# Galaxy Zoo DESI: large-scale bars as a secular mechanism for triggering AGN

Izzy L. Garland,1 Mike Walmsley,2,3 Maddie S. Silcock,1,4 Leah M. Potts,1 Josh
Smith,1 Brooke D. Simmons,1 Chris J. Lintott,5 Rebecca J. Smethurst,5 James M.
Dawson,6,7 William C. Keel,8Sandor Kruk,9 Kameswara Bharadwaj Mantha,10,11
Karen L. Masters,12 David O’Ryan,1 Jürgen J. Popp,13Matthew R. Thorne1
1Physics Department, Lancaster University, Lancaster, LA1 4YB, UK
2Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50
St. George Street, Toronto, ON M5S 3H4, Canada
3Jodrell Bank Centre for Astrophysics, Department of Physics & Astronomy,
University of Manchester, Oxford Road, Manchester, M13 9PL, UK
4Centre for Astrophysics Research, University of Hertfordshire, College Lane,
Hatfield, AL10 9AB, UK
5Oxford Astrophysics, Department of Physics, University of Oxford, Denys
Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK
6South African Radio Astronomy Observatory (SARAO), Black River Park North, 2
Fir St, Cape Town, South Africa, 7925
7Department of Physics and Electronics, Rhodes University, PO Box 94, Makhanda
6140, South Africa
8Department of Physics and Astronomy, University of Alabama, 206 Gallalee
Hall, 514 University Blvd. Tuscaloosa, AL 35487-0324, USA
9European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino
Bajo del Castillo s/n, 28692, Villaneuva de la Cañada, Madrid, Spain
10School of Physics and Astronomy, University of Minnesota, Minneapolis,
Minnesota, 55455, USA
11Minnesota Institute for Astrophysics, University of Minnesota, Minneapolis,
Minnesota, 55455, USA
12Departments of Physics and Astronomy, Haverford College, Lancaster Avenue,
Ardmore, PA, 19041 USA
13School of Physical Sciences, The Open University, Milton Keynes, MK7 6AA, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Despite the evidence that supermassive black holes (SMBHs) co-evolve with
their host galaxy, and that most of the growth of these SMBHs occurs via
merger-free processes, the underlying mechanisms which drive this secular co-
evolution are poorly understood. We investigate the role that both strong and
weak large-scale galactic bars play in mediating this relationship. Using
72,940 disc galaxies in a volume-limited sample from Galaxy Zoo DESI, we
analyse the active galactic nucleus (AGN) fraction in strongly barred, weakly
barred, and unbarred galaxies up to $z=0.1$ over a range of stellar masses and
colours. After controlling for stellar mass and colour, we find that the
optically selected AGN fraction is $31.6\pm 0.9$ per cent in strongly barred
galaxies, $23.3\pm 0.8$ per cent in weakly barred galaxies, and $14.2\pm 0.6$
per cent in unbarred disc galaxies. These are highly statistically robust
results, strengthening the tantalising results in earlier works. Strongly
barred galaxies have a higher fraction of AGNs than weakly barred galaxies,
which in turn have a higher fraction than unbarred galaxies. Thus, while bars
are not required in order to grow a SMBH in a disc galaxy, large-scale
galactic bars appear to facilitate AGN fuelling, and the presence of a strong
bar makes a disc galaxy more than twice as likely to host an AGN than an
unbarred galaxy at all galaxy stellar masses and colours.

###### keywords:

galaxies: active – galaxies: bar – galaxies: disc – galaxies: Seyfert

††pubyear: 2023††pagerange: Galaxy Zoo DESI: large-scale bars as a secular
mechanism for triggering AGN–B

##  1 Introduction

Supermassive black holes (SMBHs) reside in the centre of the majority of
galaxies, gaining most of their mass during active phases, where the accretion
systems are known as active galactic nuclei (AGNs). Yet what triggers the
“switch on” of an AGN is equivocal. This question is critical to understanding
the interplay between AGNs and their host galaxies, including the
effectiveness of AGN feedback and SMBH-galaxy co-evolution (see e.g., Kormendy
& Ho, 2013; Heckman & Best, 2014, for a review).

Recent simulation studies have shown that the majority of SMBH growth occurs
via secular (merger-free) mechanisms (Martin et al., 2018; McAlpine et al.,
2020; Smethurst et al., 2023), meaning that mergers are not the primary
drivers of the relationships known to exist between SMBHs and their host
galaxies. Disc-dominated, bulgeless galaxies have had a history free from
major mergers (1:10 mass ratio) since at least $z\sim 2$ (Martig et al.,
2012), and so by exclusively looking at a population of disc-dominated,
bulgeless galaxies and the kiloparsec scale structures within them (such as
large-scale galactic bars), we can gain a better understanding of AGN
triggering in the absence of major mergers. The bulge present in some disc-
dominated galaxies could be merger-formed, but it could also be formed through
a number of other mechanisms, including minor mergers, and potentially bars.
By looking at a population of disc-dominated galaxies as a whole, we can
investigate structures such as bars across the entire disc-dominated galaxy
population.

Large-scale strong bars are observed at optical wavelengths in the Sloan
Digital Sky Survey (SDSS; York et al., 2000) in $29.4\pm 0.5$ per cent of disc
galaxies at redshift $0.01<z<0.06$ (Masters et al., 2011), and when using
either a deeper optical survey or one with better seeing, such as DECaLS, this
increases to around 45 per cent when combining galaxies with either weak or
strong bars (Géron et al., 2021). This distinction between strong and weak
bars is important, despite their being on a continuum, since work has shown
that they may have different formation mechanisms (e.g., Géron et al., 2023),
although separating out strong and weak bars consistently poses a challenge.
In general, a bar is classified as strong if it dominates the galaxy flux, and
weak as containing a smaller fraction of the total flux (Nair & Abraham,
2010). These bars can cause transfers of a disc’s angular momentum, leading to
gas being transported down to the central kiloparsec region (Friedli & Benz,
1993; Athanassoula, 2003), where it could be accreted onto a black hole. Thus,
by tracing these kiloparsec-scale structures, we can gain insight into the
dynamics within a galaxy that facilitate the transfer of angular momentum, and
hence the fuelling which gives rise to the AGN characteristics that we
observe.

Simulations have shown that it is physically possible for bars to provide the
necessary inflow of gas to match the accretion rates we see in AGNs (Sakamoto,
1996; Maciejewski et al., 2002; Regan & Teuben, 2004; Lin et al., 2013), and
this is mirrored in observational work by Smethurst et al. (2021). Several
other studies have pointed to either an increase in the bar fraction of AGN
hosts compared to inactive galaxies, or an increase in AGN fraction in barred
galaxies compared to unbarred (Knapen et al., 2000; Laine et al., 2002; Coelho
& Gadotti, 2011; Oh et al., 2012; Galloway et al., 2015; Alonso et al., 2018;
Silva-Lima et al., 2022; Garland et al., 2023). However, many of these
previous studies have suffered from low statistical significance or
sensitivity to methodology and selection effects.

There are also a number of studies finding no correlation (e.g., Cheung et
al., 2013; Goulding et al., 2017). Thus, in this work, we aim to revisit this
correlation between large-scale bars and AGNs, using Galaxy Zoo DESI (Walmsley
et al., 2023a) to obtain robust morphologies from deeper imaging, and observed
emission lines from SDSS MPA-JHU DR7111Available at https://www.mpa-
garching.mpg.de/SDSS/DR7/ to determine the activity category of the systems in
our sample.

Section 2 discusses our sample selection and classification. We present our
results in Section 3, followed by our discussions and conclusions in Sections
4 and 5. Throughout this work, we consider AGNs and LINERs (low-ionisation
nuclear emission line regions) to be two distinct categories, rather than
LINERs being a subset of AGN. We use WMAP9 cosmology (Hinshaw et al., 2013),
incorporated via Astropy, where we assume a flat universe,
$H_{0}=69.3\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ and
$\Omega_{m}=0.287$.

##  2 Data Collation

In the subsections below, we describe the use of multiple surveys to obtain
the data required for this study. We collate a sample of disc-dominated
galaxies (divided into strongly barred, weakly barred and unbarred) which are
either AGN hosts, star-forming, or undetermined.

###  2.1 Sample Selection

Galaxy Zoo DESI (GZD; Walmsley et al., 2023a) uses machine learning to
identify the morphology of 8.7M galaxies in the DESI Legacy Imaging Surveys:
DECaLS, MzLS and BASS, plus DES. Given the improved seeing on DESI compared to
SDSS, we can push reliable morphology classifications to higher redshifts.
Full details of the methodology can be found in the release paper, and we
summarise briefly here.

Given the size of the DESI Legacy Imaging Surveys, it was not feasible to
collect morphological classifications from volunteers alone (such as in Galaxy
Zoo 2), as this would take around 200 years at current classification rates.
Thus more efficient techniques are required. Walmsley et al. (2023a) trained
deep learning models (Walmsley et al., 2023b) on 10M Galaxy Zoo volunteer
votes over 401k galaxies from the DESI Legacy Surveys to classify galaxy
morphology based on this training data. Their models can typically predict
what fraction of volunteers would give a particular answer to each question to
within a mean vote fraction error of 10 per cent.

We match GZD within a 3” radius to the MPA-JHU SDSS DR7 catalogue (to obtain
stellar masses, $M_{\ast}$, colours and emission line fluxes; Kauffmann et
al., 2003; Salim et al., 2007) and the NYU-VAGC catalogue (to obtain
$k$-corrections; Blanton et al., 2005), resulting in 793,824 galaxies. Fig. 1
shows absolute $r$-band magnitude versus redshift for the entire sample, as
well as the volume-limited disc galaxy sample (described below).

Figure 1: Absolute $r$-band magnitude against redshift, showing our volume
limited sample. The grey-scale 2D histogram represents all galaxies in GZ
DESI, and the teal points represent disc-dominated, not edge-on, merger-free
galaxies within our volume limit. These teal points make up our full sample.
The red lines at $M_{r}=-19.2$ and $z=0.10$ delineate our redshift and
$r$-band magnitude limits. Figure 2: Classification of disc-dominated sources
on a trio of emission line ratio diagrams (Baldwin et al., 1981; Veilleux &
Osterbrock, 1987). The grey histogram represents anything classed ‘star-
forming’, ‘LINER’ or ‘composite’, the teal points represent optically
classified AGN and the red triangles represent WISE-classified AGN. From panel
(a), we assume that any source falling below the Ka03 line (Kauffmann et al.,
2003) is purely star-forming. Anything above the Ke01 line (Kewley et al.,
2001) is either an AGN or a LINER, and thus any source lying between those two
lines is classed as composite. To distinguish between AGNs and LINERs, we use
Panels (c), then (b), then (a) in that order. This is because where a source
has S/N > 3 in $\mathrm{\left[O\textsc{i}\right]}$, then Panel (c) is the most
reliable, and we consider any source lying above the Ke06 (Kewley et al.,
2006) line to be an AGN, whereas a source below is a LINER. Where a source has
S/N < 3 in $\mathrm{\left[O\textsc{i}\right]}$, but S/N > 3 in
$\mathrm{\left[S\textsc{ii}\right]}$, we use Panel (b). Again, a source lying
above the Ke06 line is classified as an AGN and below is a LINER. Where both
$\mathrm{\left[S\textsc{ii}\right]}$and $\mathrm{\left[O\textsc{i}\right]}$in
a source have S/N < 3, we use $\mathrm{\left[N\textsc{ii}\right]}$. Any source
lying above the S07 line (Schawinski et al., 2007) is classified an AGN, and
below is a LINER.

###  2.2 Morphology Classification

In order to examine the secular growth, we select galaxies which have a
substantial disc component using GZD. The first classification the model must
perform is to select whether the galaxy is “smooth and featureless”, has
“features or a disc”, or contains (or is) an “artefact”. To select disc
galaxies, we require that the vote fraction for “features or disc” is
$f_{\mathrm{smooth-or-featured\\_featured-or-disk}}\geq 0.27$.

We also require that any discs must not be edge-on so that a bar can be
identified if present, since in an edge-on galaxy, the bar can be obscured.
GZD must categorise each featured galaxy as “edge-on” or “not edge-on”, and
for our purposes, we require $f_{\mathrm{disk-edge-on\\_no}}\geq 0.68$.
Galloway et al. (2015) examine the relationship between inclination angle and
observed bar fraction, and show (their Fig. 2) that the exact threshold used
for “not edge-on” does not have a significant effect. These limits follow
those used in Géron et al. (2021) and Walmsley et al. (2022).

To complete our sample, we require that the galaxy in the image does not
appear to be merging with another galaxy. GZD classifies every image with a
merger class of “merger”, “major disturbance”, “minor disturbance”, or “none”.
We consider galaxies with any significant level of disturbance to be potential
contaminants to a sample of discs undergoing secular evolution. Thus we create
a parameter we refer to as merger prominence, $\zeta_{\mathrm{avg}}$,
analogous to the bulge prominence parameter in Masters et al. (2019). We
define $\zeta_{\mathrm{avg}}$ as:

$\displaystyle\zeta_{\mathrm{avg}}=0.2\times$ $\displaystyle
f_{\mathrm{merging\\_minor-disturbance}}$ $\displaystyle+0.8\times$
$\displaystyle f_{\mathrm{merging\\_major-disturbance}}$ (1) $\displaystyle+$
$\displaystyle f_{\mathrm{merging\\_merger}}$

We require our sample to contain only galaxies which are not merging, which we
identify as $\zeta_{\mathrm{avg}}<0.3$. This value has been visually checked
to be consistent with undisturbed galaxies, via spot checking of $\sim 50$
galaxies around the selected cut-off value.

In order to reduce selection effects, we select a volume-limited sample having
$z\leq 0.10$ and and $M_{r}\leq-19.2$, as shown in Fig. 1. The 48,871 galaxies
that form our final, complete sample (i.e., within the volume limit, disc-
dominated, not edge-on, not merging) are shown in teal.

Within this volume-limited sample, we subsequently identify whether each of
our galaxies has a bar, and the strength of that bar. GZD asks the models to
distinguish between “strongly barred”, “weakly barred”, and “not barred”. We
classify a galaxy as unbarred if $f_{\mathrm{strong-bar+weak-bar}}<0.5$. We
then divide the barred galaxies into strong and weak bars in order to
investigate the effect of bar strength on AGN presence. We define a barred
galaxy as strongly barred if $f_{\mathrm{strong-bar}}\geq f_{\mathrm{weak-
bar}}$, and weakly barred if $f_{\mathrm{strong-bar}}<f_{\mathrm{weak-bar}}$.
These limits follow the criteria used successfully in Géron et al. (2021,
2023). This means that every galaxy in our volume-limited disc sample is
categorised as unbarred (Ubar, 27,391 galaxies), strongly barred (Sbar, 7,069
galaxies) or weakly barred (Wbar, 14,411 galaxies).

As with any measurement, the GZD vote fractions do have errors associated with
them. When the vote fractions are varied within their errors (assumed to be
Gaussian) using a bootstrapping method iterated 1000 times with replacements,
our results do not change.

###  2.3 Activity Classification

We use emission line ratio diagrams (Baldwin, Phillips & Terlevich, 1981;
Veilleux & Osterbrock, 1987) to classify the galaxies in our sample as either:
undetermined, star-forming, composite, AGN, or LINER. We use the emission
lines from MPA-JHU DR7 to place galaxies on the diagram, and we show the
distribution in Fig. 2.

In order for a source to be classifiable according to this method, we require
that the signal-to-noise ratio (S/N) in $\mathrm{\left[O\textsc{iii}\right]}$,
H$\upbeta$, $\mathrm{\left[N\textsc{ii}\right]}$ and H$\upalpha$ be
$\mathrm{S/N}\geq 3$, in order to ensure good quality emission lines. If a
galaxy does not fulfil this first requirement, it may still be classifiable
depending on where the limits lie (e.g., Brinchmann et al., 2004; Salim et
al., 2007; Rosario et al., 2016), which we discuss below. If a source fulfils
this requirement only in H$\upalpha$ it is classified as undetermined.

If a galaxy does fulfil all the S/N requirements, we use Panel (a) in Fig. 2,
to classify a galaxy as star-forming if it falls below the Ka03 line
(Kauffmann et al., 2003). If a galaxy falls between the Ka03 line and the Ke01
line (Kewley et al., 2001), then it is classified as composite, and the
emission is likely due to a combination of star formation and AGN/LINER
activity. If a galaxy falls above the Ke01 line, we classify it as either an
AGN or a LINER (Low-Ionisation Nuclear Emission-line Region). We explain how
these two objects are differentiated below.

If a source only fulfils the S/N criteria in H$\upalpha$,
$\mathrm{\left[O\textsc{iii}\right]}$ and H$\upbeta$, this provides an upper
limit on the H$\upalpha$/$\mathrm{\left[N\textsc{ii}\right]}$ ratio. Thus if
this source falls below the Ka03 line, it is still classifiable as star-
forming. Else, it is classified as uncertain. If a source only fulfils the S/N
criteria in H$\upalpha$, $\mathrm{\left[N\textsc{ii}\right]}$ and H$\upbeta$,
this provides an upper limit on the
$\mathrm{\left[O\textsc{iii}\right]}$/H$\upbeta$ ratio. Thus if this source
falls below the Ka03 line, it is still classifiable as star-forming. Else, it
is classified as uncertain. If a source only fulfils the S/N criteria in
H$\upalpha$, $\mathrm{\left[N\textsc{ii}\right]}$ and
$\mathrm{\left[O\textsc{iii}\right]}$, this provides a lower limit on the
$\mathrm{\left[O\textsc{iii}\right]}$/H$\upbeta$ ratio. Thus if this source
falls above the Ke01 line, it is still classifiable as either an AGN or a
LINER. Else, it is classified as uncertain. If a source only fulfils the S/N
criteria in H$\upalpha$ and H$\upbeta$, this provides a lower limit on the
$\mathrm{\left[N\textsc{ii}\right]}$/H$\upalpha$ ratio, and an upper limit on
the $\mathrm{\left[O\textsc{iii}\right]}$/H$\upbeta$ ratio. Thus if this
source below above the Ke01 line, it is still classifiable as starforming.
Else, it is classified as uncertain.

There are three different emission lines we can use to distinguish AGNs from
LINERs – $\mathrm{\left[S\textsc{ii}\right]}$,
$\mathrm{\left[O\textsc{i}\right]}$ and $\mathrm{\left[N\textsc{ii}\right]}$.
The most reliable line is $\mathrm{\left[O\textsc{i}\right]}$ (Kewley et al.,
2006) and this should be used where possible, so if
$\mathrm{S/N}_{\mathrm{\left[O\textsc{i}\right]}}\geq 3$, we can use Panel
(c), and classify any source falling below the Ke06 line (Kewley et al., 2006)
as a LINER. This results in the hard cut-off line we see in Panel (c) that is
not present in (a) or (b) for distinguishing between AGNs and LINERs. Where
$\mathrm{S/N}_{\mathrm{\left[O\textsc{i}\right]}}$ is too low,
$\mathrm{\left[S\textsc{ii}\right]}$ is the next best emission line, and so if
$\mathrm{S/N}_{\mathrm{\left[S\textsc{ii}\right]}}\geq 3$ we can use Panel
(b), and classify any source falling below the Ke06 line as a LINER. Where
both $\mathrm{S/N}_{\mathrm{\left[S\textsc{ii}\right]}}$ and
$\mathrm{S/N}_{\mathrm{\left[O\textsc{i}\right]}}$ are too low, we can resort
to Panel (a), and use the S07 line (Schawinski et al., 2007), since a source
must have $\mathrm{S/N}_{\mathrm{\left[N\textsc{ii}\right]}}\geq 3$ in order
to be classified as either an AGN or a LINER at all. Anything both below this
line and above the Ke01 line can be classified as an LINER.

This leaves our volume-limited disc sample with: 712 undetermined galaxies,
2,518 uncertain galaxies, 28,807 starforming galaxies, 8,669 composite
galaxies, 4,843 LINERs and 3,160 optically classified AGNs. When the fluxes
are varied within their errors (assumed to be Gaussian) using a bootstrapping
method iterated 1000 times with replacements, our results do not change.

Some AGNs are not optically classifiable, and are instead observable primarily
in the infrared regime. We match our catalogue to the Wide-Field Infrared
Survey Explorer (WISE) AGN catalogue (Assef et al., 2018), and any AGNs which
are present in this catalogue, but not classified as AGNs according to the
method described above, we add to our sample. There are 5 WISE AGNs which
appear in our volume-limited galaxy sample, 1 of which is classified as an AGN
using emission line ratio diagrams, so we can reclassify an additional 4
galaxies as AGN, to give us a total of 3,164 AGNs.

Examples of different bar strengths in star-forming, AGN-host, and
undetermined galaxies are shown in Fig. 3. For a complete breakdown of how
many galaxies are in each morphology category, and in each activity category,
see Table 2 in Appendix A. Note that whilst the classification of LINERs,
uncertain sources and composite galaxies is important, it is simply so they
can be confidently removed from our sample for analysis. We do not explicitly
make use of these galaxies, as they are possible contaminants in our otherwise
pure sample of Seyferts.

Figure 3: Examples of each morphology and activity classification. The left-
hand column shows unbarred galaxies, the middle shows weakly barred, and the
right-hand shows strongly barred galaxies.The top row shows AGN-host galaxies,
the middle row shows star-forming galaxies, and the bottom row shows
undetermined galaxies according to classification using emission line ratio
diagrams. The undetermined galaxies are predominantly red spirals.

##  3 Results

Figure 4: The distributions in $M_{\ast}$ and $(g-r)_{0}$ colour for a variety
of subsamples, with strongly barred galaxies in solid lines, unbarred in
dashed lines, AGNs in red, star-forming in teal, composite in orange and
undetermined in navy blue. Weakly barred galaxies are not shown for
simplification, but lie between the strongly barred and unbarred samples.

We look at the variation in stellar mass ($M_{\ast}$) and $(g-r)_{0}$ colour,
where the 0 indicates we have corrected the $(g-r)$ colour for galactic
absorption, between strongly barred (Sbar), weakly barred (Wbar) and unbarred
galaxies (Ubar), and the results are shown in Fig. 4. For visualisation
purposes, we omit the results for Wbar since they lie between the two other
samples (however this inclusive plot is shown in Fig. 8 in Appendix B). As
expected, the star-forming galaxies are less massive and slightly bluer than
the AGN hosts. The composite galaxies have overlap with both star-forming and
AGN hosts, which confirms that their activity is due to a mixture of star
formation and AGN. This is very similar to the undetermined galaxies, whose
signal to noise is too low to classify their activity. The undetermined
galaxies are predominantly a mix of quenching and fully quenched disc
galaxies. There are also some small differences between the barred and
unbarred samples, with bars tending to reside in more massive, redder discs,
particularly in both the star-forming samples and the undetermined samples, in
agreement with previous studies (e.g., Masters et al., 2011).

For further analysis, we limit our sample to only star-forming, undetermined
and AGN host galaxies to avoid any ambiguity from the LINER and composite
samples.

We divide our sample of star-forming, undetermined and AGN host galaxies into
our Sbar, Wbar and Ubar samples. Within these three samples, we divide the
$M_{\ast}$ and colour each into 15 bins of equal width (over
$7.0\leq\log(M_{\ast}/\rm{M}_{\odot})\leq 12.0$ and $0.4\leq(g-r)_{0}\leq
2.0$), and assign weights to each galaxy such that the weighted distributions
of $M_{\ast}$ and colour are matched between the Sbar, Wbar and Ubar
subsamples. This is because AGN presence is known to correlate with $M_{\ast}$
and colour, and we want to reduce selection effects, and ensuring that the
distributions are the same will aid this. We can then determine the fraction
in each bar category of AGN, star-forming and undetermined galaxies. However,
when determining the activity fractions, we introduced a more conservative
mass cut of $10.0\leq\log(M_{\ast}/\rm{M}_{\odot})\leq 12.0$ before weighting
the distributions. This is because AGN are more easily observable in higher
mass galaxies (Aird et al., 2012), and this conservative mass cut reduces this
selection bias. Changing this lower mass limit between
$10^{9.0}\rm{M}_{\odot}$ and $10^{10.0}\rm{M}_{\odot}$ does not change our
results. The weighted results are shown in Fig. 5, and Table 1. Errors arise
from the binomial distribution (Cameron, 2011).

Figure 5: The distribution of activity classification within each bar category, as shown in Tab. 1 AGN fraction is shown as positive diagonal in red, star-forming (SFing) is shown as teal square hatching, and Undetermined is shown as navy blue negative diagonal. Whilst in all three bar categories, the AGN fraction is smaller than the inactive fraction, the strongly barred galaxies have a noticeably greater fraction of AGN than the weakly barred galaxies, which in turn have a greater AGN fraction than the unbarred galaxies. Table 1: The percentage of each activity category within each bar classification, as shown in Fig. 5. AGN presence in strongly barred galaxies is around twice as prolific as in weakly barred or unbarred galaxies. | Strongly Barred | Weakly Barred | Unbarred
---|---|---|---
AGN | $31.6\pm 0.9$ | $23.3\pm 0.8$ | $14.2\pm 0.6$
Star-forming | $63.6\pm 0.9$ | $73.6\pm 0.8$ | $83.9\pm 0.6$
Undetermined | $4.7\pm 0.4$ | $3.1\pm 0.3$ | $1.9\pm 0.2$

Given the small errors on each of these fractions within each bar category, it
is highly unlikely that any of these subsamples are drawn from the same parent
distribution. These initial results indicate that AGNs are more likely to
reside in strongly barred galaxies than in weakly barred, and even less likely
to reside in unbarred. However, given the ranges of $M_{\ast}$ and colour, we
endeavour to examine these fractions as a function of both, whilst
simultaneously examining how the AGN fractions may vary across
$M_{\ast}$-colour space. Given that we cannot classify the star-formation rate
of the AGN hosts, we do not consider SFR as a parameter, and thus we combine
star-forming galaxies and undetermined galaxies into one category, which we
refer to as ‘inactive’ galaxies.

###  3.1 AGN-bar correlation with stellar mass and colour

We divide our sample of AGN hosts and inactive galaxies into nine bins in
$M_{\ast}$ and nine bins in $(g-r)_{0}$ colour. Within each bin, we calculate
the AGN fraction in strongly barred galaxies,
$f_{\mathrm{AGN,\textsc{Sbar}}}$, and the AGN fraction in unbarred galaxies,
$f_{\mathrm{AGN,\textsc{Ubar}}}$. We then find the difference in these two
fractions, and this is shown in Fig. 6a. It can be assumed that there is one,
real, intrinsic value for this difference, but when we sample it, we get some
scatter. Thus, by sampling it multiple times by varying the binning, we should
get an approximately Normal distribution that centres around the true value.

Figure 6: The difference between the AGN fraction in two bar categories for
every combination of Sbar, Wbar and Ubar with $M_{\ast}$ on the x-axis and
$(g-r)_{0}$ colour on the y-axis. In each case, the label along the top is
written as ‘Category 1 v. Category 2’. The black contours indicate the
population of disc galaxies (AGN-host and inactive) within the volume limit.
The 2D histogram indicates the distribution of AGN-host disc galaxies, where
there are a minimum of 17 AGN in a bin. Where the bin is more green, this
indicates that the fraction of Category 1 galaxies hosting AGNs is greater
than the fraction of Category 2 galaxies hosting AGNs. Where the bin is more
purple, the reverse is true.

The difference between the two fractions is shown as a colour bar, where green
indicates that the fraction of strongly barred galaxies which host AGNs is
greater than the fraction of unbarred galaxies which host AGNs. In order to
reduce noise, we only show bins where there are at least 17 AGNs in a bin.
Varying the minimum AGN count per bin within reasonable values does not change
our qualitative result. Every bin is green, with approximately
$f_{\mathrm{AGN,\textsc{Sbar}}}-f_{\mathrm{AGN,\textsc{Ubar}}}\approx 0.18$.
This is a small but significant increase in the number of AGNs in strongly
barred galaxies.

When we repeat this analysis for Wbar and Ubar our result is qualitatively
similar, in that weakly barred galaxies appear more likely to host an AGN. But
the signal is much less strong, indicating that any effect that weak bars have
on AGN presence is less pronounced than for strong bars, which is also
reflected in the lower fractions over the full $M_{\ast}$-colour-matched
sample. This is plotted in Fig. 6b. Overall the AGN fraction is still higher
in weakly barred galaxies than in disc galaxies without bars.

We can directly compare Sbar and Wbar across the $M_{\ast}$-colour diagram as
well (Fig. 6c), and we find that the fraction of strongly barred galaxies
hosting AGNs is significantly greater than the fraction of weakly barred
galaxies hosting AGNs although again this is less pronounced than in Fig. 6a.

Figure 7: The distributions of the median difference between AGN fractions in
Panel (a), and the distribution of the fraction of bins where Category 1 bars
are greater than Category 2, in a sample consisting of AGN-host galaxies and
inactive galaxies. In each case, the legend is written as ‘Category 1 v.
Category 2’. The black dashed-dotted lines indicate the expected mean of the
distributions if bar presence did not affect AGN presence. The navy blue,
solid lines represent Sbar v. Ubar. The teal, dashed lines represent Wbar v.
Ubar. The red, dotted lines represent Sbar v. Wbar. The further to the left of
the expected null result the histograms lie, the greater the tendency for AGN
to lie in bar Category 1 galaxies.

Given that any increase in AGN fraction is small, we check that this value is
not overly dependent on binning, and we repeat these calculations for every
$M_{\ast}$ and colour bin combination from 5 bins to 17 bins, for a total of
169 bin combinations. For each binning combination, we calculate the median
difference in AGN fraction (e.g.,
$f_{\mathrm{AGN,\textsc{Sbar}}}-f_{\mathrm{AGN,\textsc{Ubar}}}$), and we plot
these medians in Fig. 7a. We assume that the different binning choices each
sample the true value of the difference in AGN fraction between subsamples,
such that the distribution of values recovered from all binning choices
represents the measured value and its uncertainty. This means that the
histogram for each comparison does not consist of independent measurements,
but rather is expected to peak around the true value of the difference in the
AGN fraction.

If there was no difference in the likelihood of hosting an AGN between these
three subsamples, we would expect the histograms to centre around 0 (e.g.,
$f_{\mathrm{AGN,\textsc{Sbar}}}-f_{\mathrm{AGN,\textsc{Ubar}}}=0$). We always
take the weaker bar category from the stronger bar category, so if the centre
of the histograms is greater than 0, the stronger bar category is more likely
to host an AGN than the weaker, and vice versa if the centre of the histogram
is less than 0.

Fig. 7a shows that the stronger bar category is more likely to host an AGN
than the weaker bar category in every case: strongly barred galaxies are more
likely to host an AGN than weakly barred galaxies, which are in turn more
likely to host an AGN than unbarred galaxies. Yet this excess of AGNs we see
is very small. The difference in AGN fraction between strongly barred and
unbarred galaxies is $0.17\pm 0.01$, between strongly barred and weakly barred
is $0.07\pm 0.01$, and between weakly barred and unbarred is $0.09\pm 0.01$.

A Shapiro-Wilk test (Shapiro & Wilk, 1965) shows that we cannot reject
Normality for any of the distribution of the medians, with p-values in each
case greater than $p_{\mathrm{SW}}>0.21$ ($<1.3\upsigma$). Given these
distributions are consistent with the Normal distribution, we can perform a
simple T-test (Student, 1908) to quantify the significance of this excess of
AGN. In each of these cases, the p-value resulting from a T-test is
$p_{\mathrm{T}}\ll 1\times 10^{-6}$ ($\gg 5\upsigma$), and thus we reject the
hypothesis that the likelihood of each of these bar categories hosting an AGN
are identical to each other. Furthermore, we can say that the galaxies in Ubar
are less likely to host an AGN than the galaxies in Sbar or Wbar category to a
$5\upsigma$ confidence.

For each binning combination, we also calculate the fraction of bins where the
stronger bar category hosts a greater AGN fraction than the weaker bar
category
(e.g.$f_{\mathrm{AGN,\textsc{Sbar}}}>f_{\mathrm{AGN,\textsc{Ubar}}}$), and we
plot these values in Fig. 7b.

For example, if we have 5x5 bins, and 20 bins have
$f_{\mathrm{AGN,\textsc{Sbar}}}>f_{\mathrm{AGN,\textsc{Ubar}}}$, we would
report a value of 0.8 for this bin combination. If there was no difference in
the likelihood of hosting an AGN between our three subsamples, we would expect
the distributions to centre around 0.5 – half of the bins would show a greater
fraction of AGNs in one bar category than the other. This point is signified
by a dash-dotted line.

Comparing Sbar and Ubar, the fraction of bins where Sbar has a greater AGN
fraction is $0.95\pm 0.03$. For Sbar and Wbar this is $0.78\pm 0.06$, and for
Wbar versus Ubar, the fraction of bins where Wbar has a greater AGN fraction
is $0.90\pm 0.04$, where errors arise from the standard deviation.

Again, we perform a Shapiro-Wilk test for Normality. For the combination Wbar
vs. Ubar, we obtain a p-value of $p_{\mathrm{SW}}=0.10$ ($1.6\upsigma$), and
thus for this comparison, we can use a T-test to quantify the significance of
the excess of bins containing a higher AGN fraction in the weakly barred
category. The p-value resulting from this T-test is $p_{\mathrm{T}}\ll 1\times
10^{-6}$ ($\gg 5\upsigma$).

Since we can reject Normality for Sbar vs. Ubar, and Sbar vs. Wbar,
($p_{\mathrm{SW}}\leq 0.03$), we must use the more conservative method of
calculating the number of standard deviations between the mean and the null
result of 0.5. For Sbar v. Ubar, with a mean value of 0.95, and a standard
deviation, $\sigma_{\mathrm{SD}}$, of 0.03, we can say that the mean is
$15\sigma_{\mathrm{SD}}$ away from 0.5, and therefore is not in agreement. For
Sbar v. Wbar, with a mean value of 0.78, and a standard deviation,
$\sigma_{\mathrm{SD}}$, of 0.06, we can say that the mean is
$13\sigma_{\mathrm{SD}}$ away from 0.5, and therefore is not in agreement.
Thus, in each case, the stronger bar category has an AGN fraction that is
greater than the weaker bar category. This occurs across the $M_{\ast}$-colour
regime, meaning that there is not one specific combination of $M_{\ast}$ and
colour driving this relationship, further justifying that our results are not
sensitive to the choice of binning.

The trends between bar strength and AGN activity are likely a mix of
relatively straightforward and more complex results. We discuss these further
below.

##  4 Discussion

Our overall result, with AGN activity in both unbarred and barred disc
galaxies, confirms that a large-scale bar is not _required_ to feed an AGN in
the secular-evolution regime. There are multiple secular channels by which
matter from the kiloparsec-scale disc can flow into the SMBH sphere of
influence, according to both simulations (e.g., Ciotti et al., 1991; Friedli &
Benz, 1993; Sakamoto, 1996; Maciejewski et al., 2002; Regan & Teuben, 2004;
Hopkins & Hernquist, 2006; Ciotti & Ostriker, 2012; Lin et al., 2013; Slater
et al., 2019) and observations (e.g., Davies et al., 2007; Smethurst et al.,
2021). However, our primary result also shows clear evidence for an increase
in AGN activity in both strongly and weakly barred systems, and we focus on
discussing this result below.

As described in Section 3, we find that strong bars are clearly linked to a
higher incidence of AGN activity, and that weak bars show a more subtle, but
still positive, correlation. These results clarify the debate over the last
few years regarding whether (and how much) bars are associated with AGN
activity. They also highlight an emerging consensus regarding the link between
bars and AGN. For example, our results agree with Silva-Lima et al. (2022),
who counter for selection effects and find that barred galaxies have a higher
accretion parameter than unbarred, and that AGNs are found more commonly in
galaxies with a bar. Since we are looking at incidence rather than luminosity
or accretion, this study is particularly complementary. Given that recent
studies have shown that strong and weak bars must be considered separately
(Géron et al., 2023), this could also be responsible for some of the
discrepancies seen in contradicting previous studies (e.g., Cheung et al.,
2013; Goulding et al., 2017; Zee et al., 2023), who find no correlation
between bars and AGNs.

It is natural at this point to consider inverting the question we have been
considering throughout this work, and instead examine the fraction of AGN that
have bars. Within our sample and the defined mass limits, $40.0\pm 1.0$ per
cent of AGN have strong bars, $34.7\pm 1.0$ per cent have weak bars, and the
remaining $25.3\pm 1.0$ per cent are unbarred. $24.5\pm 0.5$ per cent of
inactive galaxies have strong bars, $43.2\pm 0.6$ per cent have weak bars, and
the remaining $32.3\pm 0.5$ per cent are unbarred. This agrees qualitatively
with multiple previous studies, such as Oh et al. (2012) and Garland et al.
(2023), although the exact numbers can vary with the study because of the
sample selection and bar sensitivities of each individual study. This is the
largest study so far to examine the disc galaxy population for evidence of
correlation between bars and AGN and there is clearly a positive correlation
in both directions. AGN are most likely to host strong bars, and strongly
barred galaxies are most likely to host AGN, within the disc galaxy
population.

Our findings are consistent with recent evidence that strong and weak bars
have different formation mechanisms (e.g., Géron et al., 2023): strong bars
are triggered by global disc instabilities, whereas weak bars are formed
through tidal interactions. These formation mechanisms could both be
responsible for triggering an AGN. Thus, the AGN’s presence may not be
directly due to the bar, but rather to the same mechanisms that caused the bar
to form. If the physical mechanisms are different for strong and weak bars,
this leads to a different co-incidence between AGN and the two different bar
strengths. Tidal interactions may be more efficient at depleting gas from the
centre of the galaxy than secular processes that do not lead to the
development of a bar, such that by the time the disc galaxy has evolved to a
higher stellar mass, the AGN has been deprived of fuel and shut down, although
the weak bar still remains in place.

The gas content is significantly higher in starforming discs than red spirals
(Masters et al., 2012). We postulate that were a strong bar present in a
galaxy, this bar is efficient at fuelling this additional gas down to the
central kilo-parsec, where it can be accreted onto an AGN. If weak bars are
less efficient at driving gas to the centre of the galaxy, this would explain
why we see a much weaker correlation between weakly barred galaxies and AGNs
than strongly barred galaxies and AGN.

There is recent evidence from IllustrisTNG that an AGN could drive changes in
the bars (Łokas, 2022). This is due to the AGN switching to kinetic-mode
feedback, causing depletion of gas in the inner regions, leading to quenching
and bar-formation. However given the physical scales that we are looking at,
this seems unlikely to be occurring on a large-scale in our sample, due to the
differences in AGN and bar lifetimes.

It is important to assess the potential contribution to this result of any
selection biases. As described in Section 2, we take various steps to minimise
these biases, such as controlling for $M_{\ast}$ and $g-r$ colour, and using a
volume limit to ensure completeness. We also consider the effects of the
changing physical resolution across the sample. At higher redshifts, the
minimum size of bar we can detect increases: we lose smaller bars at higher
redshift. Weak bars tend to be shorter than strong bars proportional to the
size of the galaxy. This means we preferentially lose weaker bars as we
increase in redshift, especially in lower mass galaxies. In a low mass galaxy,
a weak bar may be missed, and that galaxy classified as unbarred.

We do not have individual bar lengths and widths for all the galaxies in this
sample, and thus we cannot fully compensate for this potential source of
selection bias at an individual galaxy level. However, one way to examine how
strong this effect is likely to be in our sample is to remove the high
redshift sources, because this will significantly reduce the overall
difference between minimum resolved bar size between the lowest and highest
redshifts of the subsample. We have thus examined the subset of our volume-
limited sample with $z<0.05$ (12,251 disc galaxies), and the overall trends
seen in Fig. 6 still persist.

Further work could be done to investigate the inflow rates that each of these
bar types could sustain, and combining this with the gas availability could
show why weak bars do not correlate with AGN presence as much as strong bars,
as if they cannot provide as high an inflow rate as strong bars, they require
more gas to trigger an AGN. High-resolution IFU data will allow us to measure
star-formation rates of the AGN-host galaxies, and thus draw comparisons
between AGN and non-AGN hosts, both in starforming and quiescent galaxies.
X-ray data will allow investigation of black hole accretion rates, and
spectroscopy along the axes of the bar will allow bar inflow rates to be
obtained.

This phenomenon will also be investigated at more distant redshifts (Margalef-
Bentabol, 2023, Margalef-Bentabol et al., in prep.), along with how these AGNs
are fuelled as the bar fraction decreases out to higher redshifts. Facilities
such as Euclid will provide us with greater sky coverage at better resolution
than currently available, and so with an increase in data, we should be able
to reduce noise in our samples.

##  5 Conclusions

We have investigated the influence of large-scale bars on the likelihood of
AGN signals in a volume-limited sample of 48,871 disc galaxies by analysing
data from the DESI catalogue, Galaxy Zoo DESI morphologies, and SDSS emission
line strengths. We have taken care to control for differences in stellar mass
and galaxy colour distributions between subsamples of strongly barred, weakly
barred, and unbarred galaxies.

99.9 per cent of our 3164 AGN in disc galaxies are identified via optical
emission line diagnostics, with a mere 4 AGN only detectable via WISE infrared
colours within our volume limit. We divide galaxies without clear AGN activity
into multiple categories based on the detection of emission lines in SDSS
spectra, and focus our comparison with the AGN host galaxies on two inactive
categories: 28,807 star forming galaxies with detected nebular emission lines
below the “composite” limit on an emission line ratio diagram, and 712
“undetermined” galaxies where nebular emission lines are not robustly detected
in the central fibre spectra. These latter galaxies are, on visual inspection,
predominantly red spirals, with a smaller fraction being discs that have
red/quenched inner regions and bluer outer regions.

Our key findings can be summarised as follows:

  * •

Strongly barred galaxies are more likely to host an AGN than weakly barred
galaxies, which in turn are more likely to host an AGN than unbarred galaxies.

  * •

This effect is very slight, with the fraction of AGN in each bar category
being: $f_{\mathrm{AGN,\textsc{Sbar}}}=31.6\pm 0.9$,
$f_{\mathrm{AGN,\textsc{Wbar}}}=23.3\pm 0.0$ and
$f_{\mathrm{AGN,\textsc{Ubar}}}=14.2\pm 0.6$.

The high levels of statistical significance achieved here even after
controlling for the confounding effects of colour, stellar mass, and flux
limits, have been facilitated by the advent of large sample sizes from the
latest generation of extragalactic surveys and the highly accurate and
detailed morphological identifications of strongly barred, weakly barred, and
unbarred disc galaxies. In the near future we expect to use data from surveys
such as Euclid and LSST to extend these analyses to higher redshift and
further refine our understanding of the interplay between various types of
disc instabilities and growing supermassive black holes.

## Data Availability

The data from GZD is available in Walmsley et al. (2023a)

Other catalogues used are publicly available from the following locations:

  * •

MPA-JHU:
https://wwwmpa.mpa-garching.mpg.de/SDSS/DR7/

  * •

NYU-VAGC:
http://sdss.physics.nyu.edu/vagc/

  * •

DESI Legacy Surveys:
https://www.legacysurvey.org/dr10/description/

## Acknowledgements

We would like to thank the referee for their valuable comments and insight,
which improved the quality of this paper

ILG acknowledges support from an STFC PhD studentship [grant number
ST/T506205/1] and from the Faculty of Science and Technology at Lancaster
University. MW is a Dunlap Fellow. The Dunlap Institute is funded through an
endowment established by the David Dunlap family and the University of
Toronto. MSS acknowledges support from an STFC PhD studentship [grant
ST/V506709/1]. LMP acknowledges support from the Faculty of Science and
Technology at Lancaster University. BDS acknowledges support through a UK
Research and Innovation Future Leaders Fellowship [grant number MR/T044136/1].
CJL acknowledges support from the Sloan Foundation. RJS acknowledges support
through the Royal Astronomical Society Research Fellowship. KBM acknowledges
support from the National Science Foundation under grants OAC 1835530 and IIS
2006894. DOR acknowledges the support of the UK Science and Technology
Facilities Council (STFC) under grant reference ST/T506205/1.. JP acknowledges
funding from the Science and Technology Facilities Council (STFC) Grant Code
ST/X508640/1. MRT acknowledges support from an STFC PhD studentship [grant
number ST/V506795/1].

Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the
Participating Institutions, the National Science Foundation, and the U.S.
Department of Energy Office of Science. The SDSS-III web site is
http://www.sdss3.org/.

SDSS-III is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS-III Collaboration including the
University of Arizona, the Brazilian Participation Group, Brookhaven National
Laboratory, Carnegie Mellon University, University of Florida, the French
Participation Group, the German Participation Group, Harvard University, the
Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA
Participation Group, Johns Hopkins University, Lawrence Berkeley National
Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for
Extraterrestrial Physics, New Mexico State University, New York University,
Ohio State University, Pennsylvania State University, University of
Portsmouth, Princeton University, the Spanish Participation Group, University
of Tokyo, University of Utah, Vanderbilt University, University of Virginia,
University of Washington, and Yale University.

The Legacy Surveys consist of three individual and complementary projects: the
Dark Energy Camera Legacy Survey (DECaLS; Proposal ID #2014B-0404; PIs: David
Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID
#2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy
Survey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS, BASS and MzLS
together include data obtained, respectively, at the Blanco telescope, Cerro
Tololo Inter-American Observatory, NSF’s NOIRLab; the Bok telescope, Steward
Observatory, University of Arizona; and the Mayall telescope, Kitt Peak
National Observatory, NOIRLab. Pipeline processing and analyses of the data
were supported by NOIRLab and the Lawrence Berkeley National Laboratory
(LBNL). The Legacy Surveys project is honored to be permitted to conduct
astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with particular
significance to the Tohono O’odham Nation.

NOIRLab is operated by the Association of Universities for Research in
Astronomy (AURA) under a cooperative agreement with the National Science
Foundation. LBNL is managed by the Regents of the University of California
under contract to the U.S. Department of Energy.

This project used data obtained with the Dark Energy Camera (DECam), which was
constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES
Projects has been provided by the U.S. Department of Energy, the U.S. National
Science Foundation, the Ministry of Science and Education of Spain, the
Science and Technology Facilities Council of the United Kingdom, the Higher
Education Funding Council for England, the National Center for Supercomputing
Applications at the University of Illinois at Urbana-Champaign, the Kavli
Institute of Cosmological Physics at the University of Chicago, Center for
Cosmology and Astro-Particle Physics at the Ohio State University, the
Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M
University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho
de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de
Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de
Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia,
Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the
Collaborating Institutions in the Dark Energy Survey. The Collaborating
Institutions are Argonne National Laboratory, the University of California at
Santa Cruz, the University of Cambridge, Centro de Investigaciones
Energeticas, Medioambientales y Tecnologicas-Madrid, the University of
Chicago, University College London, the DES-Brazil Consortium, the University
of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi
National Accelerator Laboratory, the University of Illinois at Urbana-
Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC), the Institut de
Fisica d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig
Maximilians Universitat Munchen and the associated Excellence Cluster
Universe, the University of Michigan, NSF’s NOIRLab, the University of
Nottingham, the Ohio State University, the University of Pennsylvania, the
University of Portsmouth, SLAC National Accelerator Laboratory, Stanford
University, the University of Sussex, and Texas A&M University.

BASS is a key project of the Telescope Access Program (TAP), which has been
funded by the National Astronomical Observatories of China, the Chinese
Academy of Sciences (the Strategic Priority Research Program “The Emergence of
Cosmological Structures” Grant # XDB09000000), and the Special Fund for
Astronomy from the Ministry of Finance. The BASS is also supported by the
External Cooperation Program of Chinese Academy of Sciences (Grant #
114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant #
12120101003, # 11433005).

The Legacy Survey team makes use of data products from the Near-Earth Object
Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet
Propulsion Laboratory/California Institute of Technology. NEOWISE is funded by
the National Aeronautics and Space Administration.

The Legacy Surveys imaging of the DESI footprint is supported by the Director,
Office of Science, Office of High Energy Physics of the U.S. Department of
Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility under the
same contract; and by the U.S. National Science Foundation, Division of
Astronomical Sciences under Contract No. AST-0950945 to NOAO.

The data in this paper are the result of the efforts of the Galaxy Zoo
volunteers, without whom none of this work would be possible. Their efforts
are individually acknowledged at http://authors.galaxyzoo.org.

### Software

This research has made use of Topcat (Taylor, 2005), an interactive graphical
tool for analysis and manipulation of tabular data.

This research has made extensive use of the following Python packages:

  * •

Astropy, a community-developed core Python package for Astronomy (Astropy
Collaboration et al., 2013, 2018, 2022).

  * •

Matplotlib, a 2D graphics package for Python (Hunter, 2007).

  * •

Numpy (Harris et al., 2020), a package for scientific computing.

  * •

Scipy (Virtanen et al., 2020), a package for fundamental algorithms in
scientific computing.

## References

  * Aird et al. (2012) Aird J., et al., 2012, ApJ, 746, 90
  * Alonso et al. (2018) Alonso S., Coldwell G., Duplancic F., Mesa V., Lambas D. G., 2018, A&A, 618, A149
  * Assef et al. (2018) Assef R. J., Stern D., Noirot G., Jun H. D., Cutri R. M., Eisenhardt P. R. M., 2018, ApJS, 234, 23
  * Astropy Collaboration et al. (2013) Astropy Collaboration et al., 2013, A&A, 558, A33
  * Astropy Collaboration et al. (2018) Astropy Collaboration et al., 2018, AJ, 156, 123
  * Astropy Collaboration et al. (2022) Astropy Collaboration et al., 2022, ApJ, 935, 167
  * Athanassoula (2003) Athanassoula E., 2003, MNRAS, 341, 1179
  * Baldwin et al. (1981) Baldwin J. A., Phillips M. M., Terlevich R., 1981, PASP, 93, 5
  * Blanton et al. (2005) Blanton M. R., et al., 2005, AJ, 129, 2562
  * Brinchmann et al. (2004) Brinchmann J., Charlot S., White S. D. M., Tremonti C., Kauffmann G., Heckman T., Brinkmann J., 2004, MNRAS, 351, 1151
  * Cameron (2011) Cameron E., 2011, Publ. Astron. Soc. Australia, 28, 128
  * Cheung et al. (2013) Cheung E., et al., 2013, ApJ, 779, 162
  * Ciotti & Ostriker (2012) Ciotti L., Ostriker J. P., 2012, in D.-W. Kim & S. Pellegrini ed., Astrophysics and Space Science Library Vol. 378, Astrophysics and Space Science Library. p. 83 (arXiv:1104.2238), doi:10.1007/978-1-4614-0580-1_4
  * Ciotti et al. (1991) Ciotti L., D’Ercole A., Pellegrini S., Renzini A., 1991, ApJ, 376, 380
  * Coelho & Gadotti (2011) Coelho P., Gadotti D. A., 2011, ApJ, 743, L13
  * Davies et al. (2007) Davies R. I., Müller Sánchez F., Genzel R., Tacconi L. J., Hicks E. K. S., Friedrich S., Sternberg A., 2007, ApJ, 671, 1388
  * Friedli & Benz (1993) Friedli D., Benz W., 1993, A&A, 268, 65
  * Galloway et al. (2015) Galloway M. A., et al., 2015, MNRAS, 448, 3442
  * Garland et al. (2023) Garland I. L., et al., 2023, MNRAS, 522, 211
  * Géron et al. (2021) Géron T., Smethurst R. J., Lintott C., Kruk S., Masters K. L., Simmons B., Stark D. V., 2021, MNRAS, 507, 4389
  * Géron et al. (2023) Géron T., et al., 2023, MNRAS, 521, 1775
  * Goulding et al. (2017) Goulding A. D., et al., 2017, ApJ, 843, 135
  * Harris et al. (2020) Harris C. R., et al., 2020, Nature, 585, 357
  * Heckman & Best (2014) Heckman T. M., Best P. N., 2014, ARA&A, 52, 589
  * Hinshaw et al. (2013) Hinshaw G., et al., 2013, ApJS, 208, 19
  * Hopkins & Hernquist (2006) Hopkins P. F., Hernquist L., 2006, ApJS, 166, 1
  * Hunter (2007) Hunter J. D., 2007, Computing in Science and Engineering, 9, 90
  * Kauffmann et al. (2003) Kauffmann G., et al., 2003, MNRAS, 346, 1055
  * Kewley et al. (2001) Kewley L. J., Dopita M. A., Sutherland R. S., Heisler C. A., Trevena J., 2001, ApJ, 556, 121
  * Kewley et al. (2006) Kewley L. J., Groves B., Kauffmann G., Heckman T., 2006, MNRAS, 372, 961
  * Knapen et al. (2000) Knapen J. H., Shlosman I., Peletier R. F., 2000, ApJ, 529, 93
  * Kormendy & Ho (2013) Kormendy J., Ho L. C., 2013, ARA&A, 51, 511
  * Laine et al. (2002) Laine S., Shlosman I., Knapen J. H., Peletier R. F., 2002, ApJ, 567, 97
  * Lin et al. (2013) Lin L.-H., Wang H.-H., Hsieh P.-Y., Taam R. E., Yang C.-C., Yen D. C. C., 2013, ApJ, 771, 8
  * Łokas (2022) Łokas E. L., 2022, A&A, 668, L3
  * Maciejewski et al. (2002) Maciejewski W., Teuben P. J., Sparke L. S., Stone J. M., 2002, MNRAS, 329, 502
  * Margalef-Bentabol (2023) Margalef-Bentabol B., 2023, in Galactic Bars: Driving and Decoding Galaxy Evolution. p. 32, doi:10.5281/zenodo.8130460
  * Martig et al. (2012) Martig M., Bournaud F., Croton D. J., Dekel A., Teyssier R., 2012, ApJ, 756, 26
  * Martin et al. (2018) Martin G., et al., 2018, MNRAS, 476, 2801
  * Masters et al. (2011) Masters K. L., et al., 2011, MNRAS, 411, 2026
  * Masters et al. (2012) Masters K. L., et al., 2012, MNRAS, 424, 2180
  * Masters et al. (2019) Masters K. L., et al., 2019, MNRAS, 487, 1808
  * McAlpine et al. (2020) McAlpine S., Harrison C. M., Rosario D. J., Alexander D. M., Ellison S. L., Johansson P. H., Patton D. R., 2020, MNRAS, 494, 5713
  * Nair & Abraham (2010) Nair P. B., Abraham R. G., 2010, ApJS, 186, 427
  * Oh et al. (2012) Oh S., Oh K., Yi S. K., 2012, ApJS, 198, 4
  * Regan & Teuben (2004) Regan M. W., Teuben P. J., 2004, ApJ, 600, 595
  * Rosario et al. (2016) Rosario D. J., Mendel J. T., Ellison S. L., Lutz D., Trump J. R., 2016, MNRAS, 457, 2703
  * Sakamoto (1996) Sakamoto K., 1996, ApJ, 471, 173
  * Salim et al. (2007) Salim S., et al., 2007, ApJS, 173, 267
  * Schawinski et al. (2007) Schawinski K., Thomas D., Sarzi M., Maraston C., Kaviraj S., Joo S.-J., Yi S. K., Silk J., 2007, MNRAS, 382, 1415
  * Shapiro & Wilk (1965) Shapiro S. S., Wilk M. B., 1965, Biometrika, 52, 591
  * Silva-Lima et al. (2022) Silva-Lima L. A., Martins L. P., Coelho P. R. T., Gadotti D. A., 2022, A&A, 661, A105
  * Slater et al. (2019) Slater R., et al., 2019, A&A, 621, A83
  * Smethurst et al. (2021) Smethurst R. J., et al., 2021, MNRAS, 507, 3985
  * Smethurst et al. (2023) Smethurst R. J., et al., 2023, MNRAS, p. stad1794
  * Student (1908) Student 1908, Biometrika, 6, 1
  * Taylor (2005) Taylor M. B., 2005, in Shopbell P., Britton M., Ebert R., eds, Astronomical Society of the Pacific Conference Series Vol. 347, Astronomical Data Analysis Software and Systems XIV. p. 29
  * Veilleux & Osterbrock (1987) Veilleux S., Osterbrock D. E., 1987, ApJS, 63, 295
  * Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
  * Walmsley et al. (2022) Walmsley M., et al., 2022, MNRAS, 509, 3966
  * Walmsley et al. (2023a) Walmsley M., et al., 2023a, MNRAS, p. stad2919
  * Walmsley et al. (2023b) Walmsley M., et al., 2023b, The Journal of Open Source Software, 8, 5312
  * York et al. (2000) York D. G., et al., 2000, AJ, 120, 1579
  * Zee et al. (2023) Zee W.-B. G., Paudel S., Moon J.-S., Yoon S.-J., 2023, ApJ, 949, 91

##  Appendix A Subsample Counts

Tab. 2 presents the full set of number counts of all subsamples in this work.
While our analysis is confined to the volume-limited sample, we also present
numbers for the full set of GZD classified galaxies which also have ancillary
data presented in the MPA-JHU and NYU-VAGC catalogues.

Table 2: Full breakdown of the number of galaxies in each activity class, and
each bar category, both in the volume limited sample, and before volume
limiting. Note that the numbers in some of the sub-sub-categories may have
duplicates. For example, there is one WISE AGN that is also an optical AGN,
and thus the numbers do not completely add, and we show the total numbers for
clarity. Subsample Counts
---
|  |  | Total | In Volume Limit
Is Disc | 112699 | 48871
Is Undetermined | 218101 | 25004
Is Uncertain | 124990 | 29355
Is Star-forming | 280867 | 86917
Is Composite | 75872 | 30540
| LINER $\mathrm{\left[O\textsc{i}\right]}$ | 11724 | 5597
| LINER $\mathrm{\left[S\textsc{ii}\right]}$ | 26699 | 13189
| LINER $\mathrm{\left[N\textsc{ii}\right]}$ | 11298 | 4366
Is LINER | 49721 | 23152
| Optical AGN | 37558 | 13394
| WISE AGN | 98 | 14
Is AGN | 37651 | 13406
Is Disc and: |  |
| Is Undetermined | 2652 | 712
| Is Uncertain | 11011 | 2518
| Is Star-forming | 66335 | 28807
| Is Composite | 17754 | 8669
|  | LINER $\mathrm{\left[O\textsc{i}\right]}$ | 2035 | 1420
|  | LINER $\mathrm{\left[S\textsc{ii}\right]}$ | 4219 | 2835
|  | LINER $\mathrm{\left[N\textsc{ii}\right]}$ | 1172 | 588
| Is LINER | 7426 | 4843
|  | Optical AGN | 6625 | 3160
|  | WISE AGN | 16 | 5
| Is AGN | 6639 | 3164
Is Unbarred Disc and: |  |
| Is Undetermined | 1377 | 350
| Is Uncertain | 5432 | 1234
| Is Star-forming | 42045 | 18829
| Is Composite | 7051 | 3628
|  | LINER $\mathrm{\left[O\textsc{i}\right]}$ | 861 | 592
|  | LINER $\mathrm{\left[S\textsc{ii}\right]}$ | 1724 | 1155
|  | LINER $\mathrm{\left[N\textsc{ii}\right]}$ | 501 | 256
| Is LINER | 3086 | 2003
|  | Optical AGN | 2448 | 1237
|  | WISE AGN | 7 | 1
| Is AGN | 2455 | 1238
Is Weak Barred Disc and: |  |
| Is Undetermined | 860 | 212
| Is Uncertain | 4078 | 844
| Is Star-forming | 19791 | 7903
| Is Composite | 6545 | 2914
|  | LINER $\mathrm{\left[O\textsc{i}\right]}$ | 595 | 391
|  | LINER $\mathrm{\left[S\textsc{ii}\right]}$ | 1374 | 881
|  | LINER $\mathrm{\left[N\textsc{ii}\right]}$ | 436 | 182
| Is LINER | 2405 | 1454
|  | Optical AGN | 2497 | 1048
|  | WISE AGN | 8 | 4
| Is AGN | 2504 | 1051
Is Strong Barred Disc and: |  |
| Is Undetermined | 415 | 150
| Is Uncertain | 1501 | 440
| Is Star-forming | 4499 | 2075
| Is Composite | 4158 | 2127
|  | LINER $\mathrm{\left[O\textsc{i}\right]}$ | 579 | 437
|  | LINER $\mathrm{\left[S\textsc{ii}\right]}$ | 1121 | 799
|  | LINER $\mathrm{\left[N\textsc{ii}\right]}$ | 235 | 150
| Is LINER | 1935 | 1386
|  | Optical AGN | 1680 | 875
|  | WISE AGN | 1 | 0
| Is AGN | 1680 | 875

##  Appendix B Full stellar mass and colour distributions

Fig. 8 shows an identical plot to that in Fig. 4, with the addition of the
distributions in mass and colour for weakly barred galaxies for completeness.

Figure 8: The distributions in stellar mass and $g-r$ colour for a variety of
subsamples, with strongly barred galaxies in solid lines, unbarred in dotted
lines and weakly barred in dashed lines. AGNs are in teal, star-forming in
red, composite in orange and undetermined in navy blue.


* Swersky et al. (2016) Swersky L, Marques HO, Sander J, Campello RJ, Zimek A (2016) On the evaluation of outlier detection and one-class classification methods. _2016 IEEE international conference on data science and advanced analytics (DSAA)_ , 1–10 (IEEE).
  * Tanno et al. (2019) Tanno R, Saeedi A, Sankaranarayanan S, Alexander DC, Silberman N (2019) Learning from noisy labels by regularized estimation of annotator confusion. _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 11244–11253.
  * Torabi et al. (2018) Torabi F, Warnell G, Stone P (2018) Behavioral cloning from observation. _arXiv preprint arXiv:1805.01954_ .
  * Wang et al. (2019a) Wang G, Li J, Hopp WJ, Fazzalari FL, Bolling SF (2019a) Using patient-specific quality information to unlock hidden healthcare capabilities. _Manufacturing & Service Operations Management_ 21(3):582–601.
  * Wang et al. (2019b) Wang L, Bai Y, Bhalla A, Joachims T (2019b) Batch learning from bandit feedback through bias corrected reward imputation.
  * Wang and Saar-Tsechansky (2020) Wang T, Saar-Tsechansky M (2020) Augmented fairness: An interpretable model augmenting decision-makers’ fairness. _arXiv:2011.08398_ .
  * Wilder et al. (2020) Wilder B, Horvitz E, Kamar E (2020) Learning to complement humans. _arXiv_ .
  * Wolczynski et al. (2022) Wolczynski N, Saar-Tsechansky M, Wang T (2022) Learning to advise humans by leveraging algorithm discretion. _arXiv:2210.12849_ .
  * Yan et al. (2011) Yan Y, Rosales R, Fung G, Dy JG (2011) Active learning from crowds. _ICML_ , volume 11, 1161–1168.
  * Zheng et al. (2017) Zheng Y, Li G, Li Y, Shan C, Cheng R (2017) Truth inference in crowdsourcing: Is the problem solved? _Proceedings of the VLDB Endowment_ 10(5):541–552.
  * Zou et al. (2019) Zou Y, Yu Z, Liu X, Kumar B, Wang J (2019) Confidence regularized self-training. _Proceedings of the IEEE International Conference on Computer Vision_ , 5982–5991.

##  Appendix A Implementation Details of Human Behavior Models in Synthetic
Dataset

Black-Box Human Behavior Model To create a black-box human behavior model, we
use a random forest classifier on a random 30% subset of the data with full
ground-truth labels. After the model is trained, at both training (data
generating) and testing time, our synthetic expert will make a random action
based on the output probability of the classifier. Since our task is a multi-
label classification, the output probability is multiple predictive
probabilities of each class which do not sum to 1. In order to make a choice,
we use softmax with temperature to normalize the probabilities across classes,
the temperatures are set to 10 and 20 for Scene and TMC respectively. When the
number of actions increases, the probability output by softmax decreases and a
larger temperature can ensure high confidence actions are still selected with
relatively high probabilities. Here, high-confidence refers to the original
output probability of the black-box model.

Decision Noise Human Behavior Model For decision noise human behavior models,
motivated by learning from noisy labels under classification setup Tanno et
al. (2019), we assume each expert will follow a uniform decision accuracy
parameterized by $\rho$. Here we give an example in Table 19. Assume a
customer specialist has $\rho=0.6$, and needs to offer compensation plan to
two customers, Customer 1 and Customer 2, with optimal compensation plan A and
B respectively. Table 19 shows the decision probability for this decision
maker under $\rho$. In such a setting, the decision maker is equally likely to
make wrong decisions and have a decision accuracy of 0.6. Similarly, if there
are multiple optimal actions, $\rho$ is the summation of the decision accuracy
of these actions and they all have the same decision accuracy.

In our experiments, we simulate three expert decision-makers and set $\rho$ to
be 0.6, 0.7, 0.8 respectively to see whether our personalization objective can
help when experts have diverse skills Huang et al. (2017).

Optimal Action | A | B | C
---|---|---|---
Customer 1 - A | 0.6 | 0.2 | 0.2
Customer 2 - B | 0.2 | 0.6 | 0.2
Table 19: Decision Accuracy Example

##  Appendix B Dataset Statistics

| # Features | # Labels | Dataset Size
---|---|---|---
Scene | 294 | 6 | 2407
TMC | 30438 | 22 | 28596
MLC | 1248 | 6 | 695
Focus | 292 | 2 | 1000
Hate Speech | 100 | 2 | 1471
CIFAR10-H | 32x32 | 10 | 10000
MNIST | 32x32 | 10 | 70000
SVHN | 32x32 | 10 | 99289

Table 20: Dataset statistics.

##  Appendix C Significance Test

We report two sample t-tests for our experiments reported in the main paper.
The significance level is set to 0.05 and the null hypothesis is that the
difference between two methods is 0.

###  C.1 Main Result and Additional Experiments

The significance test results for our experiments are reported in Table 1,
Table 2 on the Focus dataset are shown in Table 21 and Table 22 respectively.
When the human cost is low, the hybrid system demonstrates a significant
improvement over the AO baseline. As the human cost increases, the benefit of
hybrid team performance over the human-only baseline becomes increasingly
significant, and the difference between the hybrid team and the AO baseline
becomes insignificant.

Table 21: Significance Tests on Focus dataset with Cost of 0.05. True means the averages are significantly different with confidence level of 0.05. For all settings, JC has competitive performance against all baselines, demonstrating the possibility of human-AI complementarity in lcp-hai. Data | Human/AO | Human/TS | Human/JC | AO/TS | AO/JC | TS/JC
---|---|---|---|---|---|---
Focus | False | False | False | False | True | True
Table 22: Significance Tests on Focus dataset with different expert costs. We examine the effect of human cost and set it from 0 to 0.5. True means the averages are significantly different with confidence level of 0.05. For all settings, JC has competitive performance against all baselines. When the human costs are too high, the human-AI complementarity decreases and the deferral system chooses to only use algorithm decisions. Data (cost) | Human/AO | Human/TS | Human/JC | AO/TS | AO/JC | TS/JC
---|---|---|---|---|---|---
Focus (0) | True | True | False | False | True | True
Focus (0.05) | False | False | False | False | True | True
Focus (0.1) | True | True | True | False | True | True
Focus (0.3) | True | True | True | False | False | False
Focus (0.5) | True | True | True | False | False | False

The significance tests for the additional results reported in Table 3 are
shown in Table 23. For joint collaboration, it always has a significant
benefit over the human team and often has a significant benefit in reward over
AO and TS.

Table 23: Significance Tests for different Human Behavior Models and MLC. True means the averages are significantly different with confidence level of 0.05. Model refers to the Black Box human behavior model and Noise refers to the uniform human behavior model. lcp-hai with joint collaboration is superior to all other alternatives.  Data (HBM) | Human/AO | Human/TS | Human/JC | AO/TS | AO/JC | TS/JC
---|---|---|---|---|---|---
Scene (Model) | True | True | True | False | True | True
Scene (Noise) | True | True | True | False | True | True
TMC (Model) Two Stage | True | True | True | False | False | False
TMC (Noise) | True | True | True | False | False | False
MLC | True | True | True | True | True | False

###  C.2 Personalization

Similarly, the significant tests of the synthetic human responses and MLC in
Table 3 and Table 5 are reported in Table 23 and Table 24 respectively.

Table 24: Significance Tests on different datasets for different Human Behavior Models. True means the averages are significantly different with confidence level of 0.05. Model means the Black Box human behavior model and Noise represents the uniform human behavior model. Personalization is never worse than JC and sometimes significantly outperforms JC. Method | Scene (Model) | Scene (Noise) | TMC (Model) | TMC (Noise)
---|---|---|---|---
JC / JCP | False | False | False | True
Table 25: Significance Tests on Focus datasets with different expert costs. True means the averages are significantly different with confidence level of 0.05. We examine the effect of the human cost and set it from 0 to 0.5. For all settings, JCP has competitive performance against JC, which is the best method over other baselines. When the human costs are too high, the human-AI complementarity decreases and the deferral system chooses to only use algorithm decisions. Method | $C=0$ | $C=0.05$ | $C=0.1$ | $C=0.3$ | $C=0.5$
---|---|---|---|---|---
JC / JCP | True | True | True | False | False

###  C.3 Leveraging Expert Consistencies under Deterministic Actions

For simplicity, here we only conduct significance test for the EC variant and
its corresponding algorithms here. The results for Table 7, Table 8, Table 9
and Table 10 are shown in Table 26, Table 27, Table 28 and Table 29
respectively.

The qualitative conclusions drawn from the main paper are further collaborated
with the significance tests. We can observe the significance results do not
change with the increasing bias level, a higher human cost leads to less
significant human-AI complementarity, and our conclusions are stable with
varying training set size.

Table 26: Significance Tests on Synthetic Data with Deterministic Actions. True means the averages are significantly different with confidence level of 0.05. We vary the fraction of examples that humans will apply deterministic actions on. Due to the expert consistency assumption, human decision makers’ performance increases when the fraction of deterministic action increases, while it accompanies by a significant drop in algorithm’s and JC’s performance. Leveraging expert consistency can significantly help increase human-AI complementarity. Ovp | AO/AO-EC | TS/TS-EC | JC/JC-EC
---|---|---|---
0.1 | False | False | False
0.2 | False | True | True
0.3 | False | True | True
0.4 | True | True | True
0.5 | True | True | True
Table 27: Significance Tests on Synthetic Data with Deterministic Actions and Varying Expert Bias. True means the averages are significantly different with confidence level of 0.05. We vary the bias level in deterministic actions from humans. With more consistent human bias, human decision performance decreases. While the deferral collaboration system’s performance also decreases (which is expected due to our theoretical analysis in Theorem 1), the human-AI complementarity is still possible, suggesting the deferral collaboration leveraging expert consistency may be robust to the human bias in deterministic decisions. Bias Level | AO/AO-EC | TS/TS-EC | JC/JC-EC
---|---|---|---
0.0 | False | True | True
0.1 | False | True | True
0.2 | False | True | True
0.3 | False | True | True
0.4 | False | True | True
0.5 | False | True | True
Table 28: Significance Tests on Synthetic Data with Deterministic Actions and Varying Expert Cost. True means the averages are significantly different with confidence level of 0.05. We vary the cost of human decision makers. With a larger human cost, the human decision makers’ performance decreases rapidly and the human-AI complementarity also decreases. Cost | AO/AO-EC | TS/TS-EC | JC/JC-EC
---|---|---|---
0.0 | False | True | True
0.05 | False | True | True
0.1 | False | False | False
Table 29: Significance Tests on Synthetic Data with Deterministic Actions. True means the averages are significantly different with confidence level of 0.05. We vary the number of training set size from 300 to 2000. With deterministic actions, increased sample size has a limited effect on algorithm’s performance, so as on the human-AI complementarity while complementarity is still possible. By leveraging expert consistency, there is a significant improvement on human-AI complementarity. Train | AO/AO-EC | TS/TS-EC | JC/JC-EC
---|---|---|---
300 | False | True | True
500 | False | True | True
1000 | False | True | True
2000 | False | True | True

###  C.4 Non-Stationary Data under Covariate Shift

The significance tests for our experiments in Table 15, Table 16 and Table 17
are reported in Table 30, Table 31 and Table 32 respectively. JC-OD has a
significant benefit over other baselines with challenging covariate shift such
as training on MNIST and testing on SVHN and the covariate shift happened on
Focus dataset, which further validates our conclusions in the main paper.

Table 30: Significance Tests on Simulation Data with Covariate Shift. True
means the averages are significantly different with confidence level of 0.05.
We vary the level of covariate shift by adjusting $\mu$. Due to the
assumption, humans’ performance is stable while algorithm’s performance is
impacted with large $\mu$. We find human-AI complementarity is possible for
deferral collaboration under covariate shift and our out-of-distribution
extension can further improve deferral system’s performance with severe
shifting.

| H/AO | H/TS | H/JC | H/JC-OD | AO/TS | AO/JC | AO/JC-OD | TS/JC | TS/JC-OD | JC/JC-OD
---|---|---|---|---|---|---|---|---|---|---
$\mu=1$ | True | True | True | True | False | False | False | False | False | False
$\mu=3$ | True | True | True | True | False | False | False | False | False | False
$\mu=5$ | True | True | True | True | False | False | False | False | False | False
$\mu=7$ | True | True | True | True | False | False | False | False | False | False
$\mu=9$ | True | True | True | True | False | False | False | False | False | False

Table 31: Significance Tests on Semi-Synthetic Data with Covariate Shift. True
means the averages are significantly different with confidence level of 0.05.
We vary the human cost. S $\rightarrow$ M indicates learning algorithm on SVHN
and test methods on MNIST, which is a easier task compared to M $\rightarrow$
S since images in SVHN is more complex. The algorithm’s performance is worse
than humans’ for S $\rightarrow$ M when $C=0.05$ and OD extension can further
increase human-AI complementarity in the deferral collaboration system. For M
$\rightarrow$ S, due to the challenging task, the algorithm has a
significantly worse performance while methods considering OD can still achieve
robust performance.

S $\rightarrow$ M | H/AO | H/TS | H/JC | H/JC-OD | AO/TS | AO/JC | AO/JC-OD | TS/JC | TS/JC-OD | JC/JC-OD
---|---|---|---|---|---|---|---|---|---|---
C = 0.05 | True | False | False | False | True | True | True | False | False | False
C = 0.1 | False | False | False | False | False | True | True | False | False | False
C = 0.15 | False | False | False | False | False | True | True | False | False | False
M $\rightarrow$ S | H/AO | H/TS | H/JC | H/JC-OD | AO/TS | AO/JC | AO/JC-OD | TS/JC | TS/JC-OD | JC/JC-OD
C = 0.05 | True | True | True | False | False | False | True | False | True | True
C = 0.1 | True | True | True | False | False | False | True | False | True | True
C = 0.15 | True | True | True | False | False | False | True | False | True | True

Table 32: Significance Tests on Focus with Covariate Shift. True means the
averages are significantly different with confidence level of 0.05. We vary
the size of the training set size and report the total reward. Due to the
covariate shift, algorithm solution performs worse than humans significantly
and JC-OD can still match humans’ decision performance.

| H/AO | H/TS | H/JC | H/JC-OD | AO/TS | AO/JC | AO/JC-OD | TS/JC | TS/JC-OD | JC/JC-OD
---|---|---|---|---|---|---|---|---|---|---
train ratio = 0.3 | True | True | True | False | True | True | True | True | True | True
train ratio = 0.5 | True | True | True | False | False | True | True | True | True | True
train ratio = 0.7 | True | True | True | False | False | False | True | False | True | True

##  Appendix D Samples from Semi-Synthetic Experiment with Covarite Shifting

Samples from SVHN and MNIST are shown in Figure 5. SVHN has digit images which
is more realistic in life while MNIST has more regular digit images. It is
reasonable to assume human decision maker can identify digits in both datasets
with similar accuracy while machine learning trained on one dataset might be
hard to generalize to the other (Zou et al. 2019, Han et al. 2019).

(a) SVHN

(b) MNIST

Figure 5: Samples from SVHN and MNIST datasets. Each dataset has digit images
with 10 classes.