# Interpretable machine learning for finding intermediate-mass black holes

Mario Pasquato Département de Physique, Université de Montréal, Montreal,
Quebec H3T 1J4, Canada  Physics and Astronomy Department Galileo Galilei,
University of Padova, Vicolo dell’Osservatorio 3, I–35122, Padova, Italy  Mila
- Quebec Artificial Intelligence Institute, Montreal, Quebec, Canada  Ciela,
Computation and Astrophysical Data Analysis Institute, Montreal, Quebec,
Canada  Piero Trevisan Department of Physics, Università di Roma Sapienza,
Piazzale Aldo Moro 2, 00185 Rome, Italy  INAF-Osservatorio Astronomico di
Roma, via Frascati 33, 00040 Monte Porzio Catone, Italy  Abbas Askar Nicolaus
Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18,
00-716 Warsaw, Poland  Pablo Lemos Département de Physique, Université de
Montréal, Montreal, Quebec H3T 1J4, Canada  Mila - Quebec Artificial
Intelligence Institute, Montreal, Quebec, Canada  Ciela, Computation and
Astrophysical Data Analysis Institute, Montreal, Quebec, Canada  Center for
Computational Astrophysics, Flatiron Institute, NY, USA  Gaia Carenini
Département d’Informatique, École normale supérieure - PSL Research
University, Paris, France.  Michela Mapelli Institut für Theoretische
Astrophysik, Zentrüm für Astronomie, Albert-Ueberle-Strasse 2, D-69120,
Heidelberg, Germany  Physics and Astronomy Department Galileo Galilei,
University of Padova, Vicolo dell’Osservatorio 3, I–35122, Padova, Italy  INFN
- Padova, Via Marzolo 8, I–35131 Padova, Italy  INAF - Osservatorio
Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy
Yashar Hezaveh Ciela, Computation and Astrophysical Data Analysis Institute,
Montreal, Quebec, Canada  Département de Physique, Université de Montréal,
Montreal, Quebec H3T 1J4, Canada  Center for Computational Astrophysics,
Flatiron Institute, NY, USA

(Received XXX; Revised YYY)

###### Abstract

Definitive evidence that globular clusters (GCs) host intermediate-mass black
holes (IMBHs) is elusive. Machine learning (ML) models trained on GC
simulations can in principle predict IMBH host candidates based on observable
features. This approach has two limitations: first, an accurate ML model is
expected to be a black box due to complexity; second, despite our efforts to
realistically simulate GCs, the simulation physics or initial conditions may
fail to fully reflect reality. Therefore our training data may be biased,
leading to a failure in generalization on observational data. Both the first
issue -explainability/interpretability- and the second -out of distribution
generalization and fairness- are active areas of research in ML. Here we
employ techniques from these fields to address them: we use the anchors method
to explain an XGBoost classifier; we also independently train a natively
interpretable model using Certifiably Optimal RulE ListS (CORELS). The
resulting model has a clear physical meaning, but loses some performance with
respect to XGBoost. We evaluate potential candidates in real data based not
only on classifier predictions but also on their similarity to the training
data, measured by the likelihood of a kernel density estimation model. This
measures the realism of our simulated data and mitigates the risk that our
models may produce biased predictions by working in extrapolation. We apply
our classifiers to real GCs, obtaining a predicted classification, a measure
of the confidence of the prediction, an out-of-distribution flag, a local rule
explaining the prediction of XGBoost and a global rule from CORELS.

††journal: ApJ

##  1 Introduction

The detection of quasars at high redshift (see, e.g., Mortlock et al., 2011;
Schindler et al., 2023; Maiolino et al., 2023) requires a mechanism for the
rapid assembly of supermassive black holes. Intermediate-mass black holes
(IMBHs) bridge the gap between stellar-mass remnants and supermassive black
holes, potentially playing an important role as seeds for the latter (see,
e.g., Woods et al. 2019 and Volonteri et al. 2021 for two recent reviews).
Searches for IMBHs in the present-day Universe are being actively carried out,
with focus on high-density environments such as the Galactic center (Oka et
al., 2016; Ballone et al., 2018; Takekawa et al., 2019b, a, 2020; Kaneko et
al., 2023; The GRAVITY Collaboration et al., 2023), dwarf galaxies (Mezcua et
al., 2016, 2018), and globular clusters (GCs; see e.g. Maccarone & Servillat,
2008; Cseh et al., 2010; Strader et al., 2012; Su et al., 2022; Farrell et
al., 2009; Bachetti et al., 2014; Kains et al., 2016; Kızıltan et al., 2017;
Lin et al., 2018). Regarding Galactic GCs, the most comprehensive study
providing direct IMBH mass upper limits based on radio observations is Tremou
et al. (2018), in the following T18.

IMBHs may form via repeated black hole (BH) mergers (Miller & Hamilton, 2002;
Gerosa & Berti, 2017; Fragione et al., 2018; Rodriguez et al., 2019; Antonini
et al., 2019; Mapelli et al., 2021; Di Carlo et al., 2021). This channel is
supported by observational evidence from gravitational waves (Abbott et al.,
2016a, b, c, d, 2017, 2020a, 2020b) but there are other proposed mechanisms,
such as population III stars (Madau & Rees, 2001; Ryu et al., 2016; Tanikawa
et al., 2021; Costa et al., 2023; Santoliquido et al., 2023), runaway stellar
collisions (Bonnell et al., 1998; Portegies Zwart et al., 1999, 2004; Giersz
et al., 2015; Mapelli, 2016; Sakurai et al., 2017; Di Carlo et al., 2019,
2021; Rizzuto et al., 2022; Arca Sedda et al., 2023; Reinoso et al., 2023), or
primordial formation (Kawaguchi et al., 2008; Das Gupta & Thareja, 2017).

Lacking a smoking-gun, uncontroversial direct IMBH detection, indirect
detection methods based on IMBH dynamical effects in GCs are useful for
compiling a list of candidate hosts for follow-up. The observable effects (see
Mezcua, 2017, for a recent and quite comprehensive review of IMBH detection
attempts) include the presence of a central cusp in surface density (Bahcall &
Wolf, 1976; Newell et al., 1976) and/or velocity dispersion (Peterson et al.,
1989; Noyola et al., 2006, 2008; Lützgendorf et al., 2011, 2012, 2013b;
Feldmeier et al., 2013; Lanzoni et al., 2013; Lützgendorf et al., 2016;
Lanzoni & Cosmic-Lab Team, 2016; Lanzoni, 2016; Lützgendorf et al., 2017;
Askar et al., 2017a), mass-segregation reduction (Baumgardt et al., 2004; Gill
et al., 2008; Pasquato et al., 2009; Beccari et al., 2010; Umbreit & Rasio,
2013; Pasquato et al., 2016; Di Cintio et al., 2023), anomalous accelerations
from pulsar timing (Peuten et al., 2014; Perera et al., 2017; Gieles et al.,
2018; Kızıltan et al., 2017), high velocity stars (Meylan et al., 1991;
Lützgendorf et al., 2011; Fragione & Gualandris, 2018), and other clues
(Miocchi, 2007; Pasquato & Bertin, 2008, 2010; Leigh et al., 2014; Pepe &
Pellizza, 2016; Askar et al., 2017a).

###  1.1 Applying machine learning (ML) models to indirect IMBH detection

The current prevailing approach for indirect IMBH detection is comparing a
measured physical quantity and its prediction according to a theoretical model
of the specific phenomena affected by the presence of an IMBH. These
handcrafted physical models clearly present the advantage of full
interpretability. But it is (at least in principle) possible that more
flexible data-driven models can significantly outperform them. On the other
hand, several data-driven approaches to IMBH detection based on machine
learning (ML), while promising, have at least two clear drawbacks: first, the
general expectation is that a high-performance model must be very complex,
virtually becoming a black box111The notion of a black box, together with its
opposites white box or glass box, is now commonplace in ML literature (see,
e.g., Garrett & Rudin, 2022) and has been attested at least since Ashby
(1956), though it is uncertain when exactly it was introduced. For the
purposes of this paper, it is understood that _Black box ML models are
predictive formulas, either too complicated for human understanding, or based
on proprietary code with purposefully hidden calculations_ (Rudin, 2022).,
with few notable exceptions. Such a model could be a deep neural network or a
decision tree ensemble. These kinds of models are hard to interpret due to
their complexity, irrespective of the soundness of the statistical foundations
on which they are built. For instance proving that a neural network is a
universal function approximator (Hornik et al., 1989) is scant consolation for
the fact that humans can only make sense of the inner workings of a trained
neural network model through laborious analysis that resembles experimental
biology more than mathematics (this reverse engineering work constitutes the
newborn field of ‘mechanistic interpretability’ see, e.g., Olah et al., 2017;
Carter et al., 2019; Nanda et al., 2023). Second, the model would have to be
trained on simulations. Even in the unlikely event that simulations were to
correctly model most of the relevant physics, they are quite likely to be run
from unrealistic initial conditions, because constraining the early stages of
cluster evolution is essentially still an open problem (see e.g. Torniamenti
et al., 2022). The simulated data on which a data-driven model will be trained
can therefore be biased by containing e.g. spurious associations between
variables or by under- or over-representing instances with certain
characteristics. In this paper, we address both issues.

###  1.2 Explainability versus native interpretability

We build a baseline black-box model based on XGBoost (Chen & Guestrin, 2016).
We treat XGBoost as a black box since it is impossible for a human within a
reasonable time frame to make sense of the weighed contributions of thousands
of decision trees. Details on XGBoost are given in Section 3. We show that the
black-box model behavior can be explained locally using anchors (Ribeiro et
al., 2018). Anchors are logical rules that apply in the vicinity of a given
instance, explaining the black box model locally -in the sense that they are
faithful to the underlying model in a neighborhood of a selected instance. In
concrete, anchors are combinations of few binary choices obtained by
thresholding one or more features (for instance, relaxation time greater than
two Gyr and half-mass radius smaller than three pc), which predict the
behavior of the underlying classifier (predicting IMBH host versus non-host)
on the majority of data points around the one for which we want an
explanation. They have been shown empirically to be readily understandable by
humans (Ribeiro et al., 2018). Indeed, the anchors we find are often
physically meaningful as we discuss below.

Anchors are one of the many schemes recently introduced to explain black box
ML models. Such explanations are necessarily unable to fully capture the
behavior of the model they are meant to explain (for instance, anchors are
local explanations, and differ based on the data point we seek an explanation
for; moreover, sometimes no suitable anchor exists given our requirements). If
this were not the case, we could ditch the original model and use the
explanations directly for prediction. Following this line of argument,
natively interpretable models, i.e. models that are simple enough to be
understandable by humans directly, have the advantage of not needing a post-
hoc explanation. Rudin (2019) for instance has argued that, with properly
engineered features, supervised problems on tabular data can be tackled
effectively by interpretable models without the need to rely on black boxes.
We thus train a fully interpretable model based on decision rules, Certifiably
Optimal RulE ListS (CORELS; Angelino et al., 2017), to assess how it compares
to XGBoost in terms of performance. The CORELS model finds rules that have a
straightforward interpretation in terms of GC physics; moreover the rule list
it finds bears some resemblance to the anchor explanations for XGBoost.

###  1.3 Training on simulations, predicting on observations

A systematic discussion of how out-of-distribution generalization may impact
the application of ML methods in the context of astronomy is Acquaviva et al.
(2020), in which the authors try to measure the increase in generalization
error due to training on cosmological simulations and predicting on actual
observational data. Their discussion clearly applies to our case, but it
focuses mostly on measuring a global distance between the training data and
the data set on which the models are eventually deployed, evaluating the
effects on overall generalization error. In this paper we will focus on
deciding whether individual predictions should be trusted; thus we will be
interested in measuring how far an individual data point corresponding to an
actual, observed physical system is from the simulated training data.

##  2 Data

###  2.1 Training, validation and test datasets

To build a training and validation sample for our ML models, and to test them,
we used results from the MOCCA-Survey Database I simulations described in
detail by Askar et al. (2017b). The MOCCA-Survey Database comprises nearly
2000 star cluster simulations with different initial parameters that were
carried out using the MOCCA code (Hypki & Giersz, 2013; Giersz et al., 2013),
which is based on the Monte Carlo algorithm (Hénon, 1971; Stodolkiewicz, 1982,
1986) for treating the long-term evolution of star clusters. The orbit-
averaged Monte Carlo method combines a statistical approach for the treatment
of distant two-body interactions that drive the dynamical evolution of a star
cluster with the particle based approach of N-body methods (Giersz, 2001;
Joshi et al., 2001; Hypki & Giersz, 2013; Rodriguez et al., 2022). This
approach allows for the implementation of important physical processes and the
possibility to simulate star cluster models with several hundred thousands to
millions of stars within a few days to weeks. MOCCA incorporates prescriptions
for stellar and binary evolution based on the SSE/BSE codes (Hurley et al.,
2000, 2002). For computing the outcome of close dynamical interactions between
binary-single stars and binary-binary stars, MOCCA uses the fewbody code
(Fregeau et al., 2004) which is a direct N-body integrator for small-N
gravitational dynamics. MOCCA also implements a realistic treatment of escape
processes in tidally limited clusters based on Fukushige & Heggie (2000). In
order to model the Galactic potential, MOCCA uses a simple point-mass
approximation. The treatment of escapers from the tidally limited GC models is
based on Fukushige & Heggie (2000). MOCCA has been comprehensively tested and
compared with results from direct N-body codes (see e.g. Giersz et al., 2013;
Heggie, 2014; Wang et al., 2016; Madrid et al., 2017; Giersz et al., 2019).

The star clusters models simulated in the MOCCA-Survey Database I span a wide
range of initial parametes with different number of objects, metallicity,
binary fraction and parameter distribution, central concentration, tidal and
half-mass radii, and different natal kick prescriptions for stellar-mass BHs
(see Table 1 in Askar et al. 2017b). Initial stellar masses in each cluster
model were sampled using the Kroupa (2001) initial mass function (IMF) with
minimum and maximum stellar masses of $0.08\rm\ M_{\odot}$ and $100.0\
M_{\odot}$. The initially densest ($\rho_{c}>10^{6}\rm\ M_{\odot}\rm pc^{-3}$)
GC models form an IMBH ($\geq 500\ \rm M_{\odot}$) within a few hundred Myr of
dynamical evolution through a combination of collisions, mergers and mass
transfer events onto a seed BH. This seed typically forms from a merger
between a BH and a very massive main-sequence star ($>50\ \rm M_{\odot}$). The
latter typically form via the runaway merger of stars during the early
evolution (within 100 Myr) of these dense GC models. The rapid formation of a
seed IMBH in these MOCCA models has been described as what is known as the
_fast_ IMBH formation scenario (Giersz et al., 2015).

In other more moderately dense models ($10^{4}\rm\
M_{\odot}pc^{-3}\lesssim\rho_{c}\lesssim 10^{6}\rm\ M_{\odot}pc^{-3}$), the
IMBH forms after a few Gyr of dynamical evolution from mergers during binary
interactions involving a stellar-mass BH. This has been referred to as the
_slow_ IMBH formation scenario. Both the _fast_ and _slow_ IMBH formation
mechanisms in the MOCCA GC models have been discussed in detail in Giersz et
al. (2015). In most of these simulated cluster models, for mergers involving a
BH and a star, it is assumed that the BH accretes the entirety of the mass of
the star that it merges with (Hong et al., 2020; Askar et al., 2021).
Furthermore, gravitational wave recoil kick following the merger of two stars
are also not implemented (Morawski et al., 2018; Maliszewski et al., 2022).
Both these assumptions facilitate the formation and growth of IMBHs in these
cluster models.

At any rate, any set of simulations having a shot at realism will explore a
large space of parameters that may lead to potentially unphysical outcomes.
For instance, stellar evolution parameters such as the amount of mass fallback
in the event of supernova explosion may affect the formation of stellar-mass
black holes, leading to the simulations spanning a wider range of mass-to-
light ratios with respect to typical Galactic GCs. However our knowledge of
real star clusters is not complete: recent evidence suggests that exploring a
wide range of mass-to-light ratios (e.g. 0.8 to 5.03 in our simulated
clusters) may indeed be a feature rather than a bug, given the possibility of
e.g. hypercompact star clusters (Greene et al., 2021). The important safeguard
is that the potential lack of realism of the simulations used to train a ML
model is taken into account and mitigated appropriately, as we discuss below.

From the $1298$ GC models that evolved up to at least 12 Gyr in the MOCCA-
Survey Database I, $388$ harboured an IMBH more massive than $500\ M_{\odot}$;
in the following we adopted a more inclusive definition of IMBH, extending the
range to $100\ M_{\odot}$ and above: this choice makes the classification
problem harder and so is more conservative. Similar to the approach taken in
Askar et al. (2019), we used the simulation snapshot at 12 Gyr for each GC
model to determine the following features: Spitzer (2014) half-mass relaxation
time corresponding to the half-mass radius (HRT), total luminosity of cluster
in units of solar luminosity (TVL), mass-weighted central velocity dispersion
(CVD) in units of $\rm{km}/\rm{s}$, central surface brightness in units of
$\rm L_{\odot}/\mathrm{pc}^{2}$ from King (1962) fitting of cumulative
luminosity profile (CSB), core radius in $\mathrm{pc}$ obtained from King
(1962) (OCR), Half-light radius obtained from King (1962) fitting in
$\mathrm{pc}$ (OHLR). Fig. 1 shows a pair plot of these features on the
adopted dataset from MOCCA-Survey Database I.

The simulation data was partitioned into a train-validation dataset, where we
used k-fold cross validation as described below, and an internal test dataset
comprising $10\%$ of the initial sample. This was stratified with respect to
the label, i.e. each subset was forced to contain approximately the same
proportion of IMBH hosts as the whole sample.

Figure 1: Pair plot showing our GC simulations in feature space. IMBH hosts
are shown as orange points and non-hosts as blue points. In order, the axis
are: Half-mass relaxation time (HRT), total luminosity (TVL), central velocity
dispersion (CVD), central surface brightness (CSB), core radius (OCR), and
half-light radius (OHLR). All the reported values are at $12$ Gyr.

###  2.2 Deployment dataset

After training our models on simulated data, we used them to predict the
presence of an IMBH on real GCs. To this end we use the observational features
obtained by Baumgardt & Hilker (2018) on a sample of $161$ GCs in the Milky
Way as a final dataset to deploy our models. The data was obtained by
accessing
https://people.smp.uq.edu.au/HolgerBaumgardt/globular/parameter.html.
Observational data can in principle be out-of-distribution with respect to the
simulations comprising our training set. We discuss how we are mitigating this
issue by identifying out-of-distribution points using Kernel Density
Estimation (KDE) in Section 3.4.

##  3 Methods

###  3.1 K-fold cross-validation

K-fold cross-validation is a widely used technique in ML to evaluate the
generalization performance of a model on a given dataset. The process of
k-fold cross-validation involves partitioning a dataset into k subsets, or
folds, of approximately equal size. For each iteration, k-1 of the folds are
used as training data, and the remaining fold is used as the validation set.
This process is repeated k times, with each fold serving as the validation set
once. The average performance of the model across all k iterations is then
used as an estimate of its performance on unseen data. The major advantage of
this technique is that it allows for a more robust estimate of the model’s
performance by utilizing all of the data for both training and validation,
thus reducing the risk of overfitting. In this work, we utilize $k=5$ and
stratify the sampling so to have roughly the same fraction of IMBH hosts and
non-hosts in each fold.

###  3.2 Performance metrics

We evaluate the performance of our classifiers in terms of two metrics:
precision and recall. Precision, also referred to as purity in astronomical
catalogs, is the fraction of relevant instances (real IMBH hosts) among the
retrieved instances (IMBH host claims), while recall, also known as
completeness, is the fraction of the total amount of relevant instances (real
IMBH hosts) that were actually retrieved (claimed to be hosts). If a
classifier returns a number between $0$ and $1$ as a measure of its confidence
that a given instance belongs to the designated class ($0.642$ meaning for
instance that a given GC is predicted to be an IMBH host with $64.2\%$
probability), it is possible to choose a threshold to convert such number to a
hard prediction. For any given threshold we can thus obtain a value for
precision and one for recall. Repeating this procedure for different
thresholds we obtain a precision-recall curve, which illustrates the trade-off
between precision and recall for different threshold values of a binary
classifier. This curve shows the ability of our classifier to maintain a
balance between avoiding false positives and capturing true positives as the
classification threshold is varied. If on the other hand a classifier returns
only a hard classification, this procedure cannot be carried out and only one
(precision, recall) couple is calculated.

It is worth pointing out that the nature of the problem of IMBH detection
calls for a comparison mostly on the precision metric. Precision is more
important than recall because false positives (IMBH claims in the absence of
an actual IMBH) would waste telescope time and other resources needed for an
IMBH-candidate follow-up. On the other hand, just one confirmed claim would
have momentous astrophysical implications, so false negatives (missed IMBH
hosts) are not problematic if at least some true positive is found.

###  3.3 Learning algorithms

####  3.3.1 XGBoost classifier

We used XGBoost (Extreme Gradient Boosting; Chen & Guestrin, 2016). XGBoost
found wide application in astronomy (Sesar et al., 2017; Shu et al., 2019;
Spina et al., 2021). XGBoost is a scalable and efficient implementation of
gradient boosting, which is an ensemble learning method that combines multiple
weak decision trees to form a strong predictor. Decision trees are a
supervised learning algorithm. They work by recursively splitting the dataset
into smaller sub-groups based on the features that lead to the highest
improvement in the prediction accuracy. The final result is a tree-like model
where the internal nodes represent the decisions based on the input features,
and the leaf nodes represent the prediction or outcome.

The decision trees in XGBoost are grown using a gradient-based optimization
algorithm that aims to minimize a cost function that measures the prediction
error. The final prediction is obtained by combining the predictions of all
trees through a weighted sum, where the weights are learned during training.

In comparison to random forests, which also use an ensemble of decision trees,
XGBoost grows trees sequentially and tries to correct the mistakes of previous
trees at each step. This results in a stronger predictor, but with a higher
computational cost (Chen & Guestrin, 2016). Humans can effectively track of
the decisions of only a few shallow decision trees at most (Freitas, 2014).
Therefore, all tree ensemble methods, unlike single trees, lack
interpretability as soon as the number of trees grows enough to justify their
use (Molnar, 2022), with the lone exception of the recently introduced FIGS
algorithm (Tan et al., 2022).

We trained an XGBoost classifier based on an ensemble of $1000$ trees of
maximum depth $2$ on our set of six features. We held out $20\%$ of our
simulation data set for validation. Fig. 2 shows a precision-recall curve for
our classifier on the test set. As discussed above, being based on a large
ensemble of decision trees, our classifier is not readily interpretable: it is
a black box. In Fig. 2 we compare its performance in terms of its precision-
recall curve with that of a fully interpretable classifier trained on the same
dataset, which we will discuss below. The left-hand side of Fig. 2 shows the
precision-recall curves obtained in cross-validation, while the right hand
side shows the final curve on the test set.

Figure 2: Left-hand panel: precision recall curve for our XGBoost classifier
on each cross-validation fold (solid lines). The dots represent the recall and
precision achieved by a CORELS classifier trained on the same fold. The
colours identify the fold for both XGBoost and CORELS. Right-hand panel:
Precision recall curve for our XGBoost classifier on the test set. The red dot
represents the recall and precision achieved by the CORELS classifier on the
same test set. In both panels CORELS yields a single point rather than a curve
because it produces a fixed rule list leading to a hard class label.

####  3.3.2 Explaining XGBoost classifier: Anchors

Ribeiro et al. (2018) introduced local, model agnostic explanation in the form
of anchor rules as an improvement on their previous algorithm, LIME (Ribeiro
et al., 2016). Anchors are decision rules -sequences of if-then logical
statements- that locally approximate a black box: one such rule list is
learned for each data point for which an explanation is required. The scope of
applicability of such a rule is explicitly defined in terms of the
neighborhood of the relevant data point, leading to the definition of anchors
as scoped rules. Over the data points to which the rule applies, i.e. those
that satisfy the logical conditions, the black box prediction coincides with
the prediction cast for the data point being explained, at least with a given
frequency, termed precision. The fraction of points covered by the rule is
termed coverage. Ribeiro et al. (2018) empirically show that anchors are
readily understandable to humans when used as post-hoc explanations to a black
box model. Anchors however have a few drawbacks: first, being a post-hoc
explanation they capture the behavior of the underlying black box only locally
and up to the chosen precision; second, they can get quite complex or have
very low coverage, especially in the proximity of the decision boundary, which
challenges either their usefulness or their effectiveness as human-
understandable explanations.

For instance, an application for a loan could be rejected by a black-box
decision algorithm; the relevant explanation could be a rule such as "you
carry a balance on two credit cards or more, and your FICO score is below
$X$"; this may not apply to other individuals, but is guaranteed to explain
the majority of the relevant classification outcomes in a neighborhood of the
instance for which we requested an explanation. In the vicinity of the
decision boundary, finding such rules can become challenging. The intuition
for this is that the decision is harder in such cases, making it harder to
summarize the black box behavior in simple terms.

####  3.3.3 CORELS

Angelino et al. (2017) introduced the CORELS algorithm. CORELS produces
concise rule lists based on binary features up to a user-specified depth.
These optimal rules are found by optimizing a loss function on the set of all
possible rule lists satisfying the given constraints. Unlike previously used
methods based on greedy search (e.g. Rivest, 1987), the advantage of CORELS is
that it finds the optimal rule list given the constraints. Finding optimal
rule lists of this kind is in general a computationally hard problem. CORELS
solves it by reducing the search space of rule lists relying on recent
mathematical results in discrete optimization, including those proved by the
authors (Angelino et al., 2017) and in (Nijssen & Fromont, 2010). It also
introduces specialized data structures that keep track of intermediate
computations and exploits symmetries to make the problem tractable. CORELS
loss function is made up of two terms, the first one counting
misclassifications plus an additional term penalizing the size (i.e. number of
rules) of the rule list (see Eq. 1 in Rudin, 2019). CORELS is still slower
than traditional greedy approaches for learning decision trees, but it has
been shown to be more effective than those on a variety of real-life datasets
(see e.g. Rudin & Ertekin, 2018).

The trade-off between rule accuracy and simplicity in CORELS can be adjusted
by the user by changing a hyper-parameter $\lambda$ that determines the
relative weight of the two terms of the loss. For instance, setting
$\lambda=0.01$ means that we would be indifferent to misclassifying $1\%$ of
our dataset if this means obtaining a rule list that contains one less rule.

Figure 2 shows the precision-recall curve of XGBoost on each of the five cross
validation folds. On each fold we also trained a CORELS model. Both the
performance of XGBoost and CORELS change across folds, but in the case of
CORELS it is immediately evident _why_. The only fold in which CORELS learned
a different rule from the others (corresponding to the blue dot in Fig. 2)
achieves a much greater recall at the cost of a lowered precision. We show
below that the rule learned by CORELS in this case is less strict in weeding
out GCs that have a longer relaxation time.

It is also quite evident from Fig. 2 that the folds on which XGBoost performs
best (green and black solid lines respectively), see also a higher performance
by CORELS, which achieves the best (green dot) and second best (black dot)
precision in each of them.

Table 1:  Rules found by CORELS during training in 5-fold cross-validation. The first column reports the number of folds in which a given rule was found. The corresponding rule is reported in the second column.  Number of folds | Rule
---|---
4 |  $T_{r}<2000$ Myr and $R_{h}<3.0$ pc and both $L>10^{4}$ $L_{\odot}$ and $R_{c}/R_{h}>0.2$
1 |  $T_{r}<5000$ Myr and CSB $>10^{3}$ $L_{\odot}/\mathrm{pc}^{2}$ and $L>10^{4}$ $L_{\odot}$ and $R_{c}/R_{h}>0.2$

The rules found by CORELS are reported in Tab. 1 and shown graphically (in
projection on relevant feature planes) in Fig. 3. They were learned on one of
the cross-validation folds each. Remarkably, an identical rule is learned on
four folds out of five. The rule learned on the remaining fold is still quite
similar to it. All rules stipulate that an IMBH candidate must have an
inflated core, in particular requiring that $R_{c}/R_{h}>0.2$. This is
immediately interpretable dynamically: a large core is produced by dynamical
heating due to a binary containing the IMBH. The IMBH has a high probability
of ending up in a binary system in the first place, because its large mass
ensures that exchange interactions are favoured in IMBH-binary encounters. In
addition, mass segregation makes it likely that the secondary is a stellar-
mass black hole. The resulting binary system hardens through interactions with
the other objects in the core, releasing energy that ultimately results in a
bigger core.

All rules also demand that the total luminosity of a potential candidate
exceeds $10^{4}$ $L_{\odot}$. This is a very low bar for GCs, whose typical
luminosity is about one order of magnitude larger. This rule thus filters out
GCs that are likely too small to be able to assemble an IMBH, either because
they do not have a high enough density to trigger runaway mergers or because
their escape velocity is too small to confine dark remnants that may
eventually merge to form an IMBH.

High density is also a criterion for all rules, but it is enforced in one fold
by a direct cutoff in central surface brightness, and in the other four by a
condition on the half mass radius, i.e. $R_{h}<3.0$ pc. As shown in Fig. 3
this combines with the luminosity constraint common to all rules to select
high density GCs.

Finally, all rules require that the relaxation time be short, but the cutoffs
differ, being set to either $2$ Gyr (in four folds out of five) or to $5$ Gyr
(in the remaining one). This is likely the reason for the higher recall in the
corresponding fold, as shown in Fig. 2. A short initial relaxation time
corresponds to higher interaction frequency between stars, which results in
faster mass segregation and makes both runaway mergers and stellar-mass black
hole coalescences more likely. Relaxation times observed at $12$ Gyr are
essentially a proxy for initial relaxation times, which are unobservable
(Trenti et al., 2010). Additionally, the swelling of the core due to an IMBH
binary is observable only if the system is relaxed, because the associated
dynamical heating requires time to take place. Thus a swollen core in the
absence of a short relaxation time is an unreliable indicator of IMBH
presence.

|
---|---
|
|
Figure 3: The rules found by CORELS, displayed on the planes corresponding
each to the relevant couple of features. The white area corresponds to the
region that respects the rule. The red points correspond to simulated GCs in
the training set that host an IMBH and the blue points correspond to non-
hosts.

###  3.4 Rejecting out of distribution points

There is a potential mismatch between our simulated training data and actual
GCs, which can lead to biased predictions. As we can see in Fig. 4, the data
on which our models were tested and the real clusters are not perfectly
matched. It is thus crucial to assess how far out-of-distribution the GCs we
cast predictions for are with respect to the training data. A multitude of
out-of-distribution detection approaches have been discussed in ML literature
(see Yang et al., 2021, for a recent review). In our case, we stuck to a
simple density-based approach, which is generally recognized as effective in
low-dimensional feature spaces. For this reason, we utilized KDE. KDE is a
non-parametric technique for estimating the probability density function (PDF)
of a random variable from independent, identically distributed samples. The
basic idea behind KDE is to place a smooth, symmetric kernel function at each
data point, and then sum these kernel functions to estimate the underlying
PDF. While other approaches to density estimation exist, such as e.g. Gaussian
mixture models (McLachlan & Basford, 1988), mixture density networks (Bishop,
1994), and autoregressive flows (Huang et al., 2018), KDE is a simple and time
tested method that works well in low dimension. We used a simple Gaussian
kernel KDE relying on the scikit-learn package (Pedregosa et al., 2011). We
fitted the density estimation on the training set and evaluated on both the
test set and the real data set. The distributions of the KDE score are shown
in Fig. 4. Low KDE score corresponds to weird -i.e. unlikely, out of
distribution- data points with respect to the training set. We calculated the
KDE scores for the points in our test set (randomly extracted from the
simulation data set) and used its first decile as a cutoff to reject the out-
of-distribution data points among the actual GCs. This way we remove from the
pool of IMBH host candidates those whose prediction would require us to trust
our models in regions where the data on which they were trained is scarce.

---
Figure 4: Upper panel: pair plot showing test data (a representative sample of
the simulations, shown in blue) versus real GC data (orange). In order, the
axis are: Half relaxation time (HRT), total Luminosity (TVL), central velocity
dispersion (CVD), central surface brightness (CSB), core radius (OCR), and
half-light radius (OHLR). Lower panel: distribution of the KDE score for test
data (orange) and actual GC data (blue). The vertical line is the cutoff
adopted for inclusion in the final sample of candidates, which corresponds to
the first decile of the KDE score for test data.

##  4 Results

In this study, we compared a baseline black-box method (XGBoost) to a fully
interpretable alternative method (CORELS) for classifying GCs. We obtained a
comparison between the two methods in terms of performance given the
difference in interpretability. To this end, we utilized both methods on the
simulated GCs (explained in Section 2) and on real GCs as well. Figure 2 shows
the precision-recall curve of this classifier on our test set. Additionally,
we applied CORELS, to the same data set, obtaining a precision (recall) of
0.89 (0.46). Since CORELS returns a hard classification, unlike XGBoost, we
did not obtain a precision-recall curve for CORELS, but only single points in
the precision-recall plane. The rules comprising the rule list learned by
CORELS are visualized in Figure 4.

Furthermore, we used a KDE approach to measure how much the real data deviated
from the test data distribution. We flagged real GCs as out of distribution if
they were below the first decile of the KDE score. Lastly, we present a table
of GCs that are predicted to be IMBH hosts both by CORELS and by XGBoost and
also are in-distribution based on our KDE criterion. This is shown in Tab. 2
alongside with the upper limits on IMBH mass obtained for each candidate by
T18 (if any are present) and with any reference we found presenting evidence
of IMBH presence or absence in the candidate. Candidate selection for this
table is summarized by Figs. 5 and 6.

Table 2: Final table of IMBH candidate hosts. The first column reports the cluster name, columns $3$-$8$ report the (rounded, log) values of the relevant features. Columns $9$-$11$ show the classification prediction by XGBoost, CORELS and the in-distribution criterion based on KDE and column $12$ shows the confidence of the classification by XGBoost, based on which the rows are ordered in decreasing order. Column $13$ reports the IMBH upper mass limit based on radio observations by T18. The last column lists any previous claims or other relevant literature discussing IMBHs in the cluster. The full table, including non-hosts, is presented in Appendix A. 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Cluster ID | Other ID | HRT | TVL | CVD | CSB | OCR | OHLR | CORELS | XGB | KDE | Pred. proba | T18 upper limit | References
NGC 6569 |  | 3.06 | 5.02 | 0.86 | 3.68 | 0.04 | 0.41 | True | True | True | 0.9999 |  |
Pal 6 |  | 2.67 | 4.79 | 0.72 | 3.64 | -0.02 | 0.35 | True | True | True | 0.9998 |  |
NGC 6638 |  | 2.49 | 4.9 | 0.84 | 4.21 | -0.34 | 0.27 | True | True | True | 0.9998 |  |
NGC 6333 | M9 | 3.22 | 5.18 | 0.92 | 3.9 | -0.04 | 0.44 | True | True | True | 0.9996 | $1030M_{\odot}$ | Miocchi (2007)
NGC 6712 |  | 2.73 | 4.78 | 0.71 | 3.54 | 0.03 | 0.4 | True | True | True | 0.9988 | $1150M_{\odot}$ | Miocchi (2007)
|  |  |  |  |  |  |  |  |  |  |  |  | Lützgendorf et al. (2013a)
NGC 6254 | M10 | 3.25 | 5.04 | 0.79 | 3.65 | 0.02 | 0.47 | True | True | True | 0.9985 | $740M_{\odot}$ | Beccari et al. (2010)
|  |  |  |  |  |  |  |  |  |  |  |  | Umbreit & Rasio (2013)
FSR 1735 |  | 2.63 | 4.69 | 0.67 | 3.83 | -0.29 | 0.33 | True | True | True | 0.9969 |  |
NGC 2298 |  | 2.65 | 4.47 | 0.59 | 3.55 | -0.28 | 0.38 | True | True | True | 0.9955 |  | Pasquato et al. (2009)
NGC 5986 |  | 3.25 | 5.24 | 0.91 | 3.83 | 0.07 | 0.44 | True | True | True | 0.9954 |  |
NGC 6218 | M12 | 2.94 | 4.75 | 0.68 | 3.34 | 0.08 | 0.45 | True | True | True | 0.9947 | $800M_{\odot}$ | Sollima et al. (2016)
NGC 6316 |  | 3.27 | 5.09 | 0.91 | 4.04 | -0.22 | 0.47 | True | True | True | 0.9913 |  |
VVV-CL001 |  | 2.75 | 4.54 | 0.81 | 3.65 | -0.27 | 0.34 | True | True | True | 0.9889 |  |
Ton 2 |  | 3.01 | 4.51 | 0.53 | 3.04 | 0.06 | 0.46 | True | True | True | 0.9889 |  |
NGC 6352 |  | 2.96 | 4.47 | 0.54 | 3.06 | 0.05 | 0.46 | True | True | True | 0.9871 |  |
NGC 6779 | M56 | 3.17 | 5.03 | 0.78 | 3.59 | 0.07 | 0.47 | True | True | True | 0.9845 |  |
NGC 6171 | M107 | 2.87 | 4.53 | 0.61 | 3.32 | -0.08 | 0.46 | True | True | True | 0.9843 | $990M_{\odot}$ |
NGC 6553 |  | 3.26 | 4.95 | 0.88 | 3.67 | -0.06 | 0.37 | True | True | True | 0.9837 |  | Kains et al. (2016)
NGC 6934 |  | 3.26 | 4.95 | 0.68 | 3.46 | 0.09 | 0.47 | True | True | True | 0.9815 |  |
NGC 6637 | M69 | 2.95 | 4.95 | 0.79 | 3.76 | -0.07 | 0.38 | True | True | True | 0.9724 |  | Miocchi (2007)
UKS 1 |  | 2.85 | 4.63 | 0.62 | 3.29 | 0.04 | 0.46 | True | True | True | 0.9518 |  |
Ter 1 |  | 2.54 | 4.94 | 0.91 | 4.44 | -0.49 | 0.18 | True | True | True | 0.9502 |  |
NGC 6342 |  | 2.22 | 4.36 | 0.64 | 3.85 | -0.47 | 0.17 | True | True | True | 0.9288 |  |
HP 1 |  | 2.89 | 4.58 | 0.72 | 3.18 | 0.1 | 0.46 | True | True | True | 0.8894 |  |
NGC 6401 |  | 2.86 | 4.65 | 0.81 | 3.73 | -0.28 | 0.39 | True | True | True | 0.5381 |  |
Figure 5: Candidate selection shown in the space of relaxation time, total luminosity, log half-light radius, and log core over half-light radius. The gray squares represent the training data set, the black dots the data from actual GCs compiled by Baumgardt & Hilker (2018). GCs selected as IMBH host candidates only by XGBoost (CORELS) are shown in blue (red), and candidates selected by both models are shown in purple.  Figure 6: Euler-Venn diagram summarizing the inclusion rules for GCs in the final candidate list. The dark gray set includes all GCs who fall in-distribution with respect to the simulations according to the criterion based on KDE we discussed in the text (112 in total); the light gray set includes the XGBoost candidates (88 in total); and the brown set the CORELS candidates (29 in total). The clusters in the mutual intersection of all three sets are included in the final candidate list (24 in total). Table 3: Anchors explaining XGBoost decision for real GCs. The first column reports the GC name, the second column the anchor as a list of conditions (each feature has been given a different colour for ease of comparison), the third column reports the anchor’s precision and the fourth column its coverage. Cluster | Anchor | Precision | Coverage
---|---|---|---
NGC 2298 |  [’HRT $\leq$ 741.31’, ’CSB $>$ 458.30’, ’0.39 $<$ OCR $\leq$ 1.05’, ’OHLR $\leq$ 3.15’, ’TVL $>$ 26722.22’, ’CVD $\leq$ 5.10’] | 0.992 | 0.019
NGC 5986 |  [’TVL $>$ 139719.63’, ’OHLR $\leq$ 3.15’, ’OCR $>$ 1.05’] | 1.000 | 0.006
NGC 6171 |  [’HRT $\leq$ 741.31’, ’CSB $>$ 458.30’, ’OCR $>$ 0.39’, ’2.37 $<$ OHLR $\leq$ 3.15’, ’CVD $\leq$ 5.10’, ’TVL $>$ 26722.22’] | 1.000 | 0.025
NGC 6218 |  [’HRT $\leq$ 1513.56’, ’CSB $>$ 458.30’, ’OCR $>$ 1.05’, ’OHLR $\leq$ 3.15’] | 0.990 | 0.056
FSR 1735 |  [’HRT $\leq$ 741.31’, ’CSB $>$ 4371.87’, ’0.39 $<$ OCR $\leq$ 1.05’, ’OHLR $\leq$ 2.37’, ’TVL $>$ 26722.22’, ’CVD $\leq$ 5.10’] | 0.995 | 0.006
NGC 6254 |  [’TVL $>$ 61963.19’, ’OHLR $\leq$ 3.15’, ’CSB $\leq$ 16864.96’, ’5.10 $<$ CVD $\leq$ 7.60’] | 1.000 | 0.050
NGC 6316 |  [’TVL $>$ 61963.19’, ’OHLR $\leq$ 3.15’, ’CSB $\leq$ 16864.96’, ’HRT $>$ 1513.56’] | 0.984 | 0.068
NGC 6333 |  [’TVL $>$ 139719.63’, ’OHLR $\leq$ 3.15’, ’CSB $\leq$ 16864.96’, ’HRT $>$ 1513.56’] | 1.000 | 0.037
NGC 6342 |  [’HRT $\leq$ 741.31’, ’4371.87 $<$ CSB $\leq$ 16864.96’, ’CVD $>$ 3.10’, ’OCR $\leq$ 2.15’] | 0.870 | 0.074
NGC 6352 |  [’HRT $\leq$ 1513.56’, ’0.39 $<$ OCR $\leq$ 2.15’, ’TVL $>$ 26722.22’, ’OHLR $\leq$ 3.15’, ’CVD $\leq$ 5.10’] | 1.000 | 0.056
HP 1 |  [’HRT $\leq$ 1513.56’, ’CVD $>$ 5.10’, ’0.39 $<$ OCR $\leq$ 2.15’, ’2.37 $<$ OHLR $\leq$ 4.54’, ’CSB $>$ 458.30’] | 0.970 | 0.056
Ter 1 |  [’HRT $\leq$ 741.31’, ’CVD $>$ 7.60’, ’61963.19 $<$ TVL $\leq$ 139719.63’, ’OCR $\leq$ 1.05’, ’OHLR $\leq$ 3.15’, ’CSB $>$ 16864.96’] | 0.936 | 0.019
Ton 2 |  [’HRT $\leq$ 1513.56’, ’OCR $>$ 1.05’, ’OHLR $\leq$ 3.15’] | 0.984 | 0.056
NGC 6401 |  [’HRT $\leq$ 741.31’, ’5.10 $<$ CVD $\leq$ 7.60’, ’0.39 $<$ OCR $\leq$ 1.05’, ’OHLR $\leq$ 4.54’, ’4371.87 $<$ CSB $\leq$ 16864.96’, ’TVL $>$ 26722.22’] | 0.969 | 0.019
Pal 6 |  [’HRT $\leq$ 741.31’, ’5.10 $<$ CVD $\leq$ 7.60’, ’0.39 $<$ OCR $\leq$ 1.05’, ’OHLR $\leq$ 2.37’, ’CSB $>$ 458.30’, ’TVL $>$ 26722.22’] | 0.982 | 0.019
UKS 1 |  [’HRT $\leq$ 741.31’, ’3.10 $<$ CVD $\leq$ 7.60’, ’0.39 $<$ OCR $\leq$ 2.15’, ’TVL $>$ 26722.22’, ’CSB $>$ 458.30’] | 0.945 | 0.062
VVV-CL001 |  [’HRT $\leq$ 741.31’, ’5.10 $<$ CVD $\leq$ 7.60’, ’OCR $>$ 0.39’, ’CSB $>$ 4371.87’] | 0.981 | 0.025
NGC 6553 |  [’TVL $>$ 61963.19’, ’OHLR $\leq$ 2.37’, ’CSB $\leq$ 16864.96’] | 1.000 | 0.012
NGC 6569 |  [’HRT $\leq$ 1513.56’, ’CVD $>$ 5.10’, ’OHLR $>$ 2.37’, ’CSB $>$ 4371.87’, ’OCR $>$ 1.05’] | 0.992 | 0.006
NGC 6637 |  [’HRT $\leq$ 1513.56’, ’5.10 $<$ CVD $\leq$ 7.60’, ’TVL $>$ 61963.19’, ’CSB $>$ 4371.87’, ’OHLR $>$ 2.37’] | 0.993 | 0.031
NGC 6638 |  [’HRT $\leq$ 741.31’, ’CVD $>$ 5.10’, ’OCR $>$ 0.39’, ’OHLR $\leq$ 2.37’] | 0.983 | 0.025
NGC 6712 |  [’HRT $\leq$ 741.31’, ’3.10 $<$ CVD $\leq$ 5.10’, ’0.39 $<$ OCR $\leq$ 2.15’, ’OHLR $\leq$ 3.15’, ’TVL $>$ 26722.22’] | 1.000 | 0.031
NGC 6779 |  [’HRT $\leq$ 1513.56’, ’5.10 $<$ CVD $\leq$ 7.60’, ’OCR $>$ 0.39’, ’OHLR $\leq$ 3.15’, ’TVL $>$ 61963.19’] | 1.000 | 0.037
NGC 6934 |  [’TVL $>$ 61963.19’, ’OHLR $\leq$ 3.15’, ’OCR $>$ 1.05’] | 1.000 | 0.031

###  4.1 Physical interpretation of the CORELS rule list and of the anchors

To be classified as an IMBH host by CORELS, a GC has to meet the following
conditions: $T_{r}<2$ Gyr, $R_{h}<3.0$ pc, $L>10^{4}$ $M_{\odot}$, and
$R_{c}/R_{h}>0.2$. The first rule selects dynamically relaxed GCs. The second
and third rules, in combination, correspond to a condition on cluster density.
The last rule selects GCs whose core is inflated with respect to the overall
size of the cluster. All of these rules can be understood in terms of IMBH
formation physics: mass segregation (which brings heavy stars and compact
remnants to the cluster centre where they can merge to give rise to an IMBH)
happens on a timescale proportional to the two-body relaxation time. The
typical age of a GC is of the order of $\approx 12$ Gyr, so clusters with
relaxation times of 2 Gyr or shorter had ample time to undergo mass
segregation.

High density (captured by the requirement to have a small radius for a minimum
mass) is instrumental in increasing the frequency of close encounters that can
result in IMBH formation. Finally, a dynamically puffed-up core was identified
as a sign of IMBH presence early on (Baumgardt et al., 2005; Hurley, 2007;
Heggie et al., 2007), as IMBHs tend to undergo dynamical exchange reactions
with stellar-mass black hole (BH) binaries, forming an IMBH-BH binary that is
confined to the core by its heavy mass and releases energy in the core through
close encounters, heating it up. Additionally, close interactions with the
IMBH can also result in stars merging with it, or they can also get ejected
out of the central part of the cluster due to strong encounters. These effects
contribute in depleting stars from the central part of the cluster.

There is some overlap between the CORELS rules and the anchors listed in Tab.
3, suggesting that the latter may also have a similar physical interpretation.
However, the anchors explaining the behavior of XGBoost often include a large
number of features (even all of them at times) and a disparate array of
cutoffs, making a straightforward interpretation harder. These are still a
potentially useful tool for investigating a specific candidate in the process
of planning a follow-up observation.

##  5 Summary

In the recent years, we witnessed an explosion of ML tools in astronomy.
Still, most contributions avoid or only briefly touch upon the issue of
interpretability and explainability of the models they adopt. In this paper,
we distinguish between the two approaches, with interpretability meaning that
a model is natively simple enough that a human can understand its inner
workings and explainability meaning that a complex or otherwise opaque model
is explained post-hoc in terms that a human can understand. This is a
terminological distinction that has not yet been received by the astronomical
community, and the trickle of works that address this issue at some level of
depth often conflates the two, using interpretability as an umbrella term.

With this in mind, in the present work, we have applied both kinds of
techniques, comparing a natively interpretable model -CORELS- with a black-
box-plus-explanation model (XGBoost explained using the local model agnostic
explanation rules known as anchors) on the problem of IMBH detection in GCs.
We trained and validated both models on features measured on GC simulations,
and deployed the models in prediction on actual GCs. The features we selected
are standard, easily measurable characteristics of GCs and are in fact readily
available in multiple catalogues. The meaning of each feature is clearly
understood in theoretical terms by astronomers.

We produced a list of GCs that are likely to contain an IMBH according to both
models, presenting for each one the anchor rule, the confidence score from
XGBoost, and a measure of how far out of distribution each actual GC is with
respect to the training data. We compare the rule list learned from the
intepretable CORELS model to the anchor rules and conclude that there is a
qualitative resemblance but the latter are usually more complex. Moreover, the
rule list learned by CORELS has an immediate physical interpretation, some
aspects of which are shared with a subset of the anchor rules.

We evaluated the performance of both CORELS and XGBoost on a test set
comprised of simulations not seen in training. We find that the CORELS model
we learned, based on a rule list comprised of four rules, achieves $90\%$
precision at $\approx 50\%$ recall, whereas the precision achieved by XGBoost
at the same recall is nominally perfect ($100\%$). However, this comes at the
cost of relying on a black box, which is only moderately mitigated by post-hoc
explanations.

Despite the simulator’s best efforts, our training data and actual GCs are
characterized by distributions that do not perfectly overlap in feature space.
This raises the issue of whether we can trust the predictions of models
trained on simulations and deployed on observational data. In this context,
understanding the inner workings of a model provides a crucial advantage: we
can decide whether to trust it based on how its behavior matches our subject-
matter knowledge, i.e. the physics of IMBH formation. Still, as an additional
safeguard, we removed from our final candidate list GCs that were too far out
of distribution with respect to the training data. This was achieved by
estimating the density of training data in feature space with KDE and placing
a relevant cutoff. This procedure also implicitly yields a measure of the
realism of the simulations we considered, with respect to the actual
observational data, and taken as a whole. The predictions cast by our models
with respect to actual GCs cannot be directly tested, since we do not know
which GCs host an IMBH. However, they can be used as guidance for further
investigation, knowing that they are based on an interpretable model which is
not working in extrapolation.

## Acknowledgments

M. P. acknowledges financial support from the European Union’s Horizon 2020
research and innovation program under the Marie Skłodowska-Curie grant
agreement No. 896248. This research was made possible by a generous donation
from Eric and Wendy Schmidt by recommendation of the Schmidt Futures program.
Y.H. acknowledges support from the National Sciences and Engineering Council
of Canada Discovery Grant RGPIN-2020-1505 and the Canada Research Chairs
Program. MM acknowledges financial support from the European Research Council
for the ERC Consolidator grant DEMOBLACK, under contract no. 770017 and from
the German Excellence Strategy via the Heidelberg Cluster of Excellence (EXC
2181 - 390900948) STRUCTURES. AA acknowledges support for this paper from
project No. 2021/43/P/ST9/03167 co-funded by the Polish National Science
Center (NCN) and the European Union Framework Programme for Research and
Innovation Horizon 2020 under the Marie Skłodowska-Curie grant agreement No.
945339\. For the purpose of Open Access, the authors have applied for a CC-BY
public copyright licence to any Author Accepted Manuscript (AAM) version
arising from this submission. AA also acknowledges support from the Swedish
Research Council through the grant 2017-04217 and from NCN through the grant
UMO-2021/41/B/ST9/01191.

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##  Appendix A Full list of candidates

Cluster | Final score | HRT | TVL | CVD | CSB | OCR | OHLR | CORELS | XGB | KDE | Pred. prob.
---|---|---|---|---|---|---|---|---|---|---|---
NGC_6569 | 0.9999 | 3.06 | 5.02 | 0.86 | 3.68 | 0.04 | 0.41 | True | True | True | 0.9999
Pal_6 | 0.9998 | 2.67 | 4.79 | 0.72 | 3.64 | -0.02 | 0.35 | True | True | True | 0.9998
NGC_6638 | 0.9998 | 2.49 | 4.9 | 0.84 | 4.21 | -0.34 | 0.27 | True | True | True | 0.9998
NGC_6333 | 0.9996 | 3.22 | 5.18 | 0.92 | 3.9 | -0.04 | 0.44 | True | True | True | 0.9996
NGC_6712 | 0.9988 | 2.73 | 4.78 | 0.71 | 3.54 | 0.03 | 0.4 | True | True | True | 0.9988
NGC_6254 | 0.9985 | 3.25 | 5.04 | 0.79 | 3.65 | 0.02 | 0.47 | True | True | True | 0.9985
FSR_1735 | 0.9969 | 2.63 | 4.69 | 0.67 | 3.83 | -0.29 | 0.33 | True | True | True | 0.9969
NGC_2298 | 0.9955 | 2.65 | 4.47 | 0.59 | 3.55 | -0.28 | 0.38 | True | True | True | 0.9955
NGC_5986 | 0.9954 | 3.25 | 5.24 | 0.91 | 3.83 | 0.07 | 0.44 | True | True | True | 0.9954
NGC_6218 | 0.9947 | 2.94 | 4.75 | 0.68 | 3.34 | 0.08 | 0.45 | True | True | True | 0.9947
NGC_6316 | 0.9913 | 3.27 | 5.09 | 0.91 | 4.04 | -0.22 | 0.47 | True | True | True | 0.9913
VVV-CL001 | 0.9889 | 2.75 | 4.54 | 0.81 | 3.65 | -0.27 | 0.34 | True | True | True | 0.9889
Ton_2 | 0.9889 | 3.01 | 4.51 | 0.53 | 3.04 | 0.06 | 0.46 | True | True | True | 0.9889
NGC_6352 | 0.9871 | 2.96 | 4.47 | 0.54 | 3.06 | 0.05 | 0.46 | True | True | True | 0.9871
NGC_6779 | 0.9845 | 3.17 | 5.03 | 0.78 | 3.59 | 0.07 | 0.47 | True | True | True | 0.9845
NGC_6171 | 0.9843 | 2.87 | 4.53 | 0.61 | 3.32 | -0.08 | 0.46 | True | True | True | 0.9843
NGC_6553 | 0.9837 | 3.26 | 4.95 | 0.88 | 3.67 | -0.06 | 0.37 | True | True | True | 0.9837
NGC_6934 | 0.9815 | 3.26 | 4.95 | 0.68 | 3.46 | 0.09 | 0.47 | True | True | True | 0.9815
NGC_6637 | 0.9724 | 2.95 | 4.95 | 0.79 | 3.76 | -0.07 | 0.38 | True | True | True | 0.9724
UKS_1 | 0.9518 | 2.85 | 4.63 | 0.62 | 3.29 | 0.04 | 0.46 | True | True | True | 0.9518
Ter_1 | 0.9502 | 2.54 | 4.94 | 0.91 | 4.44 | -0.49 | 0.18 | True | True | True | 0.9502
NGC_6342 | 0.9288 | 2.22 | 4.36 | 0.64 | 3.85 | -0.47 | 0.17 | True | True | True | 0.9288
HP_1 | 0.8894 | 2.89 | 4.58 | 0.72 | 3.18 | 0.1 | 0.46 | True | True | True | 0.8894
NGC_6401 | 0.5381 | 2.86 | 4.65 | 0.81 | 3.73 | -0.28 | 0.39 | True | True | True | 0.5381
NGC_6440 | 0 | 2.78 | 5.5 | 1.16 | 4.96 | -0.46 | 0.1 | True | False | False | 0.1708
Djor_2 | 0 | 3.03 | 4.54 | 0.63 | 3.04 | 0.08 | 0.63 | False | True | True | 0.9951
NGC_6539 | 0 | 3.22 | 5.04 | 0.77 | 3.46 | 0.18 | 0.57 | False | True | True | 0.9998
NGC_6535 | 0 | 2.54 | 3.92 | 0.43 | 3.88 | -1.1 | 0.43 | False | False | True | 0.0253
NGC_6388 | 0 | 3.49 | 5.76 | 1.24 | 4.96 | -0.38 | 0.41 | False | True | False | 0.6810
NGC_7006 | 0 | 3.46 | 4.93 | 0.61 | 3.19 | 0.2 | 0.63 | False | True | True | 0.6639
NGC_6528 | 0 | 2.46 | 4.5 | 0.67 | 3.93 | -0.48 | 0.28 | False | True | True | 0.9979
Laevens_3 | 0 | 2.86 | 3.07 | -0.4 | 0.95 | 0.43 | 0.85 | False | False | False | 0.1948
NGC_6522 | 0 | 2.85 | 4.94 | 0.91 | 4.19 | -0.38 | 0.39 | False | False | True | 0.0972
Djor_1 | 0 | 3.11 | 4.64 | 0.56 | 3.22 | -0.01 | 0.61 | False | True | True | 0.9980
NGC_7078 | 0 | 3.47 | 5.6 | 1.12 | 5.85 | -1.4 | 0.31 | False | False | False | 0.0020
Ter_10 | 0 | 3.22 | 4.74 | 0.94 | 4.1 | -0.59 | 0.54 | False | False | True | 0.2129
NGC_6517 | 0 | 2.64 | 4.92 | 1 | 4.85 | -0.89 | 0.23 | False | True | False | 0.9365
Ter_9 | 0 | 2.38 | 4.57 | 0.89 | 4.23 | -0.6 | 0.2 | False | False | True | 0.1856
NGC_6402 | 0 | 3.49 | 5.45 | 1 | 3.67 | 0.33 | 0.55 | False | False | True | 0.0021
NGC_6397 | 0 | 2.94 | 4.76 | 0.72 | 5.07 | -1.4 | 0.33 | False | True | False | 0.9645
Pal_13 | 0 | 3.26 | 3.19 | -0.4 | 1.02 | 0.29 | 1.11 | False | False | False | 0.4264
NGC_6496 | 0 | 3.18 | 4.63 | 0.48 | 2.69 | 0.4 | 0.64 | False | False | True | 0.0069
NGC_7089 | 0 | 3.44 | 5.55 | 1.04 | 4.34 | -0.11 | 0.48 | False | True | True | 0.9914
NGC_6540 | 0 | 3.09 | 4.39 | 0.4 | 4.23 | -1.22 | 0.42 | False | False | True | 0.0212
Cluster | Final score | HRT | TVL | CVD | CSB | OCR | OHLR | CORELS | XGB | KDE | Pred. prob.
NGC_6453 | 0 | 3 | 5.05 | 0.86 | 4.94 | -1 | 0.44 | False | True | False | 0.9967
NGC_7099 | 0 | 3.28 | 4.87 | 0.75 | 4.8 | -1.22 | 0.4 | False | False | False | 0.0417
Ter_6 | 0 | 2.07 | 4.77 | 0.92 | 4.45 | -0.51 | 0.01 | True | True | False | 0.6464
Pal_12 | 0 | 3.11 | 3.69 | -0.15 | 1.36 | 0.59 | 0.84 | False | True | False | 0.8911
NGC_6441 | 0 | 3.33 | 5.87 | 1.28 | 5 | -0.28 | 0.32 | False | True | False | 0.8584
Ter_5 | 0 | 3.35 | 5.57 | 1.19 | 4.64 | -0.29 | 0.27 | False | True | True | 0.9572
NGC_6426 | 0 | 3.43 | 4.56 | 0.45 | 2.48 | 0.45 | 0.71 | False | False | True | 0.0035
NGC_6838 | 0 | 3.32 | 4.58 | 0.45 | 2.87 | 0.17 | 0.53 | False | False | True | 0.1592
NGC_6541 | 0 | 3.29 | 5.22 | 0.93 | 5.07 | -1.05 | 0.36 | False | False | False | 0.0490
NGC_6681 | 0 | 2.71 | 4.8 | 0.85 | 5.12 | -1.15 | 0.33 | False | True | False | 0.9957
Pal_11 | 0 | 2.94 | 3.98 | 0.08 | 1.75 | 0.61 | 0.76 | False | True | False | 0.9989
Ter_8 | 0 | 4.12 | 4.54 | 0.2 | 1.47 | 1.03 | 1.18 | False | False | False | 0.0024
NGC_6809 | 0 | 3.55 | 4.97 | 0.69 | 2.95 | 0.46 | 0.66 | False | False | True | 0.0001
Arp_2 | 0 | 3.77 | 4.31 | 0.11 | 1.43 | 0.89 | 1.15 | False | False | False | 0.0093
Pal_10 | 0 | 3.35 | 4.94 | 0.66 | 3.09 | 0.32 | 0.6 | False | False | True | 0.0040
Ter_7 | 0 | 3.69 | 4.05 | 0.08 | 1.57 | 0.63 | 0.8 | False | False | False | 0.0116
NGC_6760 | 0 | 3.28 | 5.17 | 0.83 | 3.72 | 0.03 | 0.51 | False | True | True | 0.9649
NGC_6752 | 0 | 3.4 | 5.07 | 0.88 | 4.83 | -0.89 | 0.46 | False | True | True | 0.6079
NGC_6749 | 0 | 3.51 | 5.18 | 0.71 | 3.47 | 0.16 | 0.66 | False | True | True | 0.9098
NGC_6723 | 0 | 3.21 | 4.94 | 0.74 | 3.3 | 0.23 | 0.55 | False | False | True | 0.1026
NGC_6717 | 0 | 2.73 | 4.36 | 0.52 | 3.9 | -0.7 | 0.56 | False | False | True | 0.1662
NGC_6715 | 0 | 3.65 | 5.93 | 1.28 | 4.93 | -0.27 | 0.55 | False | True | False | 0.9831
NGC_6864 | 0 | 2.95 | 5.35 | 1.05 | 4.69 | -0.42 | 0.31 | False | True | True | 0.9711
RLGC_2 | 0 | 3.23 | 5.13 | 0.88 | 4.76 | -0.62 | 0.53 | False | True | True | 0.9972
Pal_8 | 0 | 3.15 | 4.03 | 0.49 | 2.25 | 0.29 | 0.54 | False | False | True | 0.0025
NGC_6544 | 0 | 2.36 | 4.55 | 0.85 | 4.49 | -0.8 | 0.18 | False | False | False | 0.2371
NGC_6656 | 0 | 3.52 | 5.37 | 0.95 | 3.99 | 0.02 | 0.5 | False | True | True | 0.9654
NGC_6652 | 0 | 2.19 | 4.43 | 0.69 | 4.1 | -0.59 | 0.17 | False | False | True | 0.3098
NGC_6642 | 0 | 1.92 | 4.45 | 0.68 | 4.41 | -0.77 | 0.13 | False | False | False | 0.3241
NGC_6626 | 0 | 2.72 | 5.15 | 1.05 | 4.67 | -0.54 | 0.2 | False | True | True | 0.9927
NGC_6624 | 0 | 2.91 | 4.87 | 0.85 | 4.37 | -0.48 | 0.37 | False | False | True | 0.3324
NGC_6584 | 0 | 3.2 | 4.88 | 0.6 | 3.24 | 0.19 | 0.54 | False | True | True | 0.7763
BH_261 | 0 | 2.92 | 3.45 | 0.32 | 2.57 | -0.47 | 0.42 | False | False | True | 0.0000
NGC_6981 | 0 | 3.15 | 4.71 | 0.49 | 2.89 | 0.32 | 0.62 | False | False | True | 0.1884
Ter_12 | 0 | 2.89 | 4.37 | 0.63 | 3.15 | -0.08 | 0.24 | True | False | True | 0.4512
IC_1276 | 0 | 3.07 | 4.61 | 0.56 | 3.12 | 0.07 | 0.5 | False | True | True | 0.9385
NGC_6558 | 0 | 1.95 | 4.36 | 0.61 | 4.33 | -0.77 | 0.17 | False | True | False | 0.6577
2MASS-GC02 | 0 | 2.31 | 3.94 | 0.34 | 2.98 | -0.2 | 0.34 | False | False | True | 0.3450
ESO_280-SC06 | 0 | 3.08 | 3.66 | -0.1 | 1.57 | 0.38 | 0.85 | False | True | False | 0.5992
Sagittarius_II | 0 | 4.23 | 4.1 | -0.22 | 0.47 | 1.31 | 1.46 | False | False | False | 0.0020
2MASS-GC01 | 0 | 2.81 | 4.29 | 0.43 | 3.28 | -0.3 | 0.55 | False | False | True | 0.0202
NGC_104 | 0 | 3.73 | 5.66 | 1.08 | 4.34 | -0.1 | 0.56 | False | True | True | 0.9550
NGC_6380 | 0 | 3.2 | 5.06 | 0.91 | 3.68 | 0.05 | 0.5 | False | True | True | 0.9987
NGC_4147 | 0 | 2.74 | 4.38 | 0.51 | 3.62 | -0.44 | 0.4 | False | False | True | 0.0800
Rup_106 | 0 | 3.58 | 4.39 | 0.2 | 1.9 | 0.71 | 0.88 | False | False | True | 0.0298
NGC_4590 | 0 | 3.53 | 4.83 | 0.57 | 2.96 | 0.29 | 0.65 | False | False | True | 0.0003
BH_140 | 0 | 3.48 | 4.38 | 0.38 | 2 | 0.68 | 0.79 | False | False | True | 0.0008
NGC_4833 | 0 | 3.22 | 5.07 | 0.79 | 3.49 | 0.19 | 0.51 | False | True | True | 0.9994
Cluster | Final score | HRT | TVL | CVD | CSB | OCR | OHLR | CORELS | XGB | KDE | Pred. prob.
NGC_5024 | 0 | 3.98 | 5.41 | 0.8 | 3.36 | 0.36 | 0.81 | False | False | True | 0.0005
NGC_5053 | 0 | 3.97 | 4.46 | 0.3 | 1.49 | 1.01 | 1.09 | False | False | False | 0.0000
NGC_5139 | 0 | 4.4 | 6.15 | 1.25 | 3.82 | 0.63 | 0.88 | False | False | False | 0.0127
NGC_5272 | 0 | 3.66 | 5.4 | 0.88 | 3.87 | 0.09 | 0.53 | False | True | True | 0.9363
NGC_5286 | 0 | 3.18 | 5.38 | 0.99 | 4.6 | -0.4 | 0.37 | False | False | True | 0.4562
AM_4 | 0 | 3.18 | 2.32 | -0.7 | 0.12 | 0.33 | 0.8 | False | True | False | 0.7439
NGC_5466 | 0 | 3.81 | 4.62 | 0.28 | 1.93 | 0.81 | 0.98 | False | False | False | 0.0001
NGC_5634 | 0 | 3.57 | 5.02 | 0.75 | 3.79 | -0.12 | 0.65 | False | True | True | 0.9919
NGC_5694 | 0 | 3.29 | 5.18 | 0.96 | 5.38 | -1.4 | 0.49 | False | True | False | 0.8416
IC_4499 | 0 | 4.02 | 4.83 | 0.48 | 2.22 | 0.71 | 1.01 | False | False | True | 0.0000
NGC_5824 | 0 | 3.66 | 5.57 | 1.05 | 4.42 | -0.18 | 0.65 | False | True | True | 0.9533
NGC_4372 | 0 | 3.68 | 4.98 | 0.65 | 2.74 | 0.58 | 0.76 | False | False | True | 0.0002
Crater | 0 | 3.78 | 3.97 | -0.22 | 0.74 | 1.1 | 1.29 | False | False | False | 0.0562
NGC_5897 | 0 | 3.82 | 4.85 | 0.54 | 2.34 | 0.74 | 0.88 | False | False | True | 0.0000
Pal_4 | 0 | 3.72 | 4.28 | -0.15 | 1.17 | 1.06 | 1.2 | False | False | False | 0.1473
NGC_362 | 0 | 3.16 | 5.29 | 0.93 | 4.49 | -0.4 | 0.34 | False | True | True | 0.7864
Whiting_1 | 0 | 3.18 | 3.6 | -0.4 | 1.36 | 0.39 | 0.98 | False | True | False | 0.5067
NGC_1261 | 0 | 3.3 | 5.05 | 0.76 | 3.79 | -0.06 | 0.51 | False | True | True | 0.8986
Pal_1 | 0 | 1.89 | 2.63 | -0.3 | 1.56 | -0.16 | 0.26 | False | True | False | 0.9337
AM_1 | 0 | 3.63 | 4.37 | -0.05 | 1.54 | 0.82 | 1.17 | False | False | False | 0.1989
Eridanus | 0 | 3.48 | 4.1 | -0.15 | 1.29 | 0.83 | 1.14 | False | False | False | 0.3368
Pal_2 | 0 | 3.67 | 5.24 | 0.71 | 3.44 | 0.21 | 0.68 | False | False | True | 0.1936
NGC_1851 | 0 | 2.95 | 5.28 | 1.04 | 5.32 | -1.1 | 0.24 | False | True | False | 0.6323
NGC_1904 | 0 | 2.81 | 5 | 0.81 | 4.3 | -0.38 | 0.4 | False | True | True | 0.8813
NGC_2419 | 0 | 4.63 | 5.69 | 0.76 | 2.66 | 0.91 | 1.29 | False | False | False | 0.0000
Pyxis | 0 | 3.9 | 4.13 | 0 | 0.93 | 1.13 | 1.23 | False | False | False | 0.0135
NGC_2808 | 0 | 3.37 | 5.76 | 1.15 | 4.62 | -0.1 | 0.39 | False | True | False | 0.8892
E_3 | 0 | 2.56 | 3.35 | -0.22 | 1.59 | 0.26 | 0.63 | False | True | False | 0.9953
Pal_3 | 0 | 3.95 | 4.11 | -0.1 | 0.8 | 1.16 | 1.31 | False | False | False | 0.0008
NGC_3201 | 0 | 3.51 | 4.87 | 0.66 | 3.33 | 0.09 | 0.58 | False | False | True | 0.0182
Pal_5 | 0 | 3.87 | 3.91 | -0.22 | 0.57 | 1.17 | 1.31 | False | False | False | 0.0028
NGC_5904 | 0 | 3.51 | 5.34 | 0.9 | 3.86 | 0.07 | 0.55 | False | True | True | 0.6028
Liller_1 | 0 | 2.85 | 5.53 | 1.34 | 5.33 | -0.72 | 0.05 | False | False | False | 0.1790
NGC_6266 | 0 | 2.91 | 5.49 | 1.18 | 4.85 | -0.41 | 0.25 | True | True | False | 0.9919
NGC_6284 | 0 | 2.89 | 4.99 | 0.77 | 4.14 | -0.33 | 0.47 | False | True | True | 0.9988
NGC_6287 | 0 | 3.01 | 4.71 | 0.8 | 4.21 | -0.68 | 0.29 | False | True | True | 0.7655
NGC_6293 | 0 | 3.16 | 4.91 | 0.88 | 4.61 | -0.89 | 0.37 | False | False | True | 0.4187
NGC_6304 | 0 | 3.1 | 4.9 | 0.74 | 4.38 | -0.66 | 0.29 | False | True | True | 0.7501
NGC_6325 | 0 | 2.24 | 4.45 | 0.72 | 4.05 | -0.51 | 0.23 | False | False | True | 0.3542
NGC_6341 | 0 | 3.39 | 5.21 | 0.92 | 4.01 | -0.11 | 0.38 | False | True | True | 0.6153
NGC_6355 | 0 | 2.86 | 4.79 | 0.74 | 4.34 | -0.68 | 0.37 | False | False | True | 0.0326
NGC_6356 | 0 | 3.72 | 5.34 | 0.94 | 3.64 | 0.17 | 0.59 | False | True | True | 0.5412
IC_1257 | 0 | 2.85 | 4.16 | 0.23 | 2.66 | 0.06 | 0.61 | False | True | True | 0.7621
Ter_2 | 0 | 3.03 | 4.78 | 0.76 | 3.95 | -0.38 | 0.39 | False | True | True | 0.8764
Ter_4 | 0 | 3.43 | 4.85 | 0.74 | 3.67 | -0.24 | 0.54 | False | False | True | 0.4998
NGC_288 | 0 | 3.47 | 4.64 | 0.49 | 2.43 | 0.56 | 0.77 | False | False | True | 0.0024
FSR_1758 | 0 | 4.41 | 5.6 | 0.76 | 2.63 | 1.03 | 1.1 | False | False | False | 0.0000
NGC_6362 | 0 | 3.42 | 4.83 | 0.59 | 2.78 | 0.46 | 0.71 | False | False | True | 0.0010
Cluster | Final score | HRT | TVL | CVD | CSB | OCR | OHLR | CORELS | XGB | KDE | Pred. prob.
NGC_6366 | 0 | 2.99 | 4.34 | 0.38 | 2.58 | 0.29 | 0.58 | False | False | True | 0.0010
NGC_6273 | 0 | 3.3 | 5.49 | 1.08 | 4.19 | 0 | 0.5 | False | True | True | 0.9990
Pal_15 | 0 | 4.1 | 4.16 | 0.11 | 0.86 | 1.16 | 1.3 | False | False | False | 0.0003
NGC_5927 | 0 | 3.31 | 5.23 | 0.83 | 3.68 | 0.15 | 0.54 | False | True | True | 0.6543
NGC_6256 | 0 | 3.15 | 4.77 | 0.73 | 4.9 | -1.4 | 0.47 | False | True | False | 0.8825
NGC_5946 | 0 | 2.59 | 4.77 | 0.78 | 4.22 | -0.51 | 0.29 | False | True | True | 0.9940
FSR_1716 | 0 | 3.01 | 3.84 | 0.52 | 2.13 | 0.28 | 0.56 | False | False | True | 0.0063
Pal_14 | 0 | 4.1 | 4.05 | -0.15 | 0.55 | 1.21 | 1.44 | False | False | False | 0.0006
Lynga_7 | 0 | 2.99 | 4.68 | 0.56 | 2.91 | 0.33 | 0.61 | False | False | True | 0.2345
NGC_6093 | 0 | 2.86 | 5.21 | 1.04 | 4.53 | -0.39 | 0.25 | True | False | True | 0.4276
RLGC_1 | 0 | 3.9 | 5.22 | 0.66 | 3 | 0.45 | 0.9 | False | False | True | 0.0000
NGC_6101 | 0 | 4.04 | 4.84 | 0.52 | 2.21 | 0.77 | 0.98 | False | False | True | 0.0000
NGC_6121 | 0 | 2.87 | 4.74 | 0.68 | 3.96 | -0.41 | 0.4 | False | True | True | 0.6485
NGC_6139 | 0 | 2.79 | 5.27 | 1.06 | 4.87 | -0.57 | 0.27 | False | True | False | 0.8153
NGC_6144 | 0 | 3.01 | 4.5 | 0.57 | 2.88 | 0.21 | 0.56 | False | True | True | 0.7837
Ter_3 | 0 | 3.17 | 4.36 | 0.34 | 2.32 | 0.47 | 0.67 | False | True | True | 0.5244
ESO_452-SC11 | 0 | 2.41 | 3.53 | 0.15 | 2.42 | -0.14 | 0.4 | False | False | True | 0.2605
NGC_6205 | 0 | 3.52 | 5.37 | 0.97 | 3.66 | 0.28 | 0.54 | False | False | True | 0.0004
NGC_6229 | 0 | 3.21 | 5.17 | 0.88 | 3.8 | 0.02 | 0.49 | False | True | True | 0.9999
NGC_6235 | 0 | 3.09 | 4.62 | 0.64 | 3.15 | 0.08 | 0.52 | False | True | True | 0.9363
NGC_7492 | 0 | 3.29 | 4.29 | 0.18 | 1.89 | 0.67 | 0.88 | False | True | True | 0.8244
Table A1: Whole sample of star clusters including deemed non-hosts. The
ordering of the star clusters in the table is decreasing with ‘final score’
(column 2). This is the XGBoost predicted probability for the star clusters
that were classified as hosts by CORELS (column 9) and XGBoost (column 10) and
were deemed in-distribution based on our KDE criterion (column 11). We also
report cluster features (columns 3–8).