A Conditional Independence Test in the Presence of Discretization
Authors: Boyang Sun
Yu Yao
Guang-Yuan Hao
Yumou Qiu
Kun Zhang
Venue: ICLR 2025 Poster (2025)
Keywords: Conditional Independence Test
Discretization
Causal Discovery
Primary area: causal reasoning
TL;DR: We propose a novel conditional independence test to deal with data shuold be inherently continuous but only discretized observations available.

Abstract: Testing conditional independence (CI) has many important applications, such as Bayesian network learning and causal discovery. Although several approaches have been developed for learning CI structures for observed variables, those existing methods generally fail to work when the variables of interest can not be directly observed and only discretized values of those variables are available. For example, if $X_1$, $\tilde{X}_2$ and $X_3$ are the observed variables, where $\tilde{X}_2$ is a discretization of the latent variable $X_2$, applying the existing methods to the observations of $X_1$, $\tilde{X}_2$ and $X_3$ would lead to a false conclusion about the underlying CI of variables $X_1$, $X_2$ and $X_3$.
Motivated by this, we propose a CI test specifically designed to accommodate the presence of discretization. To achieve this, a bridge equation and nodewise regression are used to recover the precision coefficients reflecting the conditional dependence of the latent continuous variables under the nonparanormal model. An appropriate test statistic has been proposed, and its asymptotic distribution under the null hypothesis of CI has been derived.
Theoretical analysis, along with empirical validation on various datasets, rigorously demonstrates the effectiveness of our testing methods.

Published as a conference paper at ICLR 2025
A CONDITIONAL INDEPENDENCE TEST IN THE PRES- ENCE OF DISCRETIZATION
Boyang Sun', Yu Yao", Guang- Yuan Hao', Yumou Qiu Kun Zhang?.
1 Mohamed bin Zayed University of Artificial Intelligence
2 Carnegie Mellon University
3 Peking University
4 The University of Sydney
ABSTRACT
Testing conditional independence (CI) has many important applications, such Bayesian network learning and causal discovery. Although several approaches have been developed for inferring CI relationships among observed variables, these existing methods generally fail when the variables interest cannot be directly observed and only discretized values of those variables available. For example, if X1, X₂ and X3 the observed variables, where X₂ is discretization of the latent variable X2, applying the existing methods to the observations
X1, X2 and X3 would lead to a false conclusion about the underlying CI variables X1, X2 and X3. Motivated by this, propose CI test specifically designed to accommodate the presence of discretization. To achieve this, a bridge equation and nodewise regression are used to recover the precision coefficients reflecting the conditional dependence of the latent continuous variables under the nonparanormal model. We propose distribution under the null hypothesis of CI. Theoretical analysis, along with empirical validation on various datasets, rigorously demonstrates the effectiveness of our testing methods. Our code implementation can be found in https:// gathhbbb..co//bbyyyaaaaaaaaaaa
statistic and derive its asymptotic
INTRODUCTION
Independence and conditional independence (CI) are fundamental concepts in statistics. They leveraged for exploring queries in statistical inference, such as sufficiency, parameter identification, and ancillarity (Dawid, 1979). They also play central role in emerging areas such causal discovery (Koller and Friedman 2009), graphical model learning, and feature selection (Xing et al. 2001). Tests for CI have attracted increasing attention from both theoretical and application sides.
Formally, the problem is to test the Cl of two variables X; and X, given random vector (a set of other variables) Z. statistical notation, the null hypothesis is written as Ho XX, T1 X, Z, where IL denotes "independent from". The alternative hypothesis is written H1 X, A X, Z. where AL denotes "dependent with". The null hypothesis implies that once is known, the values of X; provide no additional information about X,. and vice versa. Various tests have been designed to address different scenarios, including Gaussian variables with linear dependence (Yuan and Lin 2007] Peterson et al. 20155] Mohan et al.] 2012 Ren al.) 2015) and non-linear dependence (Fukumizu ct al.] 2004 Zhang et al. [012 Strob et al. 2019 Sen el al. 2017 Aliferis et al. 2010) (For detailed related work, please refer App. E).
Given observations of X,, Xy. and Z, the CI relationship can be effectively tested with the existing methods. However, in many scenarios, accurately measuring continuous variables of interest is challenging due to limitations data collection. Sometimes the data obtained are approximations represented discretized values. For example, in finance, variables such asset values cannot be measured and are binned into ranges for assessing investment risks (e.g., sell, hold, and strong buy) (Changsheng and Yongfeng 2012 Damodaran 2012). Similarly, in mental health, anxiety levels
"Co-corresponding authors: Yumou Qiu gguuyumouu@mmthppuuuucc Kun Zhang/kunz @cmu.edu

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often assessed using scales like the GAD-7, which categorizes responses into levels such mild, moderate, or severe (Mossman et al. 2017: Johnson et al. 2019). In the entertainment industry, the quality of movies is typically summarized through viewer ratings (Sparling and Sen 2011 Dooms
et al. 2013).
When discretization is present, existing CI tests can fail to determine the CI relationships of the underlying variables. This issue arises because existing CI tests treat discretized observations as observations of continuous variables, leading incorrect conclusions about their CI relationships More precisely, the problem lies in the discretization process, which introduces new discrete variables. Consequently, although the intent is to test the CI of the underlying continuous variables, what is being tested is the CI involving mix of both continuous and newly introduced discrete variables. In general, this CI relationship is inconsistent with the one among the underlying continuous variables.
As illustrated in Fig. [ we show different data- generative processes using causal graphical mod- els (Pearl (000) in the presence of discretization. gray node indicates an observable variable, while white node indicates latent variable. Variables denoted by X, (without tilde ~) rep- resent continuous variables, which may not be observed; while variables denoted by X, repre- sent observed discretized variables derived from X, due to discretization. In Fig. I(a) Xg is latent, and only its discrete counterpart X₂ is observed. In this case, rather than observing X1. X2, and Xg, we only observe X1, X2,, and X3.
(b)
Figure 1: We illustrate data-generative processes
with causal graphical models. The discretization process introduces new discrete variables indicated by a tilde (~).
Existing CI methods use these observations to test whether Xi 1L {X2), but what is actually being tested is whether XL {X2]. In fact, according to the causal Markov condition (Spirtes et al. 2000). it can be inferred from Fig. ((a) that X1_ {X2} and X1 {x2]- This mismatch leads existing CI methods, that employ observations to check the CI relationships between X, and Xg given X2, to reach incorrect conclusions. Due to the same reason, checking the
CI also fails in Fig [(b] and Fig [(c)
In this paper, we design a CI test specifically for handling the presence of discretization. An appropri- ate test statistic for the CI of latent continuous variables, based solely discretized observations, is derived. To develop this test, we first estimate the covariance between latent continuous variables and discretized observations. This is achieved by constructing bridge equations that enable the estimation of covariance using statistics derived from discretized observations. Subsequently, to utilize the estimated covariance of latent continuous variables for testing CI relationships, we apply a node-wise regression approach (Callot et al. 2019),, which allows us to derive test statistics for CI based on the estimated covariance. By assuming that the continuous variables follow a Gaussian distribution, we can derive the asymptotic distribution of the test statistics under the null hypothesis of CL. Our major contributions include:
We develop CI test for ensuring accurate analysis in scenarios where data has been discretized, which are common due to limitations in data collection or measurement techniques.
Our CI test can handle various scenarios including 1). Both variables X; and Xj are discretized 2).
Both variables X; and X, continuous. 3). One of the variables X; or Xj is discretized.
We compare our test with the existing methods on both synthetic and real-world datasets, confirm- ing that our method can effectively estimate the CI of the underlying continuous variables and outperform the existing tests applied on the discretized observations.
2 PROBLEM SETTING AND NEED FOR CORRECTION
Problem Setting Consider set of independent and identically distributed (i.i.d)) p-dimensional In this set, some variables, indicated by tilde (~), such as X,, follow a discrete distribution. For each such variable, there exists corresponding latent Gaussian random variable Xj. The transformation from X, X, is governed
random vectors, denoted as X = [X1 X2, Xx..., Xpp.
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by unknown monotone nonlinear function 9j and thresholding function fz- The function fg o gj : X + X maps the continuous domain of Xj onto the discrete domain of X. Specifically, for each variable Xj, there exists a finite constant vector dj [dj. increasing elements such that
..gi.....d,.... . M-M characterized by
1 gy(X,) =d,1 m dz.m-1 < gj gg(X,) < dj
X, - (x
M (X(,)) > d
This model also known as the nonparanormal model (Liu et al. 2009). The cardinality of the domain after discretization is at least 2 and smaller than infinity. Our goal is assess both conditional and unconditional independence among the variables of the vector X = x our model, we assume X (0, E). Σ only contain among its diagonal, i.e., jjj. for all j E [1.....p] p]. One should note this assumption is without loss of generality. We provide a detailed discussion of our assumption in App. B.9
In
Why the correction is needed? We aim to propose a CI test that serves as a correction to infer the correct CI relationships among the latent continuous variables of interest. One question that arises is whether the discretized variables exhibit the same conditional independence as their original continuous counterparts, i.c., the correction is not needed. This concern becomes more significant when the level of discretization is high. To show the effect of discretization, we present the following theorem, using Gaussian random variables as an example, to demonstrate that discretization inevitably introduces distortions. These distortions can lead to incorrect conclusions about CI relationships. The proof can be found in Appendix [B.]
Theorem 2.1. Let X1, and X3 be jointly Gaussian random variables that are mutually dependent. such that Xi IL X2|X2, X, 5و(ر (X2)) is the discretized observation as defined in equation) | Then the conditional independence between X, and X3 given X₂ doesn't hold, iee, X1 P X3
DCT: A DISCRETIZATION-AAAI CI TEST
Notation Throughout this work, we use X, to denote the j-th component of the vector of variables
X. We denote the sample mean of Xj by E, [X,]. and the expectation by E[X,]. The empirical probability is represented by IP, whereas the true probability denoted by P. For a matrix X, X-; represents all columns of X except the j-th column, X-j.- denotes the submatrix obtained by removing both the j-th column and row, and X-j.j represents the j-th column of X with the j-th row removed. For any parameter a, we use & to denote its estimation. [coonitioon] } is 1 if the condition holds true, 0 otherwise. For a full notation table, please refer to App.
To develop CI test, need to design a test statistic that can reflect the conditional dependence relation and be computable using observations only. Next, it is essential to derive the underlying distribution of this statistic under the null hypothesis that the tested variables are conditionally (or unconditionally) independent. By calculating the value of the test statistic and assessing if this statistic is likely to be drawn from the derived distribution (i.e., calculating the p-value and comparing it with the significance level a), we can decide if the null hypothesis should be rejected.
Our objective is deduce the independence and CI relationships within the original multivariate Gaussian variable X, based on its discretized observations X. In the context of a multivariate Gaussian model, this challenge is directly equivalent to constructing statistical inferences for its covariance matrix Σ (0t,5) and its precision matrix N (wJ,+) Σ (Baba et al. 2004). The covariance matrix Σ captures the pairwise covariances, while the precision matrix 12 encodes CI relationships. Specifically, the entry Wj.k represents the partial correlation coefficient between variables X, and Xx. which determines their CI given other variables. Technically, we are interested in two things: (1) the calculation of the covariance oj. and the precision coefficient (or the partial correlation coefficient) Wj serving the estimation of oi and wj. respectively; (2) the derivation of the distribution of ōi.j osj and Øj.k - Wj.k under the null hypothesis of independence and CI.
In the rest of this section, discuss three key components: (1) introduce bridge equations to estimate the covariance oij; (2) we derive the distribution of ōz, oz.j. showing it to be asymptotically normal; and (3) we use nodewise regression establish the relationship between
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the covariance matrix Σ and the precision matrix Z. We show that the regression parameter Bz.k serves as a proxy for the precision matrix entry wj,k- Leveraging the distribution of os,, Oij. we demonstrate that B3.A 53, is also asymptotically normal.
3.1 ESTIMATING COVARIANCE THROUGH OBSERVATIONS
Our first task is to establish the connection between the underlying covariance oij of the continuous pair X; and Xj with their observed counterparts. Due to discretization, the sample covariance matrix computed from X is inconsistent with the covariance matrix of X. To obtain the estimation ō., consistent with ozj, the bridge equation is leveraged. In general, it takes the form:
P.j
T(ōsg; A),
where øz,j the estimated covariance, Ri is statistic that can also be estimated from observations, and is a set of additional parameters required by the function T(-). The specific form of the function T()) will be derived later. Both f..j and A should be able to be calculated purely relying on observations. Then, given the calculated fr.j and Å, ōūg equation. As a result, the covariance matrix of X can be estimated, which contains information about both unconditional independence and CI (which can be derived from its inverse).
be obtained by solving the bridge
To estimate the covariance of a latent multivariate Gaussian distribution, we need to design ft.j A, and T()). Notably, bridge equations have to be designed to handle the possible cases: C1. both observed variables are discretized; C2. one variable continuous while the other discretized. For C3. both variables remain continuous, we can easily take its sample covariance as the estimated covariance. We will show that cases CI and C2 can be merged into a single form of bridge equation with different parameters and a binarization operation applied to the observations. Our bridge
equations are presented in Def. 3.1 Def. 3.2
3.1.1 BRIDGE EQUATIONS FOR DISCRETIZED AND MIXED PAIRS
Let us first address the challenging cases where both observed variables discretized or where one variable is continuous while the other is discretized. In general, different bridge equations would need to be designed to handle each case individually. However, in our analysis. we provide unified bridge equation that applies to both cases. This is achieved by binarizing the observed variables, thereby unifying both cases into binary case. As some information may be lost in the binarization process, this unification may require more data samples compared to using tailored bridge functions for each specific case. Improving sample efficiency with tailored bridge equations left for future work.
Theoretically, continuous variables and discrete variables can be further discretized into binary variables. Imagine we have the observed variable X; with the possible values "low", "medium". "high",
create dividing point: everything above becomes "very high", everything below becomes "very low". This binarization process is also applicable to the continuous variable. Note that X, is just the discretized version of its corresponding continuous variable Xj. this dividing point directly responds to specific value in the original continuous domain, which we denote as the boundary hy. Multiple choices of hy are possible. In this paper, we define h, as the boundary in the continuous domain that corresponds to the mean of its discretized counterpart Xy. Mathematically, define hy as follows: for any single discretized variable X,. there exists a constant ej such that h2 = 9j (c₁) satisfying
16 > = 0/ = 1 > hy):
10)
where 2t is the j-th sample of x,, and æz is the j-th sample of Xy-
Estimating the boundary Since the continuous variable X, follows a normal distribution according to our assumption, we can thus construct the relation F(X, > E[X,]) 1 - ((,,), where & is the cumulative distribution function (cdf) of a standard normal distribution. Although we do not have access the true probability, we easily obtain its estimation by counting how many samples drop in the region larger than its sample mean. Specifically,
hg Φ 1 1g),
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where 13 n Ei= 1 E,[X,]), serving as the estimation of P(X, E[X,]). We further
denote 2()) = (-).
the observed discrete pair (X,, The question now is to estimate the latent covariance o.j for mixed pair Xy). Leveraging the binarization process, there exists boundaries h4, hy that partition the continuous variables pair X, and X, to a 2 contingency table. The area of each cell in this table represents the joint probability of the pair (X;, X,)) falling with a specific region defined by those boundaries. In this paper, we focus on the top-right cell of the contingency table, which represents the joint probability of both variables exceeding their respective boundaries.
Intuition of estimating covariance
Let Zi and Z2 denote random variables. Mathematically, we denote (=11, 22: = ( > 21: Ze > 22). where (Z1, Z2) follows bivariate normal distribution with mean zero, variance one and covariance p. For a discretized pair of observed variables (X,, Xj), we define
TrJ == P(X; > E[X.], X; > B[X,]) = ( hy;
The above equation shows that the probability of discretized variables larger than their mean is a function of underlying covariance. serves key estimating the covariance. The probability in the above equation can be estimated by counting samples dropped into the region of both variables exceeding their sample means as follows:
n
ti. = P, E, [X,), [XIX)]
Σ
1 [XI En
72
Since (he, by; oi.)) a function of o.j. by substituting the parameters Tz.j h4, h2 their estimation, we can construct the bridge equation as follows:
Definition 3.1 (Bridge Equation for Discretized- Variable Pair). For discretized variables X, and X,. the bridge equation is defined as:
R.j = T(ōsji h,)), where T(ooy hz1) = S2,2h, JegZh, 0(21, 22; (z,)) d21 d22, and Φ is the probability density function of bivariate normal distribution with mean zero and covariance āi.j. we note that hi, hy can be simply calculated using equation and R. can be calculated using equation 4
Following the same idea, we can apply the same bridge equation to estimate the covariance mixed pairs. The only difference is there is no need to estimate the boundary hy for the continuous variable. Instead, we can incorporate its true mean of zero into the equation.
Definition 3.2 (Bridge Equation for A Continuous-Discrretzedd Variable Pair). For one continuous variable X; and one discretized variable Xj. the bridge function is defined as:
f.j = P,, 0, 0X (X)))
= n Σ
1a 0,00
E.
[X,1] T(o,,s (0, h,
and the function T(-) has the same form of Def. 3.1
3.1.2 CALCULATION OF ESTIMATED COVARIANCE
For the continuous case where there is no discretization transformation, the sample covariance provides consistent estimation of the true one. That is, for an observable pair of continuous variables (X,,X)) we can simply obtain the analytic solution of estimated covariance:
1
(5)
For the cases involving the discretized variable as proposed in Def. 3.1 and Def. 3.2 we can rely on the property that variance Σ only contains among the diagonal, which implies the covariance oid should vary from - to Thus, we can calculate the estimated covariance by solving the objective
,. = arg min %/
|| T(ol.g; (hh, h11 s.t. < <.
The f.jj one-to-one mapping with calculated ري given h, and hy, which is proved in App. B.3
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3.2 UNCONDITIONAL INDEPENDENCE TEST
The estimation covariance ōi.j can be effectively solved using the designed bridge equation. Now, focus on deriving the distribution of ō.ح di.j- These results are used independence test in the presence of discretization. Moreover, Thm. 3.3 Lem. 3.4 Lem. 3.5
unconditional
and Lem. 3.6 will be leveraged in the derivation process of the CI test in Section 3.3 The detailed
derivation steps for both the unconditional independence test and the CI test are relatively complicated.
therefore, we will provide general intuition. For a complete derivation, please refer to the App. B.4
Assume we are interested in the true parameter Ob, e.g., for discretized pairs, O, = (oi,) his h2)) We denote f as its estimation which is close to fo. and ((0) is continuous function. By leveraging Taylor expansion, we have
f(0) f(Ou) f')) +
where the second-order terms and more are omitted, which directly constructs the relationship between the estimated parameter with the true one. Rearrange the term, we get A - 0 (f(0)
f(oo)) (On)) If the denominator is constant and the numerator can be expressed sum of i.i.d samples, we can see i 0 will be asymptotically normal (Van der Vaart 2000).
Let 3/ = (g . contains a group of funetions parameterized by ê. We define the functions evaluated at one sample 2b = *bg(z'), where z' denotes the 1-th sample point. We define the sample mean of these functions evaluated at n points as E,, (v = 1. Ei= byd. similarly, En 11bb = + Ei= wh wyu and eú denotes the Jacobian matrix 0wg We now provide the main ae
result of derived distribution ōi.j oie under the hull hypothesis that tested pairs are independent.
Theorem 3.3 (Independence Test). Under the null hypothesis that the Gaussian variables (X,, Xx) statistically independent oij 0, the test statistics ōū. obtained according to Def. 3.1 for
and discretized pairs Xx), Def. 3.2 for mixed pairs (X,, Xz) and equation 5 for continuous pairs, asymptotically normal:
Vnlōuy- O2d)) 4 N (o. ((En())))
20 (En
1)
(8)
where the specific, of 2b's are presented in Lem. 3.4 Lem. 3.5
and Lem.
3.6
We now provide the specific forms of 2 Since the variables being tested for independence can be both discretized, only one being discretized, or neither being discretized form of uba varies accordingly. The specific forms of aba in these scenarios are defined as follows:
Lemma 3.4. (1 for Continuous Variable Pair). For two continuous variables X, and Xj, where θ òi, and their corresponding [-th samples 2/,, x²y:
wb/= = = E, (X, ōi.j
Lemma 3.5 (ub) for A Discretized- Variable Pair). For discretized variables X, and X,, where Ô (ōr,, hi, hy), and their corresponding 1-th samples ā².
0/1 T(āisj (hi, (hi,hst) # d(h))
46/
where #1, = 1 R2 [X.]. > F., (X,]) # = 1 [x[)].}. and similarly for +). Lemma 3.6 (tb) for A Continuous- Variable Pair). For one discretized variable X, and one continuous variable X,, where Q (ōi.g. h1), and their corresponding 1-th sample point at al.
3/b 7/4- T(ō (0,
-5-((h.)
where f/. =1 1 0, F. ( 1 Em [8,1]

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Derivation of forms of b/ for different cases and their corresponding distribution defined in Eq equa- tion 8 can be found in App. [35] App. [.6] App. [B.7] Up to this point, our discussion has been confined to the case of covariance dtj the indicator of unconditional independence. In the next section, we will present the results of our CI test.
3.3 CONDITIONAL INDEPENDENCE (CI) TEST
To construct CI test of our model, precision coefficient J. and the derivation the corresponding distribution W5 - Wj.A- While obtaining 6j,x from the Σ is straightforward, it leaves the inference problem unresolved. Thus, we leverage nodewise regression and show the regression parameter 3 serving as surrogate of testing for wy.k O, we then construct the formulation of ВJ,x - 83. as the combination of formulation of Oi oijj and show it will also be asymptotically normal.
interested in two matters: calculation of the estimated
The following lemma formalizes the properties of nodewise regression that enable this approach:
Lemma 3.7. [Nodewise Regression Properties] For a p-dimensional multivariate normal variable X -X Xp) ~ [00 Σ) with covariance matrix E and precision matrix Ω2 --1 (Win)) For any j E (l,...,,3 P]. consider the nodewise regression where each X, is regressed on all other variables:
x XkSg c legh
where 3j,k is the regression coefficient of Xx in predicting X,, 3, (Bg,))ay E RRP- is the vector of all coefficients, and Ej is the residual term. Then the following relationships hold:
B, -- R"11 R-
Wiik
B3
$ k:
Wj.j"
The derivation can be found in App. B.8.1 The lemma establishes the deterministic relationships between the regression coefficient 33.k and the entry of precision matrix Wj.k. Since Wjy will never be zero (due to the positive definiteness 12). we can conclude Bj.k serves as an effective surrogate of Wj.k- Moreover, Bj can be expressed in terms of the submatrices of the covariance matrix Σ. We
can further conduct its estimation B (3 ΣΣ Σーコي where the estimated covariance terms can be obtained using Def. 3.1 and equation
Statistical Inference for 3y Nodewise regression offers a direct solution for the estimation
problem. A pertinent inquiry pertains to the construction of the distribution of 3, By. It is crucial to recognize that the distribution of t., osj is already established. Therefore, if we can conceptualize
â - 3, as a linear combination of o., aij. the problem is directly solved, i.e., the 33 - B3 is linear combination of dependent Gaussian variables. The underlying relationship between these variables is as follows:
B 3, -- - -,,, -3 -.i.))
E
The derivation is provided in App. 8.8.2 For ease of notation, we further express the distribution of the difference between the estimated covariance and the true covariance as
n
وs, Oijj
(10)
The specific form of cl,y is given App. B.5 convenience, we express E j.
B.6
for different cases, respectively. For notational 2 E ر and E- E-jj Σ E بدري -J
B.7
Σ E
where ز is the element of the matrix E at the position indexed by (i,j). We propose the statistic and its asymptotic distribution for the CI test in the following theorem.
Theorem 3.8 (Conditional Independence test). Under the null hypothesis that Gaussian variables
X, and X₃ are conditional statistically independent given all other variables X_ (jM}, Lee, 33 = 0, the testing statistic
B = ($-),, [:)
(11)
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Discrete
Dym
Naasbo Coreppics
Nurtheraf
Cardmilty cocoliiingggg xt
Cuddiclitty of conditoning scl
Miced
cood
Dyre umer
Narthrra
Cadisally sondiiiing
Cardinality of condiioning set
Continoous
Dyne Lape
Type
Nusbber Imappics
Cardinaity aondituiing
Cadicality ofcendroning
(00 Typo Typo
291 Type and Type arrr Dey 23345) me200
DCT
Tiderr
Figure 2: Comparison of results of Type and calibrated Type II error (1 power) for all three types of tested data (continuous, mixed, discrete) and different number of samples and cardinality of conditioning set. The suffix attached to test's name denotes the cardinality of discretization; for example, "Fisherz_4" signifies the application of the Fisher-z test to data discretized into four levels. Chi-square is only applicable for the discrete case.
where [k] denotes the element corresponding to the variable Xx in Σ_) - has the asymptotic distribution:
vn(ß3,x B3.1) S NO, a T
veal B¹ ) (oeec(BB,)) (B²,)² aa!
n
-(S xr
where B' = E
[
vec (
and [3 is Bg whose 35,k O: is row-wise vectorization of matrix, and (Σ }.=),>))m denotes the row in -ー),- that corresponds Xg.
In practice, we can plug in the estimation of regression parameter B and set 83, as the substitution of 3, to calculate the variance and do the CI test. Specifically, we can obtain the 35. using equation where the estimated covariance terms can be calculated by solving the bridge equation Eq. U Under the null hypothesis that By. = 0 (conditional independence). we can take the calculated fy into the distribution defined in Thm. 3.8 and obtain the p-value. If the p-value is smaller than the predefined significance level a (normally set at 0.05), we will infer the tested pairs conditionally dependent; otherwise, do not. The detailed derivation of the Thm. 3.8 can be found in App. B.8.2 The pscudocode of DCT is provided in App. D
EXPERIMENTS
We applied the proposed method DCT to synthetic data to evaluate its practical performance and compare it with Fisher-Z. test (Fisher 1921) (for all three data types) and Chi-Square test (FR.S.. 2009) (for discrete data only) baselines. Specifically, and its application in causal discovery. The experiments investigating its robustness, performance in denser graphs and effectiveness in a real-world dataset can be found in App.
investigated its Type and Type II error
4.1 ON THE EFFECT OF THE CARDINALITY OF CONDITIONING SET AND THE SAMPLE SIZE
Our experiment investigates the variations in Type and Type error (1 minus power) probabilities under conditions. In the first scenario, focus the effects of modifying the sample size,
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denoted as n (100, 500, 1000, 2000), while conditioning single variable. In the second, the sample size is held constant at 2000, and we vary the cardinality of the conditioning set, represented as D = (1, ...... 5). It is assumed that every variable within this conditioning set is effective, i.e., they influence the CT of the tested pairs. We repeat each test 1500 times.
We use Y, W to denote the variables being tested and use Z to denote the variables being conditioned The discretized versions of the variables are denoted with tilde symbol (e.g., Z). For both con- ditions, we evaluate three distinct types of observations of tested variables: continuous observations for both variables (Y; W), discrete observations for both variables (Ý, W) and a mixed type (Y, WW) The variables in the conditioning set will always be discretized observations (Z).
To see how well the derived asymptotic null distribution approximates the true one. we verify if the probability of Type I error aligos with the significance level a preset in advance. We generate true continuous multivariate Gaussian data Y, W from Z₁ (single
for the first scenario, and summed over n. for the second), structured as a; Z + E and Σ a. + E, where ag is sampled from U (0.5, 1.5) and E follows a standard normal distribution, independent of all other variables. This ensures Y TL W Z. The data are then discretized into K = (2, 4, 8, 12) levels, with boundaries randomly set based on the variable range. The first column in Fig. (a) (b) shows the resulting probability of Type I errors at the significance level a 0.05 compared with other methods.
good test should have as small a probability of Type II error as possible, i.c., a larger power. To test the power of our DCT, we generate the continuous multivariate Gaussian data Z; from Y, W: constructed as Z; = a,Y + b,W + F,, where a, b, are sampled from (0.5, 1.5) and E follows a standard normal distribution independent with all others, i.e., Y AL WIZ. The same discretization approach is applied here. One should note that directly comparing the p-value with common predefined significance level is unfair since all baselines tend to produce very small p- p-vauues. Therefore, all tests are calibrated in this experiment. The second column in Fig. 2 (a) and (b) correspondingly shows the calibrated Type II error as the number of samples and the cardinality of the conditioning set change, compared to other methods.
From Fig. 2 (a), note that the Type I error rates with our derived null distribution are well- approximated at 0.05 across all three data types in both scenarios. In contrast, other testing methods show significantly higher Type I error rates, increasing with the number of samples and the size of the conditioning set. This indicates that such methods are more prone to erroneously concluding that tested variables are conditionally dependent. Additionally, while alternative tests demonstrate considerable power with smaller sample sizes, our approach requires sample size of 1000 to achieve satisfactory power, particularly in mixed and continuous cases. A possible explanation for this phenomenon is that our method binarizes discretized data, which may not effectively utilize all observations. This aspect warrants further investigation in future research. Moreover, our test shows remarkable stability in response changes in the number of conditioning sets.
4.2 APPLICATION IN CAUSAL DISCOVERY
Causal discovery aims at looking for the true causal structure from the data. Under the assumption of causal Markov condition that the causal relationships among variables can be expressed by a Directed Acyelic Graph (DAG) G and its statistical independence is entailed in this graphic model, faithfulness ensures that the statistical independencies observed in the data can be reliably used to infer the causal structure. Given both assumptions, constraint-based causal discovery, e.g., PC algorithm (Spirtes et al. 2000) recovers the graph structure relying on testing the conditional independence of observation. Apparently, in the presence of discretization, the failures of testing conditional independence will seriously impair the resulting DAG.
To evaluate the efficacy of the DCT, we construct the true DAG G utilizing the Bipartite Pairing (BP) model as detailed in (Asratian et al. 1998). with the number of edges being one fewer than the number of nodes. The subsequent generation of true multivariate Gaussian data involves assigning causal weights drawn from uniform distribution U ~ (0.5, 2) and incorporating noise via samples from a standard normal distribution for each variable. Following this, we binarize the data, setting the threshold randomly based on each variable's 's range. Our experiment is divided into two scenarios:
CCalibration is the process of empirically finding the decision threshold to match the desired significance level, ensuring accurate control of Type error.
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F] soore (Skelcton)
1.0
Precision (Skeleton)
1.0
Recall (Skeleton)
1.0
Sxeleton)
10
0.5-
0.6-
S11
0.
Numb of samples
Numb of samples
Numb of samples
Numb samples
(a) fixed nodes p 8, changing sample size
(500, 1000, 5000, 10000)
Fl score (Skoleton)
1.0-
Procision Skeleton)
1.0
(Skeleton)
1.0-
SHD (Skelcton)
0.5
0.6
0.6
84
10
Numb of variables
Numb of variables
Numb variables
Numb of variables
DCT
Fisherz
Fisherz nodis
Chis-q
(b) fixed sample size 5000, changing node (4, 6, 8, 10)
Figure 3: Experimental result of skeleton discovery on synthetic data for changing sample size (a) and changing number of nodes (b). Fisherz nodis the Fisher-z test applied original continuous data. We evaluate Fj (f), Precision (1), Recall (f) and SHD (L).
In the first, we set the number of samples n = 5000, with the number nodes p varying across 4, 6, 8, and 10. In the second scenario, we fix the number of nodes and explore sample sizes n (500, 1000, 5000, 10000).
comparative analysis is performed using the PC algorithm integrated with various testing methods. Specifically, we compare DCT against the Fisher-z test applied to discretized data, the Chi-Square test, and the Fisher-z test on the original continuous data, the latter serving theoretical upper bound. Since the PC algorithm only returns completed partially directed acyclic graph (CPDAG), we apply the same orientation rules from Dor and Tarsi (1992). as implemented by Causal-DAG (Chandler Squires) 2018), to convert CPDAG into a DAG for easier comparison. We evaluate both the undirected skeleton and the directed graph using structural Hamming distance (SHD), score, precision, and recall as evaluation metrics. For each setting, we run 10 graph instances with different seeds and report the mean and standard deviation for skeleton discovery Fig. 3 and DAG discovery in Fig. in App.
According to the result, DCT exhibits performance nearly on par with the theoretical upper bound across metrics such as Fl score, precision, and Structural Hamming Distance (SHD) when the number of variables (p) is small and the sample size (n) large. DCT significantly outperforms both the Fisher-Z test and the Chi-square test. Notably, in almost all settings, the recall of DCT is lower than that of the baseline tests, which is reasonable since these tests tend to infer conditional dependencies, thereby retaining all edges given the discretized observations. For instance, fully connected graph, would achieve recall of 1.
CONCLUSION
In this paper, we present new testing method tailored for scenarios commonly encountered in real-world applications, where variables, though inherently continuous, are only observable in their discretized forms. Our method distinguishes itself from existing CI tests by effectively mitigating the misjudgment introduced by discretization and accurately recovering the CI relationships of latent continuous variables. We substantiate our approach with theoretical results and empirical validation, underscoring the effectiveness of our testing methods.
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REFERENCES
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Appendix for
"A Conditional Independence Test in the Presence of Discretization" Appendix organization:
Notation Table
15
Proof and Derivations
17
Proof of Thm
17
B.2
Proof of / 4 0
17
B.3
Proof of one-to-one mapping between Fz.j and 0
18
44
Proof of Thm.
18
B.5
Derivation of Lem.
3.5
19
B.6
Derivation of Lem. 3.6
21
7
Derivation Lem 3.4
21
B.8
Proof of T.m.
21
B.8.1 Proof of Lem.
B.8.2 Detailed derivation of inference for 3
22
B.9
Discussion of assumption
25
B.9.1 zero mean and identity variance
25
B.9.2 Discussion of Linear Gaussian Assumption
25
Figure of main experiments: causal discovery
26
Pseudo Code
27
Related Work
28
Resource Usage
28
Limiation and Broader Impacts
28
Additional experiments
29
Linear Gaussian and nonlinear
29
H.2
Denser graph
29
H.3
Multivariate Gaussian with nonzero mean and non-unit variance
29
Real-world dataset
30
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A
NOTATION TABLE
Category
Description
Number and Indices
n
Number of samples
p
Number of variables
Index of a variable i, k: E (1,..., Index of sample (I....,
Random Variables
X
A vector of Gaussian variables
X
A vector of variables whose partial variables are discretized versions of X
Σ
Covariance of
Σ-j.-j
Submatrix of E with j-th row and j-th column removed j-th column of X with j-th row removed
Σ-jj
Ω
Precision matrix of X, equals Σ-1 j-th component of the X
X
All other variables of X with X, and X2 removed Covariance between X; and X;
Wisk
Precision coefficient Wj.k
I-th sample of X, 1-th sample of X
hy
The boundary in the continuous domain that corresponds to the mean of X, Probability of X; larger than its mean: P(X, > B[X;])
Regression coefficient of Xg in predicting X
B
vector of all coefficients regressing Xj
4
Influence function component, represents the influence of the I-th obser- vation on the covariance estimation error
E'
Matrix form of g
Estimation of Variables
Estimation of dijj calculated using equation 6 equation
Σ
Estimation of Σ, matrix form of ف;. Estimation of wj.A
hy
Estimation of hy. calculated using equation
fj
Estimation of Ty. calculated as 1 - 1 { E, (X,))
Estimation of Bj. calculated as Σ - D-
Functions and Operators
P
True probability
P.
Sample probability
EiZ]
Expectation of random variable Z Sample mean random variable Z over n samples
E, [2]
1
condition: if the condition is true, 0 otherwise
(2)
Cumulative distribution function of a standard normal distribution
((2)
1 (22), corresponding to the P(Z 2)
((21, #21
P(Z1 > 21, Ze 22). where (Zz, Z2) follows bivariate normal distribu- tion with mean zero, variance one and covariance
vb
A group of functions parametrized by 0
tb evaluated at sample Jacobian matrix of 0 aó
4/
For Diseretized Pair X,
Probability of both X, and X, larger than their mean: P(X; > E[X,), E[X,]) Estimation of T. - 16 EX E.,
Ti.
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Category
Description
A sample of R... 1{ E, (X,),,, E, [,5}}
For Mixed Pair X,, Xj
Probability of both X; and X3 larger than their mean: P(X, > 0,
B[X;])
Estimation of T.. 1|=1 1a > 0, [X,
A sample P..z [ > 0, ay > [x13
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B PROOF AND DERIVATIONS
B. PROOF OF THM 2.1
If the X2 and X3 are jointly Gaussian and 1 T1 X3|X2, we have
,, 0.
To test X1, X3 are conditional independent given X2, we are interested if Cou(X1, X3l X2) equals zero. Using the law of total covariance, we have
1o
ElCov(Xi, X3/X2. X2) X2] + Coo(B[Xi E[X3 E X2)
(12)
Since X2 is the deterministic function of X2, X2 will be conditional independent with X₁ and X3 given X2. Therefore,
X3 X2) Cov(X1, X3 0.
The first term of equation 12 is zero. We now focus on the second term. Similarly, we have
||Xi|Xx X2] xx]; ||X3/XX, X2] E[Xa|X2];
due to the conditional independence. One can see
Cou(X1, X3 X2) Cov(E [[)
Without loss of generality, we assume the mean of X1. X2 and X3 are zero. Then [X]] and E[X3[Xx are scaled versions of X2. The original equation becomes
Cov(Xn X3/x2, X2) = c Var(X2|X.))
where c is a constant. We know that
Var(X, (X2 X2) [X2 E|X; x22))
which will be zero if and only if X₂ is almost surely function of X2. That means given X2, the value of X2 determined exactly without any randomness, which clearly doesn't hold true in our discretization framework. Thus, X₁ f. X3 X2, which completes the proof.
B.2 PROOF OF ò & Oo
Lemma B.1. For the estimation = (ōr.j his hy), (ōr5, hT), (ōr.)) ) for discretized pairs, mixed pairs and continuous pairs respectively, calculated using bridge equation 3 and equation 6 will converge in probability to 0, = (or.g h4, h2), (a..), hy), (ar.) respectively.
Proof According to the law of large numbers, for the estimated boundary h; and h, whose calculations are defined as hy --1 11 tg)) we should have
n 00, f,
1
Σ
1
(
E,1 2 P(X; B[X,)).
1=1
Recall the definition P(X, E(X,) 1) = 1 - Φ (h,)) according to continuous mapping theorem
Vaart 1998a), as long as the function 1 -.. is continuous, we should have h, 2 hy- And thus b4 2.
14, h3 2 hj-
We further note that न.. (h,, hy, ) and the estimation ह.j can be obtained through solving the function. Similarly, we also have
n + x0,
R.j
Σ 1 E.JE&ELL > BALL B. P( ELX,, E(X,))
Ts.
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According to the continuous mapping theorem, we have ōzs B Gz.z- Thus, the parameter (ōs.. hi, hT) S (dz.j hss h2) for the discretized pair case.
Apparently, the result above could easily extend to the mixed case where we fix h = h = 0. Using the same procedure, we should have (ōi.j: & ho))
For the continuous case whose estimated variance is calculated as ويg 1== dG4a D 11 Ei=1 z.y. according to law of large numbers, we should have n - x0, ōis
my B E|X,X,1 E[ = Oz.j.
1=1
B.3 PROOF OF ONE-TO-ONE MAPPING BETWEEN PJ., AND O..
Lemma B.2. For any fixed h4 and hy. Ti the h,j) Saish, Juash, o(r,, Zzi )d,, is a strictly monotonically increasing function on o' E (-1,1).
(ot for o.j E (--1,1)
Proof To prove the lemma, we just need to show the gradient
80
(T(ai)); (hi, 0ok
1 exp
(h 22 1,j h,, + h2)) 2(1 o
2 25 (1-333)
which is obviously positive for d' E (-1, Thus, we have one-to-one mapping between fig with the calculated ōi. for fixed hy and hj-
B.4 PROOF OF TIM. 3.3
In this section, provide the proof of Thm. 3.3 which utilizes regular statistical tool: Z-estimator (Vart 1998b). Specifically, we are interested the parameter 0 and have it estimation Ó. Let T1, are sampled from some distribution, we can construct the function characterized by the parameter 0 related the æ as (xx). ). As long we have observations, we can construct the function as follows
(
th (xx) = E., [vbo].
1=1
We further specify the form
(0) / vo(x)) E[
Assume the estimator Ô is zero of V i.e., V. = 0 and will converge in probability to O.. which is zero of L i.e., (O,) 0. Expand v (0) in Taylor series around dg. should have
- V () v (0n) + (Ø 0 (0)) + 2 -0 2 (0o)) +...
Rearrange the equation above, we have
Vn,
Ô 0
V/, (00) ( 2TT
1 "E= vo (x)
T (0.) + ( 0 1 (0))
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According the central limit theorem, the numerator will be asymptotic normal with variance 1/ as the mean (0)) = 0 is zero. The first term of denominator V (00) will converge in probability to (On)) according to the law of large numbers. The second term 0 On 0p(1). As long as the denominator converges in probability and the numerator converges in distribution, according to Slusky's 's lemma, we have
E E
Vā 0o)
S
N
0,
(13)
E
Extend into the high-dimensional case should have
0 0, V (0o) v (06)
where the second order term is omitted, further assume the matrix E( is invertible, we have
vn - 0o) 4 (0, (E(+%,)) El (E(())))))
(14)
Specifically, in our case 0g = (oi. A), where A is another parameter set influencing the estimation of Øi.j (will discuss case in case later proof). In the practical scenario, we only have access to the estimated parameter and the empirical distribution P,,, which will converge to their true counterparts. Thus, we have
ōi - -is approx N (0, (e,
I. (I 11.1)
Under the null hypothesis of independent, ot, O. We provide the proof that 1 - 0g of our in App. B.2 Thus, E,, E, (bbg). the function parameterized by ô, should also converge in E,, [ when + Besides, by the law of large numbers, E, E, will converge to E o. Thus, the equation above will converge to equation 14 when n co.
We note that the construction of Z-estimator above require two necessary ingredients: 1. The estimated parameter O should be the zero of the sample mean of criterion function V, 2. The estimated parameter ê should converge in probability to 0g. the zero of the true mean of criterion function V. For different cases (diseretized, mixed, continuous), the construction of criterion function varies. We provide their corresponding derivation in Lem. 3.5 3.6 3.4 respectveely
B.5 DERIVATION LEM. 3.5
Let's first focus the most challenging case where both variables are discretized observations and interested parameter will include Ô = (ā h4g hyy (Although only care about the distribution of d.,j oij. the estimation of boundary haand hy will influence the estimation ōs.j. thus we need to consider all of them).
The next step will be to construct appropriate criterion function ub such that V (0) 0. Given observations (0 2², ........? "....,,",,,, J"), which are discretized version of {x xx? ......, x") we should have
V (((os))) Vn (h:) Vn (hz)
F1.
-T ;; (h, h13))
1
(0)
0
7/ (h+)
0. (15)
1=1
71
1=1
7) ((,,)
Nr, (n((rs))) (n4 vn(h))
1 T(ori,si si fhi, h₂
n1 bo (x')
(Oo) =
क (h+))
(16)
क5 (h₂)
We will not provide proof of this in this paper; however, interested readers may refer
Vaart
1998h
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The difference between the estimated parameter with the true parameter can be expressed
( Bouy av.(he) Jos,) av.((k.)
oVn(ora) The av,.th) The (0,,
(ara) Øhg (v.((he) Bhg av av,)(h;) Bhg
0o-
h; hip
hj h
Bors
T(Ti.g (hi h1))) (hh))
(17)
(hş)
where the specific form of each entry of the gradient matrix is expressed
NV (oi.))
(h₂ 20 + h3))
dor,
2T /11 02,1)
2Tv(1
exp
2(1 -0%,)
Ov, (nss)))))
100
1
11 20, 20, + 122
Ohi
2 /
Dz.j
exp
2(1 [11-)
dx
a (nnss)))
1
h/3 20, + xi
exp
dz
Ohi
2T Oz.
2(1 Oi.j"
(h:))
0:
Ooss
2 mn,,)()))
112
(18)
exp
Oh:
/2T
2
2v ,.,s) 0:
Ohg
ay (ne)) 0:
doi.
mn O;
Oh;
(N,,.)))
h22
exp
ahy 2m
For simplicity of notation, we define
T Oi.j
1=1
where the specific form is of (El,) is defined in equation 17 We should note that {ct,} are i.i.d random variables with mean zero (this property will be the key the derivation of inference of CT). As long as our estimation 9 converge in probability to Og as proved in B.2 we have
Vn 0 0o) 4 N (0, ((E(%,, - 4 (E (4
where vho is defined in equation 16 However, in practice, we don't have access either or the true expectation. In this scenario, we can plug in the sample mean of [ y to get the estimated variance, i.e., the actual variance used in the calculation of
ãiن -iij is
(E., -
F.. (Fn (46 1111))
(19)
m
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B.6 DERIVATION OF LEM. 3.6
Use the same procedure as the derivation of Lem. 3.5 for mixed pair of observations where X; is continuous and X, is discrete, we can construct the criterion function
(v(()ns)) (n((h)))
1
R - T( (0,
(n)
Σ 1=1
w 48
(+))
O.
(20)
fj (b₂,)
1=1
(V((os)) Vn(hi))
Al - To FJ ((2y)
(00)
=
3 (T))
(21)
Iml
The difference between the estimated parameter with the true parameter can be expressed as
( Shg av (wnn She
-1
0u = [i.. - o3. (0% h3 h
Σ
F/. - T( (0, (0) (22) 1 (hz)).
Bols
where the specific form of each entry of the gradient matrix can be found in equation 18 the same procedure, should have the same formation of the variance calculated equation [9
Using exactly
with a different definition of the and bbg defined equation 21 equation 20
B.7 DERIVATION OF LEM. 3.4
Use the same line of procedure in the derivation of Lem. can construct the criterion function
3.5 for continuous pair of variables, we
(0) ((((),)) =
Ōū. = 0.
(23)
1=1
1=
1=1
(00) = (n((ora)
Oi.i
1=1
1=1
Denote 1 ΣE af as to and 1/ Ei= 2tt as Ty. We should have
وis - Oij =
Iz Gz.j
(24)
According to equation
[3 we have
Vū(ōu, - -1s)) ~ (1,,
E E
(L
where 13 1. In practical calculation, we have the variance
E, 2 (Em
1
> (afat F,ū, - -
n²
1=1
B.8
PROOF OF THM. 3.8
B.8.1
PROOF LEM.
3.7
Consider our latent continuous variables X (X1...., N(0, Σ) and nodewise regression
X, X X-3 + 6ej
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where X-, is the submatrix of X with X, removed. We can divide its covariance Σ and its precision
matrix f = E-1 into the predictor X and outcome variable Xj in our regression:
Σ jj = Σ fJ
2 = Ω jj
Σ
s
Σ Σ
Ω s-
Just like regular linear regression, we can get
20, = = Σ - --j
From the invertibility of a block matrix
(A- -A-BD-10)-1 [
A
-D- D - C(A- cy-1 D"+ -
-(A-BD-1Q)-BB)
C
If A and D is invertible, we will have
A B
(A
BD BD
0 (D -D- 00-CA-100 B)-
-BD-1
C D
CCA- 1
Thus, we can get:
Σ 55,- Σ -
E --10)
E Σ Σ - ) E :u) E Σj 1
Move one step forward:
2 - (E,-,(E-j--))
Take transpose for both sides, as long as 12 is a symmetric matrix and s2 j.j 52 2-j we will have
-
By.
We should note testing -zjj = 0 is equivalent to testing 33 0 as the Zzj will always be nonzero. The variable 52 _2_j., captures the CI of X, with other variables. As long as the variable 22j. is just one scalar, we can get
Wj.k
B
Wij
capturing the CI relationship between variable Xj with X2 conditioning on all other variables.
B.8.2 DETAILED DERIVATION OF INFERENCE FOR (3
Nodewise regression allows to use the regression parameter 33 as the surrogate of S2-j.j. The problem now transfers to constructing the inference for B,. specifically, the derivation of distribution of [3 By. The overarching concept is that we are already aware of the distribution of oi, oie and we know that there exists deterministic relationship between 33, with Σ. Consequently, can express B3 - B3 as a composite of oG.j oi,j to establish such an inference. Specifically, we have
B, = وΣ و Σ Σー
- (مر -
- ( - E. + Enj.jBj- -5-1J) -- وー (É E-,,-j)B3 -SS --j,t).
where each entry in matrix - and (Σ-j. - E --و.)) denotes the difference between estimated covariance with true covariance.
For ease of notation, we further denote that
n Oi,
Σ
= n 1=1
Sig
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where e are ii.d random variables with specific form defined in equation 17 for discrete case, equation 22 for mixed case and equation 24 in continuous case.
Suppose that want to test the CI of the variable X1 with other variables, 1. We then have
02.2.... .2.
02. 22P
222
cl
$2.
9-1,-1-E11,
pp.2 ... p.p-
0.2 Opp.
n1
pp. ... Sp.p.
02,1
n
$2,1
Σ111 E-1,1
0p p.
p.1
n
Sp.1
Put them together:
31,2 - 81,2
$2,2
62
<2.p
1
B1, -1,33
n
el
Sa,3
63.p
,13
Ea,1
81 B1
Σ
72
BIp - BI
Sp,2
Br.p
[Sp
As - S is asymptotically normal, who vector of B1 - G1 is a linear combination of Gaussian distribution. However, We cannot merely engage in linear combination of its variance as they are dependent with each other. For example, if Y1, Y2 are dependent and we are trying to find out Var(aYi bY2), should have
Var(Yi) Cou (Y,, Y2)
Cou(Y, Y2) Var(Y2)
[ b
Var(aY, + bY2) [a
b]
(25)
Now, suppose we are interested in the distribution of 31.2
B1,2, we have
22
2.
$1,2
$2.1
81.2 - -1,2 = 1
$3.3
S3,p
f13
$5,1
1=1
S2
[
where (É=1,-1)2 1),; is the row of index of X2 É ([2) denotes the index the variable, e.g.,
( 1)2); represents the first row of Σー,ー1 since the row of first variable removed. ). For ease of notation, we define
222
223
2
2
Y
$3,p
E 1xp-1
o' = E!
$
E RRP-1
$p.3
Sp.p_
1
and
$1.2
$1.3
u = ( - E IPP-1
E RP-1
w =
9
We can rewrite the equation as
81.2 - B1.2 =
Σ
u u(Y w - v²).
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We note that Y' o are variables, and constants (just like the example aY1 + bY2). We further let m = p - to simplify the notation. We can thus write the equation above as vector form:
81.2 B1.2
U1.
W1, W2, Wm
Y12
yi mm.
-v' |oo(()))
w T vec(uw))
where u, represents the i-th element vector u and Yi represents the entry in i-th row and j-th column of matrix y vec represents the row-wise vectorization of matrix, e.g.
Yi
Y12
vec(Y') =
Yia
E IRm²
Ymm
Similar as equation 25 the variance is calculated
Var (Vi(B122 -112)
1
uu vec(uw")))
v² vec(Y))
vl vvc()))
(ve((wo"))
1=1
Now we go back to use the notations of g and E. Under the null hypothesis that X1 11 X2 X Xotherss ie., B1,2 We thus use 31 to denote Bi where 31,2 = 0. Let
53,1
$p.1
2, 22
B
8322
53,3
S3.p
$2.2
Sp,3
and
$ -
-
[2]
vec
Similarly equation 25| The variance is calculated as
Var (vn(O1.2 B1,2)) [122T vec vec(BL, vee
B' TT Tal?
n 1=1
Simply replace the index 2 as general index j, the distribution of 33,N Bi
By 3r $ (0, a [4T
1 n
veec(B))) (aa)))
n²
In practice, we can plug in the estimates of 3, to estimate the interested distribution and do the CI test by hypothesizing 3j,a- 0.
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B.9 DISCUSSION OF ASSUMPTION
In this section, we first justify why the assumption of zero mean and identity variance can be made without loss of generality. Then, we explain the rationale behind the linear Gaussian assumption.
B.9.1 ZERO MEAN AND IDENTITY VARIANCE
In this section, we engage in a more thorough discussion regarding our assumptions about X. Specifically, we demonstrate that this assumption of mean and variance does not compromise the generality. In other words, the true model may possess different mean and variance values, but we proceed by treating it as having mean of zero and identity variance.
The key ingredient allowing to assume such model is, the discretization function 9j is unknown nonlinear monotonic function. Suppose the g' maps the continuous domain to binary variable, and we have the "groundtruth" variable, denoted Xj, with mean a and variance b. Assume the cardinality of the discretized domain is only 2, i.e., observation X, can only be 1. We further have the constant d' as the discretization boundary such that we have the observation
x, = (o) d') = (X; 9 (d)))
We can always produce our assumed variable X, with mean 0 and variance 1, such that Xj a and the same observation with different nonlinear transformation 9j and decision boundary d,, such that
v Xj vi
X, = 1 = 1 9j (())) vbg, (d,) + a).
As long as the observation X, is the same, we should have 2b + y/ (dz). Our assumed model X, clearly mimics the "groundtruth" X/. Besides, according to Lem. B.3 have one-to-one mapping between P.. with the estimated covariance for fixed h, hy. Thus, as long as the observation the same, the estimation of covariance ot, remains unaffected by our assumptions regarding the mean and variance of X, so do the following inference.
We further conduct casual discovery experiments to empirically validate our statement, which is shown in App. H.3
B.9.2 DISCUSSION OF LINEAR GAUSSIAN ASSUMPTION
Discretization continuous variables inevitably leads to information loss in the original data. Compared to the original distributional information, the recovered covariance matrix is naturally less accurate. Given this, constructing a valid statistical inference procedure, rather than solely relying on estimated covariance values for drawing conditional independence conclusions, is desirable.
One major limitation of DCT its reliance on the assumption that latent continuous variables follow multivariate normal distribution. Violations of this assumption lead to erroneous conclusions. For instance, consider a scenario where the relationship between latent variables is nonlinear, such as X, X². In this case, the covariance Js.j equals zero despite a deterministic dependency between X; and Xy. Consequently, even if the correlation is perfectly estimated, the model fails to capture the true underlying relationship, leading to incorrect inferences.
Nevertheless, although the theoretical framework of DCT requires latent continuous variables to follow a multivariate Gaussian distribution, experimental results in various settings, even in situations in which this assumption is violated, demonstrate the usefulness and robustness of DCT, suggesting the development of this technique is essential to causal discovery from discretized continuous data. Further details of the empirical validations are provided in Appendix
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C
FIGURE OF MAIN EXPERIMENTS: CAUSAL DISCOVERY
Fl score (Direction)
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DCT
Fisherz
Fisherz nodis
-
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Figure 4: Experiment result of DAG discovery on synthetic data for changing sample size (a) and changing number of nodes (b). Fisherz_nodis is the Fisher-z test applied to original continuous data. We evaluate Fi (t), Precision (f), Recall (f) and SHD (4).
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D
PSEUDO CODE
Algorithm DCT: Diseretization-Aware Test
I: Require:
Observed data matrix X' Rxxd
Tested pair indices i,j with o
Conditioning set $ _ {1....,, df {t,}}
Significance level a
2: Rearrange Data Matrix
X [1,, X (/a [:,, R
where p = 2
3: Initialize Covariance Matrix
E- I, (identity matrix of size p p)
4:
for 6- 1 to do
5:
for 1: <- q to po
if both [[: v] and [ are continuous then
7:
Compute sample covariance ōg using equation
8:
else
9:
Compute covariance 0 using Equation equation 6
end if
11:
Update covariance matrix:
E(g k] dg
Lk, g] 0g (ensuring symmetry)
12: end for
13:
end for
14:
Extract Submatrices (i and correspond the first and second column of X due the regroup)
Let 1- Rp-1xp-1 + the submatrix & without 1st column and 1st row Let E-1 E R be the 1st column Σ with first row removed
15: Compute Test Statistics
$1.2 + $1
16: Formulate Null Distribution
(2) + Cumulative distribution function of the Normal Distribution defined in Thm. 38
17: Calculate P-value
p-value 2 (1- (131.2))
18: Make Decision
19: if p-value > a then
20: Conclude: X, 11 X, Xs
21: else
22:
Conclude: X, A X, Xs
23: end
24: return The conditional independence decision
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E RELATED WORK
Testing for CI is pivotal in the field of causal discovery (Spirtes ct al. 2000), and a variety of methods exist for performing CI tests (CI tests). An important group of CI test methods involves the assumption of Gaussian variables with linear dependencies. For example, under this assumption, Gaussian graphical models are extensively studied (Yuan and Lin 2007 Peterson et al., [015 Mohan et al. 2012: Ren et al. 2015). To address CI test under Gaussian assumption, partial correlatio serves as a viable method for CI testing (Baba al. 2004). To evaluate the independence of variables X1 and X2 conditional on Z, The technique proposed by (Su and White 2008) determines CI by
comparing the estimations of Xx, 2) and pX2)
Another approach involves discretizing Z and performing independent tests within each resulting bin (argarittis] 2005). Our work, however, diverges from these existing methods in two significant ways. Firstly, equipped to handle data, where partial variables are discretized. Additionally, postulate that discrete variables are derived from the transformation continuous variables in a latent Gaussian model. With the same assumption, the most closely related study is by (Fan er al. (017)) where the authors developed a novel rank -baed estimator for the precision matrix of mixed data. However, their work stops short of providing a CI test for this method. Our research fills this gap. offering the ability to estimate the precision matrix for both discrete and mixed data and providing a rigorous CI test for our methodology.
Recent advancements in CI testing have utilized kernel methods for continuous variables influenced by nonlinear relationships. (Fukumizu et al. 2004) describes non-parametric CI relationships using covariance operators in reproducing kernel Hilbert spaces (RKHS). KCI test (Zhang et al. 2012) assesses the partial associations of regression functions linking x, y. and 2, while RCI test (Strobl et al. (219)) aims to enhance the KCI test's 's efficiency. In KCIP test (Doran et al. (214)) employs permutations of samples to emulate CI scenarios. CCI test (Sen al. (017)) further reformulates testing into a process that leverages the capabilities of supervised learning models. For discrete variable analysis, the G2 test (Aliferis et al., 2010) and conditional mutual information (Zhang et al., 2010) are commonly employed. However, their method cannot deal with our setting where only discretized version of latent variables can be observed.
F RESOURCE USAGE
All the experiments are run using Intel(R) Xeon(R) CPU E5-2680 v4 with 55 processors. It costs hours run experiments in Section 3.1.
G LIMIATION AND BROADER IMPACTS
Limitation So far, the largest limitation our method to treat discretized variables binary. which wastes the available information. Besides that, the parametric assumption limits its generaliz- ability. However, we need to point out this is pretty normal in CI test fields.
Broader Impacts The goal of our proposed method is to test the conditional independence relation- ship given discretized observation. This task essential and has broad applications. We are confident that our method will be beneficial and will not result in negative societal impacts.
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H ADDITIONAL EXPERIMENTS
H.1 LINEAR NON-GAUSSIAN AND NONLINEAR
Our model requires that the original data must adhere to the hypothesis of following a multivariate normal distribution, which appears to potentially limit the generalizability. Therefore, it is worthwhile to explore its robustness when such assumptions are violated. In this regard, we conducted several experiments, including scenarios involving linear non-Gaussian and nonlinear Gaussian.
For both cases, we follow the setting of our experiment where there are p = 8 nodes and p p- 1 edges. We explore the effect of changing sample size n (100, 500, 2000, 5000). Specifically for linear non-Gaussian case, we adhere to some of the settings outlined by (Shimizu et al.) (011), conducting experiments where the original continuous data followed: (1) a Student's 's t-distribution with 3 degrees of freedom, (2) a uniform distribution, and (3) an exponential distribution. Each variable is generated as X, = (P, noise, where noise follows the distribution in (1), (2), (3) correspondingly and f is an arbitrary linear function. The first three rows of Fig. 5 and Fig. 6 show the result of the linear non-Gaussian case.
For the nonlinear cases, we follow setting in (Li et al. 2024). where every variable X; is generated as X; = ((WPA, + noise). noise (0, and f function randomly chosen from (a) fx) = sin( (b) f(x) = x². (c) f(x tz), and (d) f(x) (u)) W is a linear function. Similarly, we set the number of nodes at p 8 and change the number of samples (500, 2000, 5000). For both cases, we run 10 graph instances with different seeds and report the result of skeleton discovery in Fig. Fig-55 and DAG in Fig. (The same orientation rules (Dor and Tarsi 1992) used in the main experiment are employed to convert CPDAG (Chandler Squires 2018 into a DAG). The last row of Fig. 3 and Fig. 6 shows the result of the nonlinear case.
Based on the experimental outcomes, DCT demonstrates marginally superior or comparable efficacy in terms of the Fl-score, precision, and SHD relative to both the Fisher-Z test and the Chi-square test when dealing with small sample sizes. Nevertheless, the sample size increases, DCT's performance clearly surpasses that of the aforementioned tests across all three evaluated metrics, especially in the linear case. Consistent with observations from the main experiment, DCT exhibits lower recall in comparison to the baseline tests. This discrepancy can be attributed to the baseline tests being prone to incorrectly infer conditional dependence and connect a large proportion of nodes. According to the results, our test shows notable robustness under the case assumptions are violated, confirming its practical effectiveness.
H.2 DENSER GRAPH
DCT primarily works on cases where CI is mistakenly judged as conditional dependence due to discretization. Consequently, its efficacy is more pronounced in scenarios characterized by relatively sparse graph, as numerous instances are truly conditionally independent. Nevertheless, the investigation of causal discovery with dense latent graph essential for evaluating the power of test, i.e., its ability to successfully reject the null hypothesis when the tested pairs are conditionally dependent. Thus, we conduct the experiment where p 8, n 10000 and changing edges (p 2.P + 4,p p + 6). Similarly, the latent continuous data follows multivariate Gaussian model and the true DAG 9 constructed using BP model. We run 10 graph instances with different seeds and report the result of the skeleton discovery and DAG in Fig.
According to the experiment results, DCT exhibits better performance in terms of the Fl-score, precision, and SHD relative to both the Fisher-Z test and the Chi-square test. As the graph becomes progressively denser, the superiority of the DCT correspondingly diminishes as there are few condi- tional independent cases in the true DAG. Due to the same reason, The recall remains lower than that of other baseline methods.
H.3 MULTIVARIATE GAUSSIAN WITH NONZERO MEAN AND NON-UNIT VARIANCE
We employed a setting nearly identical to the main experiment, with the only difference being the alteration in data generation: instead of using a standard normal distribution, we used a Gaussian distribution with mean sampled from U(-2,2) 2) and variance sampled from (0, We fix the number of variables as p = 8 and change the number of samples n = (100, 500, 2000, 5000). The Fig. 8 shows the result and demonstrates the effectiveness of our method.
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Figure 5: Experiment result causal discovery on synthetic data with p (100, 500, 2000, 5000) where the data generation process violates our assumptions. The data are generated with either nongaussian distributed (a), (b), (c) or the relations are not linear (d). The figure reports Fi (t), Precision (1), Recall (f) and SHD ({) skeleton.
n
H.4 REAL-WORLD DATASET
To further validate DCT, we employ on a real-world dataset: Big Five Personality https://openssychomettiissss org/ which includes 50 personality indicators and over 19000 data sam- ples. Each variable contains 5 possible discrete values to represent the scale of the corresponding questions, where I=Disagree, 2=Weakly disagree, 3=Neutral, 4=Weakly agree and 5=Agree, e.g.. "N3=1" means "I agree that I worry about things". This scenario clearly suits DCT, where the degree of aggremeent with certain question must be continuous variable while we can only observe the result after categorization. We choose three variables respectively; [N3: worry about things], [N10: I often feel blue J, [N4: seldom feel blue]. We then do the casual discovery using PC algorithm with DCT and compare it with the Chi-square test and Fisher-Z test. The result be found in Fig. g
Based on the experimental outcomes, despite the absence of groundtruth for reference, we observe that the results obtained via DCT appear square tests. Specifically, DCT suggests the relationship N₂ N4|NN0, which reasonable as intuitively, the answer of 'I often feel blue" already captures the information of II seldom feel blue'.
plausible than those derived from Fisher-Z and Chi-
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FI score (Direction)
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Figure 6: Experiment result of causal discovery on synthetic data with p (100, 500, 2000, 5000) where the data generation process violates our assumptions. The data are generated with either nongaussian distributed (a), (b), (c) the relations not linear (d). The figure reports F, (t), Precision (t), Recall (t) and SHD (L) on DAG.
8,
As comparison, both Fisher-Z and Chi-square return a fully connected graph. The results directly correspond to our illustrative example shown in Fig. substantiating the necessity of proposed test.
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Figure 7: Experimental comparison of causal discovery on synthetic datasets for denser graphs with p = 8, n 10000 and edges varying p + 2, p.pt + 4, p + 6. We evaluate Fi (1), Precision (1), Recall
(1) and SHD (L) on both skeleton and DAG.
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Figure Experimental comparison of causal discovery on synthetic datasets for multivariate Gaussian model with 8, n (100, 500, 2000, 5000) and where mean is not zero. We evaluate Fi (t), Precision (t), Recall (t) and SHD (L) on both skeleton and DAG.
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[N3]
worry about things
[N3]
worry about things
[N10]
[N4]
[N10]
[N4]
often blue
seldam fael blue
lafeen feel blue
soldom feel blue
(a) Fisher-Z test
(b) Chi-square test
[N3]
[N10]
[N4]
worry about things
often feel blue
seldom feel blue
(c) DCT
Figure 9: Experimental comparison of causal discovery on the real-world dataset.
33


Reviews:
[{"review_id": "xVwWyT3Tob", "reviewer": "Reviewer_mKQu", "summary": "The authors examine the problem of developing conditional independence (CI) tests when a subset of variables is discretized. This discretization introduces bias into existing CI test methods. They apply the concept of bridge equations to accommodate discretized random variables. Additionally, they demonstrate that the asymptotic distribution of their test under the null hypothesis (conditional independence) is normal and finally evaluate their test empirically.", "questions": "Please see the above comments.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The problem studied is highly relevant and significant across various fields. \nThe theoretical findings and asymptotic analysis provide valuable insights, and the empirical results are promising.", "weaknesses": "The method is developed under the assumption of Gaussianity, which is quite limiting. Although the authors argue that this assumption is not restrictive, I find their reasoning unconvincing. The analysis relies on correlation measures (covariance and precision matrices), which only reveal dependency and independence structures in the case of multivariate normal distributions.\n\nAdditionally, Equation (5) lacks precision, as a Taylor expansion includes a residual term. Specifically, the last term should be $f'(\\theta_*)$  for a $\\theta_*$ between $\\theta_0$ and $\\hat{\\theta}$ rather than $f'(\\theta_0)$.\n\nThe empirical results are limited for showing the generality of the developed method for non-gaussian settings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "T2Io6jHpZt", "reviewer": "Reviewer_mvCf", "summary": "This paper tackles the challenge of testing conditional independence (CI) with a focus on accommodating discretized data, a crucial aspect for various applications. The authors introduce bridge equations to estimate covariance and establish asymptotic normality, facilitating an unconditional independence test. For the conditional independence test, they employ nodewise regression to recover precision coefficients. The effectiveness of the proposed method is substantiated through rigorous theoretical analysis and empirical validation.", "questions": "1. In Theorem 2.1, the authors emphasize the importance of establishing correct CI relationships among the latent continuous variables of interest. However, if we can only observe $\\tilde X_2$, why is it necessary to infer the original conditional relationships? Could the authors provide practical examples to clarify this necessity?\n\n2. In line 192, multiple definitions for $\\tilde h_j$ are possible, such as substituting $E\\tilde X_j$ with other quantities. Considering that $\\tilde h_j$ impacts the asymptotic variance of the estimated covariance as noted in Theorem 3.5, is there an optimal choice for $\\tilde h_j$?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "This paper is well-written and easy to follow. The proposed CI test effectively corrects and infers the true CI relationships among latent continuous variables, recognizing that discretized variables may show different conditional independence compared to their continuous counterparts.", "weaknesses": "The authors primarily discretized variables into binary categories, which could potentially lead to some efficiency loss.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PWfhOC842a", "reviewer": "Reviewer_nGxm", "summary": "This paper addresses a specific challenge in confidence independence (CI) testing when variables of interest are discretized. In practice, accurately measuring continuous variables is often difficult due to data collection limitations, resulting in a common reliance on discretized values for these variables. To tackle this, the authors propose a unified bridge equation to estimate the covariance between two random variables by binarizing them. They then employ nodewise regression to recover precision coefficients that capture the conditional dependence among latent continuous variables. An appropriate test statistic is proposed, and its asymptotic distribution under the null hypothesis is derived.", "questions": "Major questions:\n1. The authors empirically verify the Type I and Type II errors of the test. I’m curious if the asymptotic Type I error and power of the test could also be theoretically verified as well. I'm assuming the asymptotic power or Type I error will depend on $\\mathbf{Λ}$ as well.\n2. Have the authors considered providing a quantitative rate theory to support the test’s validity for smaller datasets? Additionally, are there any debiasing corrections that could be applied for small dataset sizes?\n3. [Optional] The authors present a comprehensive framework based on the assumption that the random vector follows a multivariate normal distribution. I wonder if they have any thoughts on extending this framework to accommodate non-normal variables.\n\nMinor questions:\n1. Line 34: undefined notation $X_{j_1}$. More explanation on what ${j_1}$ means or may consider using $X_{1}$ and $X_{2}$ here.  \n2. Line 89-90: same issue as in 1.\n3. Line 122: inconsistent notations $\\mathrel{\\text{{$\\perp\\mkern-10mu\\perp$}}}$ versus $\\bot$ defined at Line 36. The former notation is more commonly referred as conditional independence in literature.\n4. Line 126: what does DCT stand for? Should introduce the abbreviation beforehand.\n5. Line 754: minor typo, should be $X_1\\mathrel{\\text{{$\\perp\\mkern-10mu\\perp$}}}X_3 | X_2$\n6. Line 137: undefined subscript $\\sigma_{j_1, j_2}$. Consider using different subscripts for $\\sigma_{j_1, j_2}$ and $\\omega_{k,l}$ to distinguish them.\n7. Line 362: remove ) after $vec(B^i_{-j})^T$.\n8. Line 369: as much as I love the theory, I do find the mathematical notations concerning. I don't really see people using $vec$ to denote the rows of a matrix.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The paper presents a well-established theoretical foundation and methodology. The authors begin with a straightforward theorem to illustrate the impact of discretization, emphasizing the potential need for correction in CI testing. The theoretical framework is clearly articulated and thorough.", "weaknesses": "The paper needs improvement in writing. There are several instances where the notations are unclear, inconsistent, and occasionally used imprecisely (see Questions for more details).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tCjYYLnRHu", "reviewer": "Reviewer_WY1e", "summary": "In this paper, a CI test has been designed specifically for handling the presence of discretization. An appropriate test statistic for the CI of latent continuous variables, based solely on discretized observations, has been derived. The key is to build connections between the discretized observations and the parameters needed for testing the CI of the latent continuous variables. To achieve this, first, the bridge equations have been developed to estimate the covariance of the underlying continuous variables with discretized observations. Then, a node-wise regression has been applied to derive appropriate test statistics for CI relationships from the estimated covariance. By assuming that the continuous variables follow a Gaussian distribution, it has been derived the asymptotic distributions of the test statistics under the null hypothesis of CI.", "questions": "1- In the numerical results for the causal discovery example, In Figure 3, with increasing the number of samples F1 score, and precision are reduced for Fisher_z and Chis-q. Usually, increasing the number of samples increases accuracy. What is your explanation? \n\n2-  The proposed CI test was developed and analytically proven for the Gaussian distribution. Although it has been applied to other distributions in the numerical results, can the proposed method be analytically extended to other continuous distributions or to handle nonlinear relationships?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1- Improving CI test accuracy is an essential and practical problem, particularly for structure learning in Bayesian networks and causal discovery.\n\n2- The proposed method is novel, and the solution provided is relatively simple and efficient.", "weaknesses": "1- In (1), the discretization model (function) is unclear. How many discretization levels are used in this formulation? In addition, While the main idea is well presented, the authors have not clearly conveyed the details of the proposed method.\n\n2- Providing a step-by-step explanation of CDT in a numerical example would enhance understanding of the proposed method.\n\n3- The proposed method has improved the accuracy of the CI tests. However, another significant challenge with CI tests is the computational burden, particularly when there are many conditioning variables and a large sample size. Including a complexity analysis or a runtime comparison with existing methods in the numerical results would help demonstrate the impact of the proposed method.\n\n4- Does not the number of discretizing levels affect the accuracy (Type I and Type II error) of the DCT method? In Figure 2 the effect of the number of discretizing levels on Fisherz and Chi2 methods has been checked and shown but the impact of it on CDT has not been reported. Why? I think the number of discretizing levels affects the accuracy of the CDT.  \n\n5- Using practical examples for the application of the CDT in causal discovery can help to indicate the effectiveness of the proposed method in practical and real problems. \n\n6- In proof of Theorem 2.1, in appendix A.1, “Var(X2| \\tilde{X2}) will never be zero since  \\tilde{X2} is the discretized version of X2 (information loss).” is intuitively correct but it is not a precise mathematical expression for proof. Is there any reference or more precise explanation for that?\n\n7- According to Figures 5, 6, and 8, CDT outperforms other methods in F1 score and Precision but generally performs worse in Recall.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DVgS2Py1NY", "reviewer": "Reviewer_i1w6", "summary": "The paper addresses errors in existing CI tests caused by an implicit discretization of the original continuous variable during measurements. The authors propose DCT to deal with this issue and establish the necessary theoretical foundations under assumptions such as having a Gaussian distribution. The method is tested using simulations.", "questions": "Minor comments: \n\nline 73) This is a very interesting idea to use a DAG to invest the discretization problem; however, the authors implicitly assume that the DAG is faithful. The fact that $X_1$ is not d-separated of $X_3$ given $\\tilde X_2$in the DAG does not imply that the $X_1\\not\\perp X_3\\mid \\tilde X_2$ in the distribution. \nline 73) extra comma\nline 89) 1). 2). 3). are not common. Consider 1) 2) 3)\nline 100) So all discretized variables are Gaussian? This is a limitation.\n[important] line 137) The notation seems to be redundantly complex. Why \\sigma_{j_1,j_2} rather than just \\sigma_{j,k}, or even simpler \\sigma_{jk}?\nline 159) better not to start a sentence with an equation.\nline 264) Consider using ^\\top for transpose.\n- Was DCT ever defined? What does it stand for?\nline 264) grammar “contain”\nline 269) “null hypothesis”\nline 270) Be more specific, what are “our settings” and “our framework”?", "limitations": "", "rating": 3, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The studied problem is important. With respect to the presentation, the problem is well motivated, formulated, and explained, and the experimental result is also written clearly. Some good effort is put into the experiments.", "weaknesses": "1) The main drawback of this paper is the overall presentation. I was not able to grasp the main idea, preventing me from properly evaluating this work. The following provides some details:\n\na) The authors were careful and patient in explaining detailed notations up to line 190. However, after that, the pace and notational complexity suddenly got of hand. For example, in line 192, the notation 1{ } is not even defined, let alone the meaning of the equation (the sentence in line 194 does not help). It was as if this part of the paper was written by a different author. I had a hard time understanding the paper from this point.\n\nb) Definition 3.1. What is “the estimated boundary”? It should be rigorously defined somewhere. If the intention of this definition is to define “the estimated boundary” then the language is not appropriate. It should be something like this: “the \\emph{estimated boundary of variables $X_j$ is defined as ….”. Currently, it reads as the estimated boundary is already defined before. \n\nc) Definition 3.4. Isn’t this simply the formula for the sample covariance? that is Cov(X,Y) = 1/n \\sum_{i=1}^n (X_iY_i) - \\mu_X\\mu_Y? If so, what’s the point of giving it a separate definition and name (bridge equation)?\n\nd) Theorem 3.5. I am confused by what is the rigorous statement in this theorem? By “this statistic is approximated to follow a normal distribution”, do the authors mean that the statistic converges to a normal distribution as N grows? \n\ne) Theorem 3.9. Where is the \\hat{\\beta}_{jk} formally defined? I suppose the answer is not eq 13 as then it should have been out of the theorem. \n\nf) The Experiment Section is back on track and written clearly. It was unclear though why the addition of W at line 450 is good for assessing the power. \n\n2) One of the main motivations of this work was its application in Bayesian network (BN) structure learning. Nevertheless, insufficient effort was put into it. As the authors stated, the experiments in Section 4.2 are only for [extremely] sparse DAGs. They do consider denser ones in C2, but that is an only minor improvement, as the number of edges increased from p-1 to at most p+6. True DAGs can be much denser. The authors should provide a fair comparison for this part (also not to move these important parts in the appendix). \n\na) I suggest to test several established datasets in BN structure learning such as ASIA, CHILD, WATER, and ANDES. \n\nb) Similar to the other experiment section, here, non-Gaussian distributions should be tested for structure learning. \n\nc) Even if the results here are weak, DCT appears to be a promising method that is worthy of publication, provided a major rewriting. \n\nd) Note that without this structure learning application, this paper perhaps better suits a statistics journal.\n\n---\nOverall, a good problem is studied in this paper with a seemingly effective solution. However, the presentation of the technical parts needs a substantial improvement.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)