baryon was
measured. As a consequence, phenomenological analyses of those data performed
in searches for (new) excited baryons and their properties should also be
updated.

###### Acknowledgements.

This work was supported by: the United Kingdom’s Science and Technology
Facilities Council (STFC) from grant number ST/P004458/1; M.D. acknowledges
support by the National Science Foundation (grant no. PHY-1452055) and by the
U.S. Department of Energy, Office of Science, Office of Nuclear Physics under
contract no. DE-AC05-06OR23177 and grant no. DE-SC0016582. This work is
supported in part by DFG and NSFC through funds provided to the Sino-German
CRC 110 “Symmetry and the Emergence of Structure in QCD” (NSFC Grant No.
11621131001, DFG Grant No. TRR110) and R.M-S acknowledges EPSRC grants
EP/M01326X/1 & EP/R018634/1. The authors gratefully acknowledge the computing
time granted through JARA-HPC on the supercomputer JURECA at Forschungszentrum
Jülich. We thank Dr H. Nickisch for helpful advice on the implementation of
coupled heteroskedastic GP models in his GPML
toolbox.111https://gitlab.com/hnickisch/gpml-matlab We also thank Ulf-G.
Meißner for his comments on the draft of this article.

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– Supplemental Material –

##  Appendix A Derivation of $\alpha_{-}$ Extraction Method

The determination of $\alpha_{-}$ and the correction factors $l,c$ raises
statistical challenges. Additional details for the statistical analysis in the
main text are provided here. To clarify the notation, random variables (RVs)
are denoted with calligraphic symbols (${\cal F,\,O,\,C}$) while the values
they can take are denoted with ordinary letters; in particular, we use $O$ for
observables with reported $\mu_{i},\,\sigma_{i}$ and $f$ for Fierz values.

Consider the measurement of observables ${\cal C}_{i,j};\,j=1,\dots,7$ for $n$
kinematic points $(W_{i},\theta_{i})$, i.e., $i=1,\dots,n$. We assume that all
measurements of ${\cal C}$ are independent and normally distributed at a given
kinematic point, ${\cal C}_{j}\sim{\cal N}[\mu_{{\cal C}_{j}},\sigma_{{\cal
C}_{j}}^{2}]$. These “raw” data correspond to the actual measurements, for
example the product $\alpha_{-}{{P}}$ in Eq. (1) of the main text. In the
subsequent experimental analysis of the CLAS data, the ${\cal C}_{j}$ are all
scaled by the (potentially wrong) PDG weak decay parameter
$\tilde{\alpha}_{-}\equiv\alpha_{-}^{\rm PDG}$ to obtain ${\cal
O}_{j}\sim{\cal N}[\mu_{j}\equiv\mu_{{\cal
C}_{j}}/\tilde{\alpha}_{-},\sigma_{j}^{2}\equiv\sigma_{{\cal
C}_{j}}^{2}/\tilde{\alpha}_{-}^{2}]$ which correspond to the published data
and which are the quantities entering the Fierz identities.

We determine corrections to $\tilde{\alpha}_{-}$ by introducing a calibration
factor $a=\tilde{\alpha}_{-}/\alpha$ to scale the observables entering the
Fierz identities, ${\cal O}_{j}\to a{\cal O}_{j}$. The parameters $a,l,c$ are
estimators, usually denoted as $\hat{a}$, $\hat{l}$, $\hat{c}$, but we leave
out the hats in favor of a simplified notation. For the formal discussion in
this supplemental material, we set the true, unknown $\alpha_{-}$ to one for
simplicity. Note that, while there are a finite number of $n$ measurements, we
have to consider the ensemble limit $m\to\infty$ to check for biases, i.e.,
the expectation values of the parameter estimators $a,l,c$ for a finite number
of kinematic points. To derive the distributions of the calibration parameters
$a,l,c$ we follow the same numbering as in the main text:

1. and 2. _Bias when data errors are scaled/ $a$-dependence of normalization constant_.

These two issues can best be discussed together. Here, we consider the inverse
of $a$, $b=1/a$ 222Considering $b$ instead of $a$ avoids undefined expressions
like $E[1/\mu_{{\cal C}_{i}}]$ that arise if only one kinematic point is
considered. For the problem with more than one point, one can work with either
$a$ or $b$, as long as the pertinent pdfs are converted correctly at the end.,
meaning that we have to show for the expectation value
$E[b]=\tilde{\alpha}_{-}^{-1}$ for any method to be unbiased so that the true
$\alpha_{-}=\alpha_{-}^{\rm PDG}b$ (all statements in the sense of pdfs, to be
formalized below). To demonstrate the issue we start with the simplest Fierz
identity, $E[{\cal O}_{i}/b]=1$, i.e., $\mu\equiv E[\mu_{{\cal C}_{i}}]=1$.
The Fierz values are given by ${\cal F}_{i}={\cal O}_{i}/b$ and distributed as
($f_{i}=O_{i}/b$, $b>0$),

$\displaystyle p^{(1)}_{{\cal F}_{i}}(f_{i},b)$ $\displaystyle=\int
dO_{i}\,p(O_{i})\delta(O_{i}/b-f_{i})=b\,p(bf_{i})\ .$ (12)

The distribution of $b$ is the conditional probability from the expected Fierz
value at $f_{i}=1$,

$\displaystyle
p^{(1)}_{i}(b){=}\frac{b}{\sqrt{2\pi}\mu_{i}\sigma_{i}}\exp\left[-\frac{\left(1-\mu_{i}/b\right)^{2}}{2\,(\sigma_{i}/b)^{2}}\right]\sim
p^{(1)}_{{\cal F}_{i}}(1,b),$ (13)

with mode at

$\displaystyle b_{i}^{(1)}=\frac{\mu_{{\cal C}_{i}}+\sqrt{\mu_{{\cal
C}_{i}}^{2}+4\sigma_{{\cal C}_{i}}^{2}}}{2\tilde{\alpha}_{-}}\ ,$ (14)

Table 3:  Ordering of observables and pertinent combinations $s_{j}$ of correction factors for the first Fierz identity. ${\cal O}_{j}$ | ${\cal O}_{x}$ | ${\cal O}_{z}$ | ${\cal T}$ | ${\cal C}_{x}$ | ${\cal C}_{z}$ | $\mathit{\Sigma}$ | ${\cal P}$
---|---|---|---|---|---|---|---
$s_{j}$ | $a^{2}l^{2}$ | $a^{2}l^{2}$ | $-a^{2}l^{2}$ | $a^{2}c^{2}$ | $a^{2}c^{2}$ | $l^{2}$ | $a^{2}$

which is equal to $\tilde{\alpha}_{-}^{-1}$ if and only if $\sigma_{{\cal
C}_{i}}\to 0$, i.e., the obtained result is biased (even without considering
the expectation value).

Next, consider the case in which $b$ is removed from the normalization but
still present for the scaling of $\sigma_{i}$ in the exponent,

$\displaystyle
p^{(2)}_{i}(b)=\frac{1}{\sqrt{2\pi}\,\sigma_{i}}\,\exp\left[-\frac{\left(1-\mu_{i}/b\right)^{2}}{2\,(\sigma_{i}/b)^{2}}\right]\
,$ (15)

with mode at $b_{i}^{(2)}=\mu_{{\cal C}_{i}}/\tilde{\alpha}_{-}$ so that
$E[b_{i}^{(2)}]=\tilde{\alpha}_{-}^{-1}$. To exclude bias, one has to consider
the measurement of at least two kinematic points, resulting in $\mu_{1}$,
$\mu_{2}$, $\sigma_{1}$, $\sigma_{2}$. Then, the expectation value of the mode
$b_{12}^{(2)}$ of of $p_{1}^{(2)}p_{2}^{(2)}$ is indeed unbiased,

$\displaystyle E[b_{12}^{(2)}]=E\left[\frac{\mu_{{\cal C}_{2}}\sigma_{{\cal
C}_{1}}^{2}+\mu_{{\cal C}_{1}}\sigma_{{\cal
C}_{2}}^{2}}{\tilde{\alpha}_{-}\left(\sigma_{{\cal C}_{1}}^{2}+\sigma_{{\cal
C}_{2}}^{2}\right)}\right]=\frac{1}{\tilde{\alpha}_{-}}\ .$ (16)

To summarize, while both $\mu_{{\cal C}_{i}}$ and $\sigma_{{\cal C}_{i}}$
scale with $b$, the scaling of the Fierz variable ${\cal F}_{i}={\cal
O}_{i}/b$ in Eq. (12) induces a $b$-dependent normalization factor which has
to be removed. This also holds true for non-Gaussian distributions and,
analogously, for more than one scaling factor. We have checked this result in
numerical simulations for linear Fierz identities of the form $E[{\cal
O}/b=1]$ and $E[{\cal O}_{1}/b_{1}+{\cal O}_{2}/b_{2}]=1$.

3. _Non-linearity of the problem_.

For the first Fierz identity given in Eq. (6) of the main text one has to
consider squares of normal distributions. For a normally-distributed ${\cal
X}\sim{\cal N}[\mu,\sigma^{2}]$, ${\cal Y}={\cal X}^{2}$ is distributed
according to a scaled, non-central $\chi^{2}$ distribution with one degree of
freedom (NC),

$\displaystyle p_{\cal Y}(y)$
$\displaystyle=\int\limits_{-\infty}^{\infty}dx\,\delta(x^{2}-y)\frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right]$
$\displaystyle=\begin{cases}\frac{\exp\left[-\frac{y+\mu^{2}}{2\sigma^{2}}\right]\cosh\left[\frac{\sqrt{y}\mu}{\sigma^{2}}\right]}{\sqrt{2\pi
y}\sigma}&\text{for }y>0\ ,\\\ 0&\text{else}\ ,\end{cases}$ (17)

with mean and variance

$\displaystyle\mu_{\cal Y}=\mu^{2}+\sigma^{2},\quad\sigma^{2}_{\cal
Y}=2\sigma^{2}(2\mu^{2}+\sigma^{2})\ ,$ (18)

Figure 2:  Reconstruction of $\tilde{\alpha}_{-}$ for Fierz identities with
synthetic data. The figures show $p^{(1)}(a)=\prod_{i}p(f_{i}^{(1)}=\Delta
f_{i}|a)$ and $p^{(2)}(a)=\prod_{i}p(f_{i}^{(2)}=0|a)$ in the top and bottom
row, respectively. Two choices of $\tilde{\alpha}_{-}$ are shown (0.55,0.85)
with the thick red lines. Different colors of the histograms represent
repetitions of the procedure to show fluctuations.  Figure 3:  Reconstruction
of $\tilde{\alpha}_{-}$ (vertical bar) from 500 sets of observables generated
from the JüBo model. Shown are pdfs using the first and second Fierz
identities. For both cases, full, nonlinear treatment according to Eqs.
(20,23) and Gaussian approximations according to the two terms in Eq. (9) of
the main text.

demonstrating that, even if $\tilde{\alpha}_{-}$ has the correct value
$\tilde{\alpha}_{-}=1$, the Fierz value ${\cal F}={\cal O}^{2}$ for the simple
identity $E[{\cal O}^{2}]=1$ will no longer have an expectation value of one
but $E[{\cal F}]=\mu^{2}+\sigma^{2}=1+\sigma^{2}$.

For the actual Fierz identity given in Eq. (6) of the main text, there is no
closed-form expression for the difference of NC distributions and thus the
distribution of the Fierz value ${\cal F}^{(1)}$, namely
$p^{(1)}(f^{(1)}|a,l,c)$, at a given point $i$, can only be obtained by
sampling the set of observables ${\cal O}_{j}$, calculating $f^{(1)}$
according to Eq. (6) of the main text for each sample, and then bin in ${\cal
F}^{(1)}$.

On the other hand, we know from Eq. (18) that for the true $\alpha_{-},l,c$
the expectation value has to be

$\displaystyle\Delta f\equiv E[{\cal
F}^{(1)}]=1+\sum_{j=1}^{7}s_{j}\sigma_{j}^{2}$ (19)

to fulfill the Fierz identity, with ${\cal O}_{j}$ and $s_{j}$ specified in
Table 3. The Fierz identity then imposes a condition on the distribution of
Fierz values at each kinematic point $i$:

$\displaystyle p_{i}^{(1)}(a,l,c)=p^{(1)}(f_{i}^{(1)}{=}\Delta
f_{i}|a,l,c)\,,$ (20)

which coincides with the first term in Eq. (8) of the main text.

For the second Fierz identity given in Eq. (7) of the main text, the situation
is similar. The distribution of the product of two independent Gaussian RVs,
${\cal Z}={\cal XY}$, cannot be written in closed form but mean and variance
can be calculated,

$\displaystyle\mu_{{\cal Z}}=\mu_{\cal X}\mu_{\cal Y}\ ,\quad\sigma_{{\cal
Z}}^{2}=\mu_{\cal X}^{2}\sigma_{\cal Y}^{2}+\mu_{\cal Y}^{2}\sigma_{\cal
X}^{2}+\sigma_{\cal X}^{2}\sigma_{\cal Y}^{2}\ .$ (21)

Compared to the squared case discussed before, there is no shift because
$E[{\cal XY}]=\mu_{\cal Z}=\mu_{\cal X}\mu_{\cal Y}$ and we simply have

$\displaystyle E\left[\mathit{\Sigma}{\cal P}+ac\,({\cal C}_{z}{\cal
O}_{x}-{\cal C}_{x}{\cal O}_{z})-{\cal T}\right]=0\ ,$ (22)

because all Gaussian RVs enter either alone (${\cal T}$) or as products, but
not as squares. Following the same arguments as for the first Fierz identity
one obtains

$\displaystyle p_{i}^{(2)}(a,l,c)=p^{(2)}(f_{i}^{(2)}{=}0|a,l,c)\ ,$ (23)

i.e., the second term of Eq. (8) in the main text.

Figure 4:  Posterior distribution of $(\mathbf{\alpha_{-}},l,c)$ as in Eq.
(11) of the main text using a Gaussian prior with $l\sim{\cal
N}(1,0.05^{2}),\,c\sim{\cal N}(1,0.02^{2})$.  Figure 5:  Posterior
distribution of $(\mathbf{\alpha_{-}},l,c)$ as in Eq. (11) of the main text
using a uniform prior with $l\sim{\cal U}(0.95,1.05),\,c\sim{\cal
U}(0.98,1.02)$.

4. _Gaussian approximation_.

For the first Fierz identity, we take account of the non-linearity of the
problem because the r.h.s. of Eq. (20) is the distribution of the Fierz value,
obtained through sampling and evaluated at $f^{(1)}=\Delta f$ (analogously,
for the second Fierz identity).

For the given data, the pdfs of the Fierz values are approximately Gaussian
for both Fierz identities which suggests that a sufficiently accurate
(partial) linearization of the problem might be possible. At a given kinematic
point $i$, the Gaussian approximation of the first Fierz identity,
$p^{(1)}(f^{(1)}|a,l,c)$, with variance $\sigma_{{\cal F}^{(1)}}^{2}$,
evaluated at $f^{(1)}=\Delta f$, is

$\displaystyle p_{G}^{(1)}(a,l,c)\sim$
$\displaystyle\,\exp\left[-\frac{\left(\Delta
f-\sum_{j}s_{j}(\mu_{j}^{2}+\sigma_{j}^{2})\right)^{2}}{2\sigma_{{\cal
F}^{(1)}}^{2}}\right]$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{\left(1-\sum_{j}s_{j}\mu_{j}^{2}\right)^{2}}{2\sigma_{{\cal
F}^{(1)}}^{2}}\right]\ ,\quad\quad$ (24)

with $\sigma_{{\cal F}^{(1)}}$ calculated from Eq. (18). Notably, all
variances in the numerator of the exponent cancel and the resulting expression
is very simple. This result corresponds to the first term in Eq. (9) of the
main text. An analogous approximation leads to the second term with
$\sigma_{{\cal F}^{(2)}}$ calculated from Eq. (21).

Note that the $a,l,c$ parameters still appear squared in the first Fierz
identity after the partial linearization of Eq. (24). In general, such
nonlinear problems are still biased. One can see this straightforwardly by
following the arguments that led to Eq. (16): the proof for $b$ being unbiased
relies on the strict linear relation of parameter and data which is not given
here anymore. Non-linear bias is usually neglected but should not be ignored.
Therefore, we will test not only the full non-linear treatment but also the
Gaussian approximation in the following through numerical simulations. As we
will find, the bias is not large within uncertainties given our data base of
around $n=300$ kinematic points.

###  A.1 Simulations with Synthetic Data

The method to extract $\alpha_{-}$ relies on a numerical sampling procedure to
construct the distributions of the Fierz values ${\cal F}^{(1)}$ and ${\cal
F}^{(2)}$ that are subsequently evaluated at the respective expectation values
$f^{(1)}=\Delta f$ and $f^{(2)}=0$. The remaining $a,l,c$ dependence
determines the distribution of these parameters. Also, for both calculations,
the $a,l,c$ dependence of normalization factors has to be removed as discussed
previously. In the following tests, we fix $l=c=1$ for simplicity but we have
also tested linear Fierz identities for two correction factors (see previous
section).

First, we numerically test the method by generating synthetic data imposing
different values of $\tilde{\alpha}_{-}$ and then checking whether the
distribution of the reconstructed $a$ has its mode at $a=\tilde{\alpha}_{-}$.
In Figs. 2 the result for both Fierz identities are shown. For each identity,
values of seven observables $j$ at 300 kinematic points $i$ were generated
that fulfill the respective Fierz identities. Gaussian noise was added using
typical uncertainties of the observables, i.e., $\\{\sigma_{{\cal
C}_{i}}|i=1,..7\\}=\\{0.2,0.25,0.2,0.15,0.3,0.25,0.2\\}$ and $\\{\sigma_{{\cal
C}_{i}}|i=1,..7\\}=\\{0.1,0.2,0.2,0.15,0.1,0.15,0.2\\}$ to first and second
Fierz identity, respectively. Then, two different values of
$\tilde{\alpha}_{-}\neq 1$ were imposed to scale the respective observables
${\cal C}_{i,j}$ leading to ${\cal O}_{i,j}$ from which $\tilde{\alpha}_{-}$
was reconstructed. For each of the seven observables, 200,000 events were
generated. The reconstruction process was then repeated a few times, as the
experimentally measured values $\mu_{{\cal C}_{i}}$ fluctuate. The results
support the claim that the $a=\tilde{\alpha}_{-}$ are indeed correctly
reconstructed for both Fierz identities.

Second, the method was tested by using 500 sets of observables calculated from
the JüBo model, where by construction the observables must obey the Fierz
identities. These pseudodata were then rescaled by a value of
$\tilde{\alpha}_{-}=1.157$ and Gaussian noise of realistic size was added to
each observable. The test result is represented in Fig. 3. Again,
$\tilde{\alpha}_{-}$ is reconstructed within 1-$\sigma$ uncertainty. Here, we
also test the Gaussian approximations from Eq. (9) of the main text together
with the full, nonlinear treatment. For both identities, the differences are
much smaller than the precision of $a$, based on the synthetic data that have
similar uncertainties as the actual data.

This numerical test also provides the opportunity to assess the size of the
different corrections applied in the extraction method of $\alpha_{-}$: If one
evaluates $f^{(1)}$ of the first Fierz identity at $f^{(1)}=1$ instead at
$f^{(1)}=\Delta f$, the result is $a=1.136\pm 0.007$, i.e., too small by about
3$\sigma$. Including the shift of the noncentral $\chi^{2}$ distribution is
therefore important (see item 3). If, in addition, one does not remove the
$a$-dependence of the normalization constant (see item 2), the result is
$a=1.120\pm 0.006$, i.e., too small by more than five $\sigma$.

As demonstrated, the linearization of the problem in form of a Gaussian
approximation provides very similar results to the full nonlinear treatment,
at least for ideal data with only statistical noise. Another question is
whether both methods are also _robust_. In an additional test, we have left
the mean values $\mu_{i,j}$ of the data unchanged but scaled all error bars
$\sigma_{i,j}$, i.e., they no longer represent the statistical noise. With
these modified data, the Gaussian approximations produce almost the same
distributions of $a$ while the results using the nonlinear method change
noticeably. Therefore, all final results for the actual data shown in the
supplemental material and main article are determined using the Gaussian
approximations.

##  Appendix B Detailed Representation of Results

Using the procedure described above we can estimate the distribution of the
Fierz values with respect to the asymmetry parameter $\alpha_{-}$ and scaling
factors $l,~{}c$ as well as their correlations. The latter two parameters
represent an unknown systematic uncertainty on which priors have to be imposed
as explained in the main text. We consider two extreme cases – Gaussian
($l,c\sim{\cal N}(1,\delta_{c,l}^{2})$ and Uniform priors $l,c\sim{\cal
U}(1-\delta_{l,c},1+\delta_{l,c})$ with $\delta_{l}=0.05$ and
$\delta_{c}=0.02$. The results are shown in Figs. 4 and 5.


# Evaluating Co-Creativity using Total Information Flow

Vignesh Gokul∗ Chris Francis∗&Shlomo Dubnov UC San Diego
∗ denotes equal contribution
{vgokul, cfrancis<EMAIL_ADDRESS>
###### Abstract

Co-creativity in music refers to two or more musicians or musical agents
interacting with one another by composing or improvising music. However, this
is a very subjective process and each musician has their own preference as to
which improvisation is better for some context. In this paper, we aim to
create a measure based on total information flow to quantitatively evaluate
the co-creativity process in music. In other words, our measure is an
indication of how “good” a creative musical process is. Our main hypothesis is
that a good musical creation would maximize information flow between the
participants captured by music voices recorded in separate tracks. We propose
a method to compute the information flow using pre-trained generative models
as entropy estimators. We demonstrate how our method matches with human
perception using a qualitative study.

##  1 Introduction

Human creativity is a complex process that stems from a wide range of
experiences, complex thoughts, and emotions. The goal of objectively
evaluating human creativity has been a long-standing research problem across
the fields of psychology and cognitive science. Most of the existing works
Brunner et al. (2018); Cífka et al. (2020); Dubnov et al. (2022); Dubnov
(2010) attempt to evaluate creativity in specific works of art such as music,
visual art, etc. This is a complex problem as creativity is an abstract
concept that even we humans do not understand completely. Humans are also
known to co-create, i.e. participate in the creation process together by
assisting each other. Some examples of co-creativity spans the fields of music
Dubnov et al. (2022); Dong et al. (2018); Jiang et al. (2020b) (musical
improvisation), arts (paintings) Rombach et al. (2022) etc. With the rise of
artificial generative models, humans have started co-creating along with
agents as well in different art forms. This has led to an interest in
evaluating co-creativity between agents and humans, which is not a simple
task.

Figure 1: Excerpt from an electronic piece “Poeme Electronique” by Edgar
Varese

Generative models have shown remarkable potential in matching humans in
creative arts. This includes tasks like image generation (painting, drawing,
etc) Rombach et al. (2022), video generation (movie making) Brooks et al.
(2022) and audio generation Dong et al. (2023); Thickstun et al. (2023)
(music, singing, and speech). While most of the frameworks focus on
autoregressive sequence generation i.e., continual generation from an input
sequence, recent works Gokul et al. (2019); Dubnov et al. (2022, 2023) have
tackled the problem of interactive creation: an agent responds to the data
rather than extending the data. In this work, we specifically focus on the
problem of musical improvisation: a human and an agent/another human plays
music together, responding to each other.

Although there has been impressive research on building generative models for
audioThickstun et al. (2023); Copet et al. (2023), there is a severe lack of
evaluative measures to understand the creative process behind the generated
data. Understanding how creativity works in humans is essential to mimic the
same behavior in artificial agents. For example, if we can come up with a
score that quantifies this creative process, then attempts can be made to use
the score to guide the model to mimic human creativity. But this is not a
trivial task. The quality of musical improvisation is difficult to assess. It
is a subjective evaluation, as many consider musical improvisation to be an
expression of personal emotions. There is no “right” or “wrong” music. This
leads to an interesting question: can we quantify this creativity preference
using a score? A possible hint towards an answer can be drawn from analyzing
an abstract score as shown in Figure 1 111 Excerpt from an electronic piece
“Poeme Electronique” by Edgar Varese. In this piece, the music comprises
multiple sonic materials notated on different lines, with pitch represented by
horizontal curves, rhythmic elements as vertical stems, and additional angled
lines specifying gradual changes in loudness. According to our approach, the
total music information in this piece is “co-created” via interaction across
multiple voices according to their individual and codependent dynamics.

Most of the existing research attempts to quantify this creative process
involve some form of subjective studiesAgarwala et al. (2017); Hadjeres et al.
(2017); Huang et al. (2019). This involves asking participants to evaluate the
score on different aspects and using statistical measures on the responses to
come up with a final measure. Recent works have attempted to use information
theory measures such as transfer entropy and mutual information to evaluate
musical improvisation/accompaniment. While these measures are a good
approximation, they are extremely hard to calculate on high-dimensional data
such as music and audio.

Figure 2: For a given input musical signal from a human musician, what
represents the best corresponding improvisation? The response must be creative
and semantically make sense.

We aim to create an evaluation measure based on the amount of musical
information passed between the two musical signals. Our hypothesis is that,
for the “most creative” signal, the amount of information flowing from the
human to the agent and vice versa is maximized. This is based on the fact that
musical improvisation is essentially a form of non-verbal communication. Each
musician tries to convey/express some information while playing and similarly
responds to the other agent/human’s music by expressing data. In other words,
we hypothesize that _the amount of information flow between two musicians is
an indication of creativity._ For example, as shown in Figure 2, given an
input musical piece, which of the three possible outputs represents the best
musical improvisation output? Based on our hypothesis, the musical score
maximizing the information flow would be the best choice. In this paper, we
present a method to calculate the amount of information flow between two
musical processes. We leverage pre-trained generative models as entropy
approximators to make computing the score tractable for long musical
sequences. Our analysis and subjective studies show that our score matches
with human perception of creativity.

Our contributions are summarized as follows: 1) We propose an evaluation score
to quantify the co-creativity process using directed information. 2) We design
a framework to compute the score using pre-trained generative models. 3)
Finally, we demonstrate the effectiveness of our score with an extensive
analysis on multiple datasets and subjective studies.

##  2 Related Work

####  2.0.1 Creativity

Research on precisely defining human creativity has spiked interest for a long
period of time. Many have attempted to define what exactly constitutes
creativity in the past Plucker et al. (2004); Simonton (2018); Taylor (1988);
Treffinger (1993). Runco and Jaeger (2012) define creativity as something that
is novel and useful. Some definitions include surprisal Simonton (2018) and
discovery of new possibilities Martin and Wilson (2017), while most of the
works still refer to novelty and usefulness.

####  2.0.2 Co-Creativity in music

Using computer systems in music production is not new. Biles and others
(1994); Marques et al. (2000). With the rapid development of AI models and
tools, the usage of artificial agents/programs in music composition, editing,
and performance has been on the rise. While most of the recent works lean
towards music generation Dong et al. (2023), music inpainting Moliner and
Välimäki (2023), there have been works to enable co-creative generation tasks
similar to music improvisation Jiang et al. (2020b, a); Wang et al. (2016).
Jiang et al. (2020a) propose a framework using Reinforcement Learning (RL) to
enable real-time human-agent interaction. Jiang et al. (2020b) introduce a
mechanism to compute intrinsic rewards to guide counterpoint music generation.
Although research has progressed a lot in building generative models,
evaluating its creativity still remains a core problem in machine learning.

####  2.0.3 Evaluating Creativity in Music

The majority of the evaluation mechanisms in music stem from some form of
subjective assessment Agarwala et al. (2017); Hadjeres et al. (2017); Huang et
al. (2019). Turing tests Turing (2009) used to be a popular form of subjective
assessment. However, it was not clear if the assessment was based on the
quality of the generated music or whether the generated music was human-made.
There has been a lot of research on the exact design of such assessments and
user studies. Dong et al. (2018) analyzed visual aspects of the generated
music, such as piano roll and spectrogram representations, to make conclusions
on the melody flow and rhythm patterns. However, all these methods provide
qualitative results, which are not great for consistent evaluation. Huang et
al. (2019) propose a quantitative measure based on the frame-wise negative
log-likelihood between the generated music and ground truth. There have also
been several music domain metrics proposed, such as pitch and rhythm-related
Yang and Lerch (2020), harmony-related Yeh et al. (2021), and style-related
Brunner et al. (2018); Cífka et al. (2020) metrics. Recently, measures that
include utilizing deep learning models have become popular. Frechet Audio
Distance (FAD) Kilgour et al. (2018) provides a score based on statistical
measures of the computed deep features of both the original and real audio.

However, most of these measures focus on evaluating the quality of the music
and not the creativity involved in the music generation process. For example,
to evaluate a music improvisation system, we can use FAD to evaluate the
quality of the generated music (amount of noise, musical structures, etc), but
it would not be possible to evaluate if the system is performing the “right”
type of improvisation for the input or if it is “creative”.

####  2.0.4 Music Information Dynamics

There have been several works to understand music structure by exploring the
information content. Music Information Dynamics is a technique inspired by
musical anticipation, i.e. quantifying the amount of information flow from
past to future Pearce and Wiggins (2012). Abdallah and Plumbley (2009) use
time-varying measures such as predictive information rate to characterize
human perception of music. Dubnov (2010) calculate information rate at varying
time scales using statistics of sound spectral features. Dubnov et al. propose
Deep Music Information Dynamics. The authors estimated Mutual Information
using Mutual Information Neural Estimators Belghazi et al. (2018) (MINE)
between two MIDI voices. They considered average MI between bars of music
without consideration of temporal or causality relationships. In this work, we
extend and reformulate the problem as information flow computation.

####  2.0.5 SymTE

One of the closest works to our paper is Dubnov et al. (2023). The authors
propose SymTE, a method to estimate information flow between two musical
processes. The authors propose to switch to a new generative model based on
which model outputs maximize the total information flow, which they call
Symmetric Transfer Entropy (SymTE). Given two processes, $X$ and $Y$, the
authors approximate the total information flow as:

$\displaystyle SymTE$ $\displaystyle\approx
I((X,Y);\overline{(X,Y)})-I(X,\bar{X})$ $\displaystyle-I(Y,\bar{Y})+I(X,Y)$

where $I$ is the predictive information. In order to compute the predictive
information, they use the generative models in Wang et al. (2016). While SymTE
shows promising directions in estimating total information flow and the
potential of using such frameworks in generative modeling, it is not tractable
to compute for long sequences like music. The framework we propose can easily
be computed for long musical signals.

##  3 Method

The main objective of our work is to calculate a score to evaluate how
semantically meaningful two musical signals are when played together. We
consider two-track musical signals denoted by $(X,Y)=[x_{i},y_{i}]$. Usually,
in co-creative tasks like music improvisation, one signal (say $X$) comes from
a human musician, while the other comes from an agent/another musician ($Y$).
Our goal is to estimate the amount of information flow between both $X$ and
$Y$. Given signals $(X,Y)$, we have:

$\displaystyle T_{X\to Y}$ $\displaystyle=I(Y;\bar{X}|\bar{Y})$ (1)
$\displaystyle T_{Y\to X}$ $\displaystyle=I(X;\bar{Y}|\bar{X})$ (2)

where $T_{X\to Y}$ represents the transfer entropy from $X\to Y$. Transfer
entropy between two sequences is the amount of information passing from the
past of one sequence to another when the dependencies of the past of the other
sequence (the sequence’s own dynamics) have already been taken into account.
$\bar{X}$ denotes the past of the signal $X$. We define the total information
flow between $X$ and $Y$ as:

Flow $\displaystyle=T_{X\to Y}+T_{Y\to X}$

Expanding on $T_{X\to Y}$, we have:

$\displaystyle T_{X\to Y}$ $\displaystyle=I(Y;\bar{X}|\bar{Y})$
$\displaystyle=H(Y|\bar{Y})-H(Y|\bar{X},\bar{Y})$
$\displaystyle=H(Y|\bar{Y})-H(Y|\bar{X},\bar{Y})-H(Y)+H(Y)$
$\displaystyle=I(Y;\bar{X},\bar{Y})-I(Y;\bar{Y})$

Similarly, we have,

$\displaystyle T_{Y\to X}$ $\displaystyle=I(X;\bar{Y},\bar{X})-I(X;\bar{X})$

From SymTE Dubnov et al. (2023), we know that

$\displaystyle T_{X\to Y}+T_{Y\to X}$ $\displaystyle\approx
I((X,Y);\overline{(X,Y)})-I(X,\bar{X})$ $\displaystyle-I(Y,\bar{Y})+I(X,Y)$

Expanding, we have

$\displaystyle T_{X\to Y}+T_{Y\to X}$
$\displaystyle=H(XY)-H(XY|\overline{XY})+I(XY)$
$\displaystyle-\left(H(X)-H(X|\bar{X})\right)-\left(H(Y)-H(Y|\bar{Y})\right)$

Since,

$\displaystyle I(X,Y)$ $\displaystyle=H(X)+H(Y)-H(XY),$

we have,

$\displaystyle T_{X\to Y}+T_{Y\to X}$
$\displaystyle=H(X|\bar{X})+H(Y|\bar{Y})$ (3)
$\displaystyle-H(XY|\overline{XY})$

In other words, the overall information flow between $X$ and $Y$ can be
approximated by removing the surprisal of the combined signal $XY$ from the
surprisal present in both $X$ and $Y$. For example, if the combined sequence
$XY$ does not make sense musically, then it will have a higher entropy or
suprisal factor, which means $H(XY|\overline{XY})$ will be higher, leading to
lower total information flow. Similarly, if the signal $XY$ is musically
pleasing, then ideally, it should be easier to predict the next note, i.e. low
surprisal or entropy. This will cause the overall information flow in equation
3 to be higher.

Note that the effectiveness of the score depends on the methods used to
estimate each of the entropy terms. If we have poor entropy estimators then
the values of the information flow would not align with our intuitive
reasoning. Another important requirement is that our estimators should work
for long musical sequences, which are high dimensional data. To summarize, we
need strong, robust entropy estimators for high dimensional data to compute
the terms in equation 3.

Algorithm 1 Calculate Information Flow

1:Input: A MIDI file with two tracks (XY)

2:Output: Total Information Flow

3:$X,Y\leftarrow\textsc{SplitTracks}(XY)$ $\triangleright$ Separate two tracks

4:for each $T\in\\{X,Y,XY\\}$ do

5: for $i\leftarrow 0$ to $seq\\_len$ do

6: $start\leftarrow T[i:\text{context}]$

7: $probs\leftarrow\textsc{MTMT.Transformer}(start)$

8: $cond\\_entropy[T][i]\leftarrow-\sum probs\times\log(probs)$

9: end for

10: $cond\\_entropy[T]\leftarrow\textsc{avg}(cond\\_entropy[T])$

11:end for

12:$InfoFlow\leftarrow cond\\_entropy[X]+cond\\_entropy[Y]-cond\\_entropy[XY]$

13:return $InfoFlow$

###  3.1 Estimating Entropy

Computing the entropy terms in equation 3 is a challenging task, especially
when the data is high-dimensional such as music. We can leverage pre-trained
generative models to approximate each entropy term. The conditional entropy
refers to the amount of surprisal in a random variable $Y$, given that another
random variable $X$ is known. We have:

$\displaystyle
H(Y|X)=-\sum_{x\in\mathcal{X},y\in\mathcal{Y}}p(x,y)\log\frac{p(x,y)}{p(x)}$
(4)

where $\mathcal{X}$ and $\mathcal{Y}$ denotes the support sets of $X$ and $Y$.
The conditional entropy can also be rewritten as:

$\displaystyle H(Y|X)=\mathbb{E}\left[-\log\frac{p(x,y)}{p(x)}\right]$
$\displaystyle=\mathbb{E}\left[-\log p(y|x)\right]$ (5)

Intuitively, the conditional entropy measures how unpredictable (amount of
surprisal) $Y$ is on average, given that we have observed $X$. Now the
conditional entropy given the past sequence becomes:

$\displaystyle H(X|\bar{X})=\mathbb{E}\left[-\log p(x|\bar{x})\right]$ (6)

Since we are dealing with temporal sequences (music audio), we take the
expectation over time. To compute $p(x|\bar{x})$, we use pre-trained music
generative models that give us the probability distribution given some context
signal (past). Similarly, we also compute $H(Y|\bar{Y})$ and
$H(XY|\overline{XY})$.

###  3.2 MIDI Transformer

In this work, we use the Multitrack Music Transformer (MTMT) model Dong et al.
(2023) pre-trained on the Symbolic Orchestral Database (SOD) dataset Crestel
et al. (2018) to estimate entropy terms in equation 6. The model is based on a
decoder-only transformer architecture Liu et al. (2018); Brown et al. (2020)
and has multi-dimensional input and output spaces similar to Hsiao et al.
(2021). We use a learnable absolute positional encoding in the model Vaswani
et al. (2017). The loss function used is the sum of the cross-entropy losses
of the six different fields in the data representation used by the model.

(a)

(b)

(c)

(d)

Figure 3: (a,b) Comparison of information flow with respect to duration and
pitch in the music for the Score dataset. (c,d) Comparison of information flow
with respect to duration and pitch in the music for the URMP dataset. We see
that for both the datasets and both pitch and duration, the information flow
in the positive samples is more than negative samples.

####  3.2.1 Data Representation

We use data in the MIDI format, which is converted to the data representation
from Dong et al. (2023) required by the transformer. The representation is
able to represent two to three times longer music samples for the same
sequence length than popular representations like von Rütte et al. (2022); Ens
and Pasquier (2020) used for multitrack music generation. Each music sample is
represented as a sequence of events, and each event $x$ is represented as a
tuple of six variables:

$x=(x^{type},x^{beat},x^{position},x^{pitch},x^{duration},x^{instrument})$

$x^{type}$ indicates the type of the event. Table 1 shows the five values it
can take and their meanings.

$\mathbf{x^{type}}$ |  Event explanation
---|---
0 |  indicates the start of the music sample
1 |  indicates an instrument used in the music sample
2 |  indicates the end of the instrument list and the start of notes
3 |  indicates a note
4 |  indicates the end of the music sample
Table 1: The five different types of events and what they represent

For start-of-music ($x^{type}=0$), start-of-notes ($x^{type}=2$), and end-of-
music ($x^{type}=4$) events, the remaining five variables are set to zero.
Instrument events ($x^{type}=1$) appear after the start-of-music event
($x^{type}=0$) and before the start-of-notes event ($x^{type}=2$) and act as a
list of the instruments used in the musical piece. For an instrument event,
the variable $x^{instrument}$ indicates the MIDI program number of the
instrument, and the remaining four variables are set to zero. For a note event
($x^{type}=3$), the remaining variables define the onset, pitch, duration, and
instrument of the note. $x^{beat}$ denotes the index of the beat that the note
lies in, and $x^{position}$ denotes the position of the note within the beat.
The onset of the note would be $x^{beat}\cdot r+x^{position}$, where $r$ is
the temporal resolution of a beat.

####  3.2.2 Workflow

Algorithm 1 explains the overall workflow of our method. Given a MIDI file
with two tracks, we first split the tracks to obtain the individual tracks $X$
and $Y$. Then for each of the individual tracks $X$, $Y$ and the combined
track $XY$, we compute the conditional entropy given its past, i.e
$H(X|\bar{X})$, $H(Y|\bar{Y})$ and $H(XY|\overline{XY})$ using equation 6. To
compute the expectation over time, we start with a small segment of the music
as our starting context. The $start$ variable in the algorithm is a sliding
window that feeds the MTMT.Transformer (Multitrack Music Transformer) with the
conditional sequence (past of the variable). For example, to calculate
$H(X|\bar{X})$, we slide over the past of sequence $X$ with some context
window. Then we obtain the probability distribution over the next output
token. Using this probability, we compute the entropy at that timestep. We
obtain the conditional entropy $H(X|\bar{X})$ by averaging the computed
entropies over time. Similarly, we compute $H(Y|\bar{Y})$ and
$H(XY|\overline{XY})$ to calculate the total information flow. Intuitively,
the transformer model (MTMT.Transformer) should provide low entropy
distributions if knowing the past sequence makes it easier to predict the next
token.

##  4 Experiments

We conduct in-depth objective and subjective experiments to evaluate the
effectiveness of our score. Below, we first introduce the datasets we use, our
results, and the studies we conducted.

###  4.1 Datasets

We conduct our experiments using two datasets:

  1. 1.

A dataset we curated using music scores from the website MuseScore. We will
refer to this dataset as the Score dataset.

  2. 2.

The University of Rochester Multi-modal Music Performance (URMP) Dataset Li et
al. (2018).

Figure 4: We compare the information flow (for the pitch variable) with the
scores obtained from the qualitative study.

####  4.1.1 Score dataset

The dataset consists of 500 musical pieces from MuseScore. Each piece has two
tracks - a piano melody and its piano accompaniment. The MuseScore files are
converted to the MIDI format using the MusPy Python library Dong et al.
(2020). We pick 200 samples from this dataset as positive (matching) melody-
accompaniment pairs for our experiments. We then replace the accompaniment
tracks in them by picking tracks uniformly at random from the remaining
samples to form corresponding negative melody-accompaniment pairs. The results
shown have less than 200 samples because we removed some samples that had
problems while converting to the data representation format described in
3.2.1.

####  4.1.2 URMP dataset

The URMP dataset Li et al. (2018) comprises 44 simple multi-instrument musical
pieces constructed from musically coordinated and separately recorded
performances of individual tracks. The dataset consists of tracks made using
thirteen different instruments. We use only the musical scores provided in the
MIDI format. The dataset also contains high-quality individual instrument
audio recordings and videos of the pieces. We removed two samples from this
dataset for our experiments due to problems while converting to the data
representation format described in 3.2.1. We created 42 positive melody-
accompaniment pairs and 42 corresponding negative melody-accompaniment pairs
using the same uniform random selection method followed for the Score dataset.

###  4.2 Analysis

We demonstrate the effectiveness of our method in identifying “good” music in
Figure 3. Ideally, musically pleasing pieces should have a higher score, while
non-coherent music (our negative samples) should have a lower score. As shown
in Figure 3, the overall score trend for positive samples is considerably
higher than for negative samples. Since MTMT uses a six-tuple representation,
we can compute our score for each variable to further understand the
creativity process involved. For example, we can analyze how much information
about pitch flows from one track to another by computing the information flow
only for the pitch variable.

####  4.2.1 Qualitative Study

In order to evaluate if our method also aligns with human perception of “good”
and creativity, we conducted a qualitative study. We chose five music samples
that included both positive and negative samples. We asked 5 expert musicians
in our social circle to evaluate each musical sample on the overall quality of
the music, novelty, complexity, and level of interaction in the music. Each
criterion was given a score out of five. We then compare the information flow
with the survey responses. Figure 4 shows the overall trend for our score and
the survey responses for the five samples. Overall, information flow aligns
with human perception of quality. Samples 2 and 3 are positive samples, and we
see a higher quality score and information flow. We also observe a higher
score for the Level of Interaction metric. This denotes how much musical
interaction is present between the two tracks in the music sample. We notice a
high level of interaction score for positive samples, while it is relatively
low for negative samples (1 and 5).

(a)

(b)

Figure 5: Self-Enhancement Bias: (a) Comparison of information flow with
respect to duration on MTMT vs AMT generated sequences, (b) Comparison of
information flow with respect to pitch. We observe that the self-enhancement
bias observed in popular LLMs is also observed when using MTMT as the entropy
estimator.

Interestingly, we found that many participants rated sample 4 (a negative
sample) with high overall quality and level of interaction scores. Although we
created sample 4 by randomly mixing two tracks from separate music pieces, a
lot of participants seemed to have preferred the combined track. This
particular example (example 4) happens to fit musically together since one of
the tracks is repetitive in a high register, with the other playing a repeated
melodic pattern in a lower register (ostinato), thus establishing a nice
separation with minimal clashes in terms of rhythm or underlying harmony. This
happy coincidence results in a mashup that is perceived as musically pleasing
and the voices meaningfully interacting. This also demonstrates the difficulty
in understanding the creative process in humans. Looking at the novelty score,
we can infer that participants found some of these negative samples as novel
and something new, but rated them as low quality (except for sample 4). Note
that none of our samples in the survey were “simple” toy examples. They were
all complex musical pieces, as indicated by the high complexity score across
all the samples.

####  4.2.2 Positional Bias

Several popular transformer-based large language models (LLMs) have been shown
to suffer from positional bias Zheng et al. (2023). Positional bias refers to
the fact that a model comes to a different conclusion based on the position of
the input. For example, Zheng et al. (2023) show that in a task of evaluating
which sentence is well structured, GPT-4 has been shown to favor one sentence
over another depending on which comes first. Since we are using transformer-
based architectures to compute our score, it is essential to evaluate our
method for such biases.

We conduct an experiment to verify if our method has positional bias. Ideally,
the score should remain the same for both orderings of tracks $XY$ and $YX$ as
they are the same music. However, since the transformer can output different
probabilities based on which track is first, there might be positional biases.
To confirm if this is true, we compare our scores with and without reversing
the tracks. We consider 166 samples for this experiment and report the Mean
Squared Error (MSE) between both the orderings of the tracks (XY and YX).
Table 2 shows that with MTMT, the entropy estimator does not suffer from
positional bias as the MSE is close to zero.

| Duration | Pitch | Position
---|---|---|---
MSE | 0.02 | 0.003 | 0.001
Table 2: Evaluating Positional Bias: We find that the position/order of the
tracks does not affect the entropy estimators.

####  4.2.3 Self-Enhancement Bias

Another bias in transformer-based architecture is the self-enhancement bias,
i.e. models tend to prefer their own generations versus other model creations
Zheng et al. (2023). For example, it is known that GPT-4 prefers text
generated by GPT-4 compared to text generated by other LLMs. This is an issue,
as we want our score to reflect the quality of the music irrespective of what
generating process was involved in creating the music. For example, since we
are using MTMT, our method should not give a higher score to musical
compositions from MTMT compared to another musical piece.

To test if self-enhancement bias exists in our method, we compute our score on
similar generated music from both MTMT and the Anticipatory Music Transformer
(AMT) Thickstun et al. (2023). AMT is a symbolic music transformer based on
the proposed method of anticipation. The model can perform infilling and
generation of multitrack symbolic music. For both models, we feed in the same
starting context and ask the model to continue generation for an equal number
of timesteps. Now, we have similar music from two different generative models.
Note that both the models have different training procedures, so the generated
music might differ vastly in spite of having the same starting sequence.
Figure 5 shows the information flow scores for eight samples from our
experiments. Interestingly, we observe that the bias observed in most LLMs is
also observed in MTMT and our framework. The model tends to prefer its own
compositions more. Theoretically, from equation 6, we see that for information
flow to be higher, $H(XY|\overline{XY})$ needs to be low. When computing this
term on its own generations, the model is easily able to predict the next
token and hence outputs a low entropy value, causing an increase in the
overall information flow. While we use data that was not part of MTMT’s
training process in our other experiments, it is important to understand the
effects of such biases in our method.

##  5 Future Work

We present a method to compute information flow and use it as a measure to
evaluate creativity in AI-generated music. In the future, we would like to
come up with optimization objectives based on this score. Ideally, we should
be able to guide a model to output music such that the information flow would
be maximized. Moreover, we would like to apply our framework in evaluating
creative processes in other domains such as text in conversational systems,
video-to-video generation systems for applications such as dance, etc. Our
method is data and application-agnostic and only needs a pre-trained
generative model to estimate the entropy. This method could also have huge
applications in the analysis of financial time series.

## Acknowledgments

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(Grant agreement #883313, project REACH : Raising Cocreativity in CyberHuman
Musicianship)

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