MANM-CS: Data Generation for Benchmarking
Causal Structure Learning from
Mixed Discrete-Continuous and Nonlinear Data
Johannes Huegle, Christopher Hagedorn, Lukas Böhme*, Mats Pörschke*,Jonas Umland*, Rainer Schlosser
Hasso Plattner Institute, University of Potsdam, Germany{firstname.lastname}@hpi.de, *{firstname.lastname}@student.hpi.de

Abstract
In recent years, the growing interest in methods of causal structure learning (CSL)
has been confronted with a lack of access to a well-defined ground truth within realworld scenarios to evaluate these methods. Commonly used synthetic benchmarks
are limited in their scope as they are either restricted to a “static” low-dimensional
data set or do not allow examining mixed discrete-continuous or nonlinear data.
This work introduces the mixed additive noise model that provides a ground truth
framework for generating observational data following various distribution models.
Moreover, we present our reference implementation MANM-CS that provides easy
access and demonstrate how our framework supports researchers and practitioners.
Further, we propose future research directions and possible extensions.

1

Introduction and Background

Methods of causal structure learning (CSL) have received widespread attention in the scientificfield as the knowledge of underlying causal structures is the basis for decision support within many
real-world scenarios [40]. In recent years, the corresponding research addressing challenges of CSL
in practice has led to a broad spectrum of different methods (see Sec. 1.1). In this context, theevaluation of CSL methods encounter requirements concerning well-defined benchmark data, e.g., for
mixed or nonlinear data (Sec. 1.2). Therefore, we propose the mixed additive noise model (MANM)
to establish a flexible yet well-defined ground-truth model, allowing the data generation under various
evaluation perspectives (Sec. 1.3).
1.1

Causal Structure Learning and Challenges

In CSL, the following standard notation is used. The causal structures between a finite set of p
random variables V = {V1 , . . . , Vp } are encoded in a causal graphical model (CGM) consisting of adirected acyclic graph (DAG) G, where directed edges Vj → Vi depict a direct causal relationshipbetween two respective nodes Vj and Vi , i, j = 1, . . . , p, and the joint distribution over the variables
V, denoted by P V , e.g., cf. [27, 40].
Within this framework, CSL aims to derive as many of the underlying causal relationships in G
from independent and identically distributed observational data as possible. Therefore, methods of
CSL either leverage probabilistic characteristics of the variables’ joint probability distribution P V orprescribe a specific functional causal model (FCM) to the relation between variables, e.g., cf. [6]. In
this context, the causal Markov condition states a coincidence between the causal structure within G
and the conditional independence (CI) characteristics of the joint distribution P V [40]. On this basis,
probabilistic methods either exploit CI tests, called constraint-based, or optimize a score function over35th Conference on Neural Information Processing Systems (NeurIPS 2021), Sydney, Australia.

the space of equivalence classes, called score-based to recover the causal structures [40]. In contrast,
algorithms based on properly defined FCMs benefit from additional but restrictive assumptions andcan distinguish between different DAGs within the same equivalence class [6]. In this context, a
direct edge Vj → Vi in G encodes that Vi is a function of its cause Vj and some noise term Ni
independent to Vj , i.e., Vi = fi (Vj , Ni ) where fi is assumed to be a function from an appropriately
constraint functional class F, e.g., linearity with additive noise [14].
While all CSL methods require that several assumptions hold, observational data of real-world
scenarios often violates the constraints made for CSL. For example, in practice, it may be impossible to observe all variables to ensure causal sufficiency, i.e., that there are no latent confounding
variables [40]. Moreover, real-world data often does not follow a simple functional form but includes nonlinear and mixed discrete-continuous relationships [6, 23]. Therefore, a wide spectrumof scientific publications focus on different extensions to improve the accuracy under weakened
constraints, e.g., given latent variables [4, 39], assuming a nonlinear function fi within FCMs [9, 46]
or considering CSL from mixed discrete continuous data [1, 10, 36, 42].
1.2

Requirements on Modeling Causal Structures for Benchmarking CSL Methods

As (novel) CSL methods are commonly evaluated against their own synthetic benchmarks andcompared within a limited scope, e.g., [30, 38, 43], it is difficult to compare individual methods
against each other; in particular, if they may require different assumptions. In this context, Glymour
et al. [6] summarized the current state as follows: "There are multiple algorithms available, manyof them are poorly tested [...] all of them have choices of parameters [...], and all of them haveconditions on the data distributions and other assumptions under which they will be informative rather
than misleading". Hence, methods for CSL should be validated within different scenarios, including avarying number of variables or sensitivity of parameters, aiming to understand the method’s behaviors
in specific edge cases, e.g., when underlying assumptions on the causal relationships are violated [17].
Therefore, a thorough evaluation of CSL methods requires the introduction of an easily customizable
framework for generating observational data supplemented with precise definitions of underlying
causal structures that connects and extends existing ideas, see Sec. 2. In particular, a data generating
model should satisfy the following requirements:(R1) be formalized as a FCM to ensure interpretability, e.g., concerning causal inference;(R2) allow for continuous, discrete, and mixed discrete-continuous causal relationships;(R3) be flexible and easily extendable, e.g., to allow for interventional data;(R4) be implemented as an easy-to-use open access package.
1.3

Contribution

In our work, we propose the mixed additive noise model (MANM) as a flexible yet easy-to-use
synthetic data generation process for benchmarking CSL methods under a wide range of conditions.
Our main contributions can be summarized as follows: First, we introduce the MANM as an FCM
to model causal structures within various distribution models from discrete over mixed discretecontinuous to nonlinear, cf. (R1)-(R3). Second, to provide easy access to the research community, we
present our reference implementation, called MANM-CS, cf. (R4). Third, we demonstrate the usabilityof MANM-CS in comparison to the well-known benchmark data sets and in a simple benchmarking
experiment on the accuracy of CSL from mixed discrete-continuous and nonlinear data.
The remainder of the paper is structured as follows. We consider related work on available benchmarking methods of CSL in Section 2. In Section 3, we introduce the MANM as a benchmarking
framework and demonstrate its application within several common scenarios. We present our reference implementation MANM-CS in Section 4 and its application for benchmarking in Section 5. In
Section 6, we conclude our work, point out limitations, and discuss future directions.

2

Related Work on Benchmarking of CSL Methods

Commonly, data for evaluating CSL methods is generated according to the following approaches:(I) predefined benchmark data sets supplemented by an expected ground truth;(II) well-established parameterized benchmark models to generate data; and(III) flexible models based upon a probabilistic or functional formalization.
2

ApproachContinuous

Mixed

Discrete(I)[24], [25]‡ , [33]

[7]

[31]

(II)

[26], [32], [35][22], [34], [44][2], [3], [5], [15], [16]

‡

§

(III)
[9] , [13][1] , [13], [18]*[29]‡
Table 1: Implementations for modeling continuous, mixed, or discrete data based upon (I) predefinedbenchmark data sets, (II) well-established parameterized models, and (III) functional or probabilistic
models (*undirected; ‡ two-variable case; § discretized auxiliary variables).

In Table 1, we recap a selection of the above approaches that has been used for evaluation within workon CSL, e.g., [1, 30, 36, 42, 43]. We do not claim completeness but focus on the most well-known
and representative data sets or models. In this context, currently used models and data sets of thethree approaches come with limitations that restrict the evaluation opportunities.
Predefined benchmark data sets (I) allow a direct comparison given a common and enclosed groundtruth model. However, they do not allow for performance comparison concerning a varying complexity
or data set size. For example, the "DREAM5 SYSGEN A - In-silico network challenge" [24] is based on
simulated gene expression data from [20] restricted to 1000 variables and sample sizes of n = 100,n = 300, or n = 999. To allow for performance evaluation of large sample properties, models from(II) sample observational data from well-established “static” parameterized models with fixed model
complexity. While this allows to evaluate and compare CSL within the provided distribution andmodel assumption, they do not allow for an examination given a varying model complexity. For
example, within the mixed case, the well-known conditional Gaussian distributed MEHRA model from[44] is restricted to 24 nodes that incorporate a mixture of 8 discrete and 16 continuous variables.
Further, within the discrete case, the ALARM model from [2] fixes the number of possible discretevalues each variable can take to the model’s assumptions. Therefore, following approach (III), mostCSL methods are evaluated within their respective scenarios, e.g., linear relationships with i.i.d.
Gaussian noise, cf. [13], the mixed graphical model (MGM) of [18], or the two-variable case, cf.,
[9, 28]. As a basis for a comprehensive evaluation, these models are quite restrictive or vary strongly
on their assumptions, e.g., the solely undirected edges of the mixed MGM model [18], and require
manual implementation overhead as they are often not open accessible.
In summary, there exist numerous different data sets and models that allow for examining theperformance of CSL methods within their specific scenarios. Apart from that, they do not allow thegeneration of observational data with varying complexity needed within a comprehensive accuracy
examination of CSL. Especially, as this requires to vary the model complexity, e.g., concerning
the number of considered variables, the ratio of discrete nodes, or the number of possible discretevalues, cf. (R2). Moreover, currently, no common framework provides a functional formalizationthat ensures interpretability cf. (R1). In contrast, we aim to establish a flexible yet well-defined andunified ground-truth model, allowing the generation of observational data under various evaluation
perspectives.

3

The Mixed Additive Noise Model (MANM)

In the following, we introduce the MANM as a framework for generating causal structures with mixed
discrete-continuous and nonlinear relationships. In Sec. 3.1, we define the MANM in its functional
form, cf. (R1), and provide some exemplary distribution models from continuous (Sec. 3.2), discrete(Sec. 3.3), and mixed discrete-continuous space (Sec. 3.4).
3.1

MANM for Modeling Mixed Discrete-Continuous and Nonlinear Data

In general, we say that variables Vi , i = 1, . . . , p, (cf. Sec. 1.1) of a probability space (Ω, A, P) are
discrete if they have a (finite) discrete domain, i.e., where Vi : Ω → Zi ⊆ R with countable subset
Zi , or continuous if they have a continuous domain, i.e., where Vi : Ω → R, such that all Vi haveLebesgue measurable domains in A. Modeling causal relationships requires to define a FCM thatgenerates a variable Vi ∈ V according to the sets of possible discrete or continuous parents of Vi
in G, denoted by P dis (Vi ) and P con (Vi ), respectively. Therefore, we introduce the mixed additive3

noise model (MANM) with mutually independent noise Ni where Ni ⊥⊥ Vj for all Vj ∈ V asX
X
Vi =fj,i (Vj ) +fk,i (Vk ) + Ni , for all Vi ∈ V,
Vj ∈P dis (Vi )(1)

Vk ∈P con (Vi )with functions fj,i : Zj → Zi and fk,i : R → Zi if Vi has a discrete domain Zi , or fj,i : Zj → R
and fk,i : R → R if Vi has a continuous domain. Moreover, we require that the independent
noise variable Ni either is a continuous distributed random variable, e.g., Ni ∼ N (0, 1), or discrete
distributed over Zi with P(Ni = 0) ≥ P(Ni = k) for all k ∈ Zi with k 6= 0 if Vi is continuous ordiscrete, respectively. Therefore, the proposed MANM of (1) extends the well-known "simple" data
generating causal models and incorporates recent work on causal discovery in the two-variable model,
which either are defined within a fully continuous space, e.g., cf. [9, 37, 46], or a fully discrete space,e.g., cf. [28, 29]. While we focus on CSL from purely observational data, the modeling of causal
structures via the MANM allows for the generation of interventional data as described by [27] aswell, e.g., see [8].
3.2

Scenario 1: MANM within the Continuous SpaceIn the following scenario, we provide intuitive and relatively well-established examples of the MANM
within the continuous domain, i.e., P dis (Vi ) = ∅ for all Vi ∈ V.

Linear Additive Noise Models Given that fk,i : R → R in (1) is linear, i.e., fk,i (x) = βk,i x, theMANM reduces to the most common form of FCMs [6]. In particular,when the additive noise Ni isP
i.i.d. standard Gaussian distributed such that (1) reduces to Vi = Vk ∈P con (Vi ) βk,i Vk +Ni with i.i.d.
Ni ∼ N (0, 1) for all Vi ∈ V. Then, V = {V1 , . . . , Vp } is multivariate Gaussian distributed withmean zero and covariance matrix Σ = (Ip −B)−1 (Ip −B)−1 , where Ip is the p×p identity matrix and
B the p × p weighted adjacency matrix with non-zero entries Bi,j = βj,i if there is an edge Vj → Vi .
Multivariate Gaussianity allows constraint-based methods to infer conditional independencies within
V by testing for zero partial correlation, which makes CSL feasible for sparse graphs with up to
thousands of variables, e.g., cf. [12]. Due to the applicability to high-dimensional settings, the linear
Gaussian model has found wide use in systems biology, e.g., to infer gene regulatory networks fromobservational gene expression data [40].
Nonlinear Additive Noise Models While linear models are well understood and easy to work with,causal structures within many real-world scenarios are not
P necessarily linear [6]. Therefore, several
CSL methods consider nonlinear FCMs of the form Vi = Vk ∈P con (Vi ) fk,i (Vk )+Ni for all Vi ∈ V,
where fk,i : R → R is not required to be linear, cf., e.g., [9, 41, 45].
3.3

Scenario 2: MANM within the Discrete SpaceIn the discrete case, a functional relationship fj,i : Zj → Zi provides an "interpretable" formalizationof causal structures and enables generating observational and interventional data. Therefore, we
consider that all variables V have a discrete domain, i.e., P con (Vi ) = ∅ for all Vi ∈ V. Then, causal
relationships between discrete variables can be modeled in two different ways, cf. [28, 29]: First, Vi
has the domain Zi = Z with support supp(Vi ), such that the MANM can be defined analogously tothe continuous case. Second, Vi has the domain Zi ⊂ Z, which allows to define + as addition within
the respective modulo ring Z/mi Z, where mi = |supp(Vi )|.

Integer Additive Noise Models Let Vi : Ω → Z, Vi ∈ V, be a discrete random
P variable with (maybe
finite) support supp(Vi ). In this scenario, the MANM of (1) reduces to Vi = Vj ∈P dis (Vi ) fj,i (Vj ) +Ni , for all Vi ∈ V, with a function fj,i : Z → Z and mutually independent noise Ni such that
P(Ni = 0) ≥ P(Ni = k) for all k ∈ Zi with k 6= 0. Note that fj,i can be any probabilistic or
deterministic assignment from Z to Z.
For illustration, consider the two variable case V2 = f1,2 (V1 ) + N2 with the following simplified
example adapted from [28]. Let V1 be uniformly distributed over {−2, −1, 0, 1, 2} and let N2 becharacterized by P(N2 = −2) = P(N2 = 2) = 0.05, and P(N2 = −1) = P(N2 = 0) = P(N2 =1) = 0.3. Then, f1,2 (x) can either be deterministic, e.g., f1,2 (x) = d0.5 x2 e or probabilistic, e.g.,Binomial(0.8, 2), if x ∈ {−2, 2}
f1,2 (x) = Binomial(0.5, 2), if x ∈ {−1, 1}(2)Binomial(0.2, 2), if x ∈ {0}.
4Cyclic Additive Noise Models Following the idea of [28], we consider the concept of m-cyclic
random variables. Therefore, let Vi : Ω → Zi = Z/mi Z, i.e., taking values in {0, . . . , mi − 1},such that the MANM incorporates functions fj,i : Z/mj Z → Z/mi Z. Contrary to the integer
additive noise model (ANM), this scenario bounds the values each variable Vi can take to be from
{0, . . . , mi − 1}, i.e., to targeted domain Z/mi Z.

For illustration, again consider the two-variable case V1 → V2 with V1 taking values {0, 1}, i.e.,
V1 : Ω → Z/2Z, and V2 : Ω → Z/3Z. Let V1 ∼ Bernoulli(0.75) and N2 be characterized by
P(N2 = 0) = 0.5, P(N2 = 1) = 0.3 and P(N2 = 2) = 0.2. We then can define V1 → V2 as
V2 = f1,2 (V1 ) + N2 with f1,2 : Z/2Z → Z/3Z as mapping 0 7→ 1 and 1 7→ 2. Moreover, the cyclic
ANM enables categorical variables with discrete values that do not have any order, cf. [29].
3.4

Scenario 3: MANM within the Mixed Space (Continuous & Discrete)

When considering CSL from mixed discrete-continuous variables, mostly, a conditional linear
Gaussian (CLG) is the assumed FCM [6]. While the CLG restricts discrete variables to have discrete
parents only [1, 30], the MANM allows for both directions of causal relationships between discreteand continuous variables, e.g., following the augmented conditional linear Gaussian (ACLG).
Conditional Linear Gaussian Models First, we examine how the MANM enables to generate V
being CLG distributed. Then, the MANM of (1) is given by(PP
fj,i (Vj ) + Vk ∈P con (Vi ) βk,i Vk + N (µi , σi ), for continuous Vi ∈ V
dis
Vi = PVj ∈P (Vi )(3)for discrete Vi ∈ V,
Vj ∈P dis (Vi ) fj,i (Vj ) + Ni ,where fj,i can be any functional assignment fj,i : Z → Zi ⊆ Z ⊂ R, continuous Gaussian noise
N (µi , σi ), and the discrete noise term Ni as defined in Sec. 3.3. As all continuous variables are
multivariate Gaussian by definition, all continuous Vi are CLG distributed given the vectors of
realisations v dis and v con of theparents P dis (Vi )and P con (Vi ), respective
P discrete and continuous
P
dis
con
i.e., we have P Vi v , v∼N
vj ∈v dis fj,i (vj ) +vk ∈v con βk,i vk + µi , σi , cf., e.g., [19].
Augmented Conditional Linear Gaussian Models To overcome the restrictions of the CLG models,[19] introduced the so-called augmented CLG model, in which discrete variables with continuousparents are generated by using the softmax function. Then, fk,i : R → Z assigns a probability to eachrealization vi within the support supp(Vi ), i.e., vi ∈ Zi with P(Vi = vi ) > 0, given the realization
vk of Vk . Therefore, let fk,i be given by the probabilistic mapping
fk,i := P(Vi = vi |Vk = vk ) = P

exp(αk,i + βk,i vk )s∈{1,...,mi −1}: vs ∈supp(Vi ) exp(αk,s + βk,s vk )(4)

with soft-max parameters αk,s and βk,s defined for all vs ∈ supp(Vi ) given the realization Vk = vk .
Then, the MANM of (1) allows for generation of data according to the ACLG model via(PP
βk,i Vk + N (µi , σi ), for continuous Vi ∈ V
con
Vj ∈P dis (Vi ) fj,i (Vj ) +PVk ∈P (Vi )(5)Vi = Pf(V)+
for discrete Vi ∈ V,
Vj ∈P dis (Vi ) j,i j
Vk ∈P con (Vi ) fk,i (Vk ) + Ni ,with fj,i as defined in the context of (3) and the discrete noise term Ni as defined in Sec. 3.3. In
this context, the MANM is flexible enough to model causal structures within discrete-continuous
mixtures by incorporating various deterministic and probabilistic mappings fj,i from Z to R, andvice versa via fk,i . For example, consider simple step functions for fk,i : R → Z.
In summary, the MANM provides a flexible functional framework to model causal structures withvarious characteristics of an edge Vj → Vi , cf. (R1) - (R2).

4

MANM-CS: A CSL Benchmarking Framework

To provide the research community easy access to the MANM for benchmarking CSL methods,
cf. (R4), we present our reference implementation MANM-CS1 It generates mixed and nonlinear
observational data (Sec. 4.1), allows for high-dimensional scenarios (Sec. 4.2), and covers the various
distribution models included in the MANM (Sec. 4.3).
1

https://github.com/hpi-epic/manm-cs

5

4.1

Implementation of Data Sampling Process

Data generation of MANM-CS follows the common two-step approach, where first, a DAG G withrespective parameterized FCM is generated, and second, each observation is sampled by iteratingover the nodes considering the functional relationships regarding their parents. Therefore, several parameters (see Table 2 in Appendix) provide easy specification of the underlying MANM
introduced in Sec. 3 as basis for generation of mixed and nonlinear data. In Algorithm 1 (in Appendix), MANM-CS generates a DAG that incorporates num_nodes number of ordered nodes withedge density edge_density . Moreover, nodes are chosen to be discrete with a number of classes
between discrete_classmin and discrete_classmax or continuous distributed according todiscrete_ratio . If the joint distribution is conditional Gaussian the first discrete_ratio ×
num_nodes are discrete, otherwise for augmented conditional Gaussian each variable is chosen to
be discrete with probability discrete_ratio . According to the MANM noise terms for discreteand continuous variables are chosen to be discrete with corresponding discrete_noise_ratio or
normal distributed with standard deviation std_continuous_noise, respectively.
Moreover, functional relationships for each edge in G are either sampled from self-chosen
functions within the continuous space (see Sec. 3.2), follow the cyclic additive noise model
within discrete (see Sec. 3.3), or defined as a soft-max and CLG model within the mixed space,respectively (see Sec. 3.4). Hence, Algorithm 1 returns a fully parameterized CGM following thespecifications of the MANM as basis for Algorithm 2 (in Appendix). In particular, Algorithm 2
implements the sampling of num_samples observations by iterating over the nodes considering thenoise terms and functional relationships regarding their parents.
4.2

Runtime Performance of Data Generation Process

To speed-up data generation within high-dimensional scenarios, MANM-CS’ data sampling can be
executed in parallel by specifying the number of processes num_processes. Although ideal speedup is not achieved, parallelization reduces data generation significantly - in particular for many nodes.
For example, the execution time of one million samples generated according to a DAG with 1 000
nodes and edge density 0.4 decreases from 4 322 seconds (16 cores), over 2 214 seconds (32 cores) to
1 367 seconds (64 cores) .
4.3

Exemplary Characteristics of Data Generated by MANM-CS

The following examples illustrate the range of causal relationships and respective data characteristics.
Example 1 First, we consider a small and sparse CGM G such that the distributional characteristics
of direct causal relationships are primarily induced through the corresponding functional mappingsof the underlying MANM. In this sense, we consider a mixed CGM (conditional_gaussian=1,
discrete_ratio =0.5) of num_nodes =10 with edge_density =0.4 that includes discrete variables (discrete_classmin =3, discrete_classmax =4) and continuous variables with nonlinear
causal relationships (functions={[0.4, linear], [0.3, quadratic], [0.3, cosine]}) and corresponding noise terms (discrete_ratio =0.5, std_continuous_noise =1.0). On this basis, thedata characteristics of num_samples=10 000 depicted in Fig. 1 follow the expected evidently linear,quadratic, discrete, and mixed causal relationships of direct edges within the sparse CGM G.
12.5

0.40

20

0.35

15

7.5

0.30

V2

1

V4

V6

2.5

10

0.4

5

0.0

0.3

−2.5

2

0

Density

0.5

5.0

V8

0.6

0

10.0

0.25

0
1
2

0.20
0.15
0.10
0.05

0.2

−5.0
−4

−2

0

2

4

V5

(a) Linear V5 → V8

−5

0.00

−4

−2

0

2

0

4

1

2

V1

V3

(c) Discrete V1 → V4(b) Quadratic V3 → V6

−4

−2

0

2

4

V3

(d) Mixed V2 → V3

Figure 1: Unconfounded data distributions: (a) scatter plots for the linear edge V5 → V8 , (b) the
nonlinear edge V3 → V6 , (c) a heatmap of conditional probabilities for the discrete edge V1 → V4 ,and (d) a density plot of V3 given the realization of a discrete parent V2 for a mixed edge V2 → V3 .

6

3000

3000

2500

2500

2000

2000

1000

500

500

0

0
0

5

10

V17

(a) Linear V17 → V21

0.00015

0.34

Density

0

0.36
1

1500

1000

0.00020

V9

1500

0.32

2

V24

V21

V5
0.38

0
1
2

0.000100.00005

0.00000
0

10000

20000

0

30000

1

2

V2

V23

(c) Discrete V2 → V9(b) Cosine V23 → V24

0

10000

20000

30000

V23

(d) Mixed V5 → V23

Figure 2: Confounded data distributions: (a) scatter plots for the linear edge V17 → V21 , (b) the
nonlinear edge V23 → V24 , (c) a heatmap of conditional probabilities for the discrete edge V2 → V9 ,and (d) a density plot of V23 given the realization of a discrete parent V5 for a mixed edge V5 → V23 .
Example 2 Next, we consider a larger and denser CGM G such that the distributional characteristics of direct causal relationships may be distorted through common confounders. Therefore, wechange Example 1 by increasing num_nodes =25 and edge_density =0.8 while retaining all otherparameters. On this basis, the data distributions of direct causal relationships depicted in Fig. 2
are now characterized by interferences of respective direct linear, cosine, discrete or mixed causal
relationships with indirect causal relationships induced through confounders within a denser CGM.
Therefore, these examples not only illustrate the achieved interpretability of a causal relationship based
upon a well-defined FCM, cf. (R1), but also demonstrate the achievable complexity of data characteristics provided by MANM-CS. Note, the contrary interference on the data distribution between edges withno direct edge through common confounders may induce a visible but not existing direct functional
relationship. Moreover, note, that variations of the noise parameters discrete_noise_ratio andstd_continuous_noise yield further statistical dispersion.

5

Benchmarking Scenarios and Experimental Evaluation

In this section, we demonstrate that MANM-CS not only covers common benchmarking approaches(Sec. 5.1), but allows for a more comprehensive examination of CSL methods, too (Sec.5.2).
5.1

Experiment 1: CSL in Comparison to Well-known Benchmark Approaches

In this experiment, we compare large sample properties of the well-known PC algorithm [40] withappropriate CI tests on observational data generated by MANM-CS that aims to mimic data sampledfrom common type (II) and (III) approaches, cf. Sec. 2. In particular, we show a coincidence inimprovements of the structural Hamming distance (SHD) regarding the learned CGMs from datawith increasing sample size in the context of different distribution models, cf. Sec. 3.2 - 3.4.
Within continuous space, the FCM of the linear ANM (see Sec. 3.2) allows for easy data generation
following approach (III), e.g., within pcalg [13]. Given that the MANM is a more general model
class, the SHD of CGMs learned by the PC algorithm with Fisher’s z-test shows a direct coincidence
on multivariate Gaussian data sampled by MANM-CS and pcalg for variations in the number ofvariables as well as in the number of observations, see Fig. 3 (a).
Within the mixed and discrete space, benchmarking of CSL methods are often restricted to data
generated by "static" type (II) approaches, e.g., parameterized models found in the bnlearn repository [33]. For comparison, we generate observational data using the MANM-CS’ capabilities forparameter adjustment to mimic the characteristics of the well-known ALARM [2] and MEHRA [44] networks. In this context, the descriptive functional restriction on modeling a causal relationship between
two discrete variables within MANM-CS becomes recognizable. For example, ALARM incorporates
probabilistic mappings fk,i between two discrete nodes Vi , Vj while MANM-CS’s implementation iscurrently restricted to the mapping of the cyclic ANM, cf. Sec. 3.3. Hence, the induced independence
characteristics of the variables’ joint distribution are empirically stronger within data sampled by
MANM-CS, which yield lower SHDs in comparison to the parameterized discrete ALARM and CLG
MEHRA networks, respectively, see Fig. 3 (b) and (c). Nevertheless, the coincidence in a decreasing
SHD of the CGMs learned by the PC algorithm with appropriate Pearson’s X 2 (discrete, cf. [33, 13])and asymptotic mutual information χ2 test (CLG, cf. [33]) for an increasing number of samples isvisible for all approaches and distribution models.
7

Structural hamming distance

Edge density = 0.4 | Nodes = 5

Edge density = 0.4 | Nodes = 10

Edge density = 0.4 | Nodes = 20

0.25
0.20
Data generation
MANM-CS
pcalg

0.15
0.10
0.05
0.00

1000

10000

100000

Number of observations

1000

10000

100000

Number of observations

1000

10000

100000

Number of observations

0.13

0.065

Structural hamming distance

Structural hamming distance(a) Continuous space: MANM-CS and pcalg
0.060
0.055
Data generation
MANM-CS
ALARM

0.050
0.045
0.040
0.035
1000

10000

0.12
0.11
Data generation
MANM-CS
MEHRA

0.10
0.09
0.08
1000

100000

10000

100000

Number of observations

Number of observations(c) Mixed space: MANM-CS and MEHRA(b) Discrete space: MANM-CS and ALARM

Figure 3: Median SHD (10 runs) of learned CGMs with the PC algorithm from (a) observational data
sampled by MANM-CS or pcalg for the continuous space with a varying number of nodes, or from (b)the Bayesian benchmark networks ALARM for the discrete and (c) MEHRA for the mixed space.
5.2

Experiment 2: CSL within Mixed and Nonlinear SpaceIn this experiment, we demonstrate MANM-CS’s capabilities for benchmarking CSL methods in thecontext of various scenarios regarding mixed discrete-continuous and nonlinear data. In particular,
we examine the decreasing accuracy of CLG model-based CSL when assumptions on the causal
relationships, e.g., linearity, are violated. Following [30], we include a discretization-based approachas nonparametric baseline but consider distribution models beyond simple CLG or Lee Hastie data.
Therfore, we compare the median SHD (10 runs) of learned CGMs with the PC algorithm, cf. [13, 33],
with asymptotic mutual information χ2 test assuming a CLG model, cf. [33], compared to Pearson’s
X 2 test, cf. [13], where continuous variables are discretized through the k-means algorithm, cf. [21],with k = 5.
Violating Linearity We consider a violation of linearity within the CLG and its implication on
the accuracy of learned CGMs. In particular, Fig. 4 (a) and (b) depict the median SHD of theparametric against the discretization-based approach for an increasing ratio of quadratic and cosine
functional relationships, respectively (num_nodes =10, edge_density =0.4, discrete_ratio =0,
num_samples=10 000). As the asymptotic mutual information χ2 test is based upon the partial
correlation within continuous space, its good accuracy within the purely linear case decreases steadily
for an increasing ratio on nonlinear functional relationships. In contrast, the discretization-based
approach’s accuracy behaves rather invariant in the context variations in terms of nonlinearity forboth cases (a) and (b). Moreover, although not true generally, cf. [23, 30], the accuracy of thediscretization-based approach even exceeds the parametric CLG-based approach in the presentededge cases of mainly quadratic or cosine functional relationships.
Violating Conditional Gaussianity We consider the implication of changing from the CLG toanother mixed model (see Sec. 3.4). In this sense, Fig. 4 (c) depicts the median SHD of theparametric CLG-based against the discretization-based approach under the assumption of an CLG
and an augmented CLG model (num_nodes =50, edge_density =0.4, discrete_ratio =0.5,
num_samples=10 000). Although the PC algorithm with an appropriate asymptotic mutual information χ2 shows a significantly better accuracy within the CLG model, the accuracy is slightly
exceeded by the discretization-based approach if discrete nodes are allowed to have continuousparents, i.e., within the augmented CLG model.
8

1.0
0.0

0.8
0.2

0.6
0.4

0.4
0.6

0.2
0.8

0.0
1.0

Share of functions linear over quadratic(a) Linear to quadratic.

0.35

Edge density = 0.4 | Nodes = 50

Edge density = 0.4 | Nodes = 10

0.30
0.25
MethodK-Means k=5
Discrete CI-Test
No discretization
Gaussian CI-Test

0.20
0.15
0.10
0.05
1.0
0.0

0.8
0.2

0.6
0.4

0.4
0.6

0.2
0.8

0.0
1.0

Share of functions linear over cosine(b) Linear to cosine.

Structural hamming distance

MethodK-Means k=5
Discrete CI-Test
No discretization
Gaussian CI-Test

Structural hamming distance

Structural hamming distance

Edge density = 0.4 | Nodes = 10
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05

0.210
0.205
0.200
0.195
0.190
0.185
0.180
0.175

MethodK-Means k=5
Discrete CI-Test
No Discretization
mi-cg CI-Test
Conditional
LinearGaussianMixed(c) CLG to mixed.

Figure 4: Median SHD (10 runs) of learned CGMs with the PC algorithm with asymptotic mutual
information χ2 test assuming a CLG model compared to Pearson’s X 2 test where continuous variables
are discretized (k-means with k = 5) given violation of the CLG model assumption for an increasing
ratio of (a) quadratic and (b) cosine functional relationships, respectively, and (c) for considering theaugmented CLG model.

The above examples demonstrate the importance of validating CSL methods assumptions in practiceand the demand for understanding the method’s accuracy within specific edge cases, e.g., whenassumptions on the causal relationships are violated, cf. [17].

6Conclusion, Limitations, and Future WorkWe introduced the mixed additive noise model (MANM) to provide a framework for generating
causal structures within mixed discrete-continuous and nonlinear data. Its functional formalization
defined in (1) provides an interpretable characterization of causal structures as demonstrated froma theoretical and empirical perspective (see Sec. 3.2 - 3.4 and Sec. 5, respectively), cf. (R1). In
particular, it connects well-established work of CSL within different distribution models, suchas CLG, and allows for the generation of continuous, discrete, and mixed discrete-continuous
observational data, cf. (R2). Due to the functional form, the MANM is flexible enough to support
further extensions, e.g., to consider the generation of interventional data similar to [8], cf. (R3).
Moreover, it allows examining methods’ accuracy in case of a misspecified choice of hyperparameters
or given invalidated assumptions, e.g., using an incomplete selection of V to model the causally
insufficient case of latent variables. To provide easy access to the research community, we present ourreference implementation MANM-CS and benchmarking scenarios, cf. (R4). In particular, MANM-CS’s
capabilities not only provide enough opportunities to mimic common benchmarking approaches but
also allows for more comprehensive evaluations with varying model complexity that exceed "static"
type (II) approaches (see Sec. 5). Further, MANM-CS can be easily integrated in pipelines for CSL
such as MPCSL [11], which allows researchers and practitioners to easily evaluate their methods.
While the restriction on the FCM allows for a formalization of various causal relationships, thefunctional constraints induce limitations that are worth to be noticed. For example, in contrast to
the assumptions of the MANM within the discrete space, cf. Sec. 3.3, there may not always bea functional representation of a causal relationship between discrete variables in practice [29], cf.
Sec. 5.1. Moreover, the embedding of discrete variables into the continuous space as defined in(1) restricts the functional relationship to be location-related, which may be violated within realworld scenarios. In this context, possible generalizations are weakened additivity concerning theindependent noise or a post nonlinearity [9, 46]. Note that if the characteristics of the data generating
mechanism do not follow the MANM, the requirements of CSL methods should be general enoughto reveal the data generating processes approximately [6].
As the MANM provides a ground truth model for generating observational data following various
distribution models with nonlinear and mixed discrete-continuous data, we work on a more comprehensive empirical evaluation of several popular CSL methods for future work. Moreover, we
aim to provide more parameters such as concerning missing values or interventional data, different
noise distributions, and functional classes to enable a more fine-grained evaluation of CSL methods.
Further, we invite the research community to participate in the extension of MANM-CS actively.
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Appendix
Parameters for Data Generation
MANM-CS currently enables to generate observational data for benchmarking CSL in the context of
mixed discrete-continuous and nonlinear causal relationships according to the parameters depicted inTable 2.
Parameter
num_nodes
edge_density
discrete_ratio
num_samples
discrete_noise_ratio
discrete_classmin
discrete_classmax
std_continuous_noisefunctionsconditional_gaussian
num_processes

Definition
Number of nodes corresponding to the number of variables p in V
Edge density of the resulting graph in range [0, 1]
Ratio of discrete nodes compared to num_nodes
Defines the number of samples that shall be generated from the DAG.
The probability of adding noise to a discrete node
Minimum of the range for the discrete domain Z of discrete nodes
Maximum of the range for the discrete domain Z of discrete nodes
Standard deviation of continuous noise
A list of sample probabilities and continuous functions.
Flag for conditional or augmented conditional Gaussian model
Number of processes used for data sampling

Table 2: Parameters and their definitions for the data generation of MANM-CS.

13

Graph Generation Algorithm
Algorithm 1 of MANM-CS generates a DAG G with respective parameterized FCM.
Algorithm 1 Graph Generation
Input:
num_nodes ,edge_density ,discrete_ratio ,discrete_classmin ,discrete_classmax , std_continuous_noise , discrete_noise_ratio , functions,conditional_gaussian
Output: DAG G
1: procedure DIS(discrete_classmin , discrete_classmax ,discrete_noise_ratio ,par)2:
discrete_class← randomUniform(discrete_classmin , discrete_classmax )3:noise ← sampleNoise(discrete_noise_ratio , discrete_class)4:
return discreteNode(discrete_class, noise, par)5: end procedure6:
7: procedure CON(std_continuous_noise, functions, par)8:
for pcon in continuousParents(parents) do
9:sampled_functions ← choose_function(functions,pcon)
10:
end for11:
noise ← N (0, std_continuous_noise)
12:
return continuousNode(sampled_functions, noise, par)13: end procedure14:
15: node_list ← [1 . . . num_nodes ]16: forward_edge_list ← [(nodeu , nodev ) in node_list × node_list | nodeu < nodev ]17: edge_list ← sampleEdges(forward_edge_list, edge_density )18: num_discrete_nodes ← discrete_ratio · num_nodes
19:
20: for node in topSorted(node_list) do
21:parents ← getParents(edge_list)
22:
if conditional_gaussian then
. Conditional Gaussian23:
if node.id ≤ num_discrete_nodes then. Create node of discrete type24:
node.set(DIS(discrete_classmin , discrete_classmax ,25:
discrete_noise_ratio ,parents))
26:
else. Create node of continuous type27:
node.set(CON(std_continuous_noise, functions, par))28:
end if29:
else. Augmented Conditional Gaussian30:
if random(0,1) ≤ discrete_ratio then. Create node of discrete type
31:
node.set(DIS(discrete_classmin , discrete_classmax ,32:
discrete_noise_ratio ,parents))
33:
else. Create node of continuous type34:
node.set(CON(std_continuous_noise, functions, par))35:
end if36:
end if37: end for
38: return constructDAG(node_list, edge_list)

14

Data Generation Algorithm
Algorithm 2 samples each observation by iterating over the nodes of the DAG G provided by
Algorithm 1 considering the functional relationships regarding their parents.
Algorithm 2 Data Generation
Input: num_samples, DAG G
Output: sampled_data_matrix
1: sampled_data_matrix ← matrix(len(G.node_list) × num_samples)
2: for node in topSorted(G.node_list) do
3:sampled_data_matrix[node.id]← node.sample(num_samples)
4: end for5: return sampled_data_matrix

15