A Sample Efficient Conditional Independence Test in the Presence of Discretization
Authors: Boyang Sun
Yu Yao
Xinshuai Dong
Zongfang Liu
Tongliang Liu
Yumou Qiu
Kun Zhang
Venue: ICML 2025 poster (2025)
Keywords: Conditional independence test
discretization
Generalized method of Moments
Primary area: general_machine_learning->causality
TL;DR: A sample-efficient CI test for variables that are inherently continuous but only discretized observations are available.

Abstract: Conditional independence (CI) test is a fundamental concept in statistics. In many real-world scenarios, some variables may be difficult to measure accurately, often leading to data being represented as discretized values. Applying CI tests directly to discretized data, however, can lead to incorrect conclusions about the independence of latent variables. To address this, recent advancements have sought to infer the correct CI relationship between the latent variables by binarizing the observed data. However, this process results in a loss of information, which degrades the test's performance, particularly with small sample sizes. Motivated by this, this paper introduces a new sample-efficient CI test that does not rely on the binarization process. We find that the relationship can be established by addressing an \textit{over-identifying} restriction problem with \textit{Generalized Method of Moments} (GMM). Based on this finding, we have designed a new test statistic, and its asymptotic distribution has been derived. 
Empirical results across various datasets show that our method consistently outperforms existing ones.

A Sample-Efficient Conditional Independence Test in the Presence of Discretization
Boyang Sun!
Yu Yao? Xinshuai Dong²
Zongfang Liu
Tongliang Liu
Yumou Qiu Kun Zhang 31
Abstract
In many real-world scenarios, interested variables are often represented discretized values due to measurement limitations. Applying Condi- tional Independence (CI) tests directly to such discretized data, however, can lead to incorrect conclusions. To address this, recent advancements have sought to infer the correct relationship between the latent variables through binarizing observed data. However, this process inevitably results in loss of information, which degrades the test's performance. Motivated by this, this paper introduces a sample-efficient CI test that does not rely on the binarization process. We find that the independence relationships of latent continuous variables can be established by ad- dressing an over-identifying restriction problem with Generalized Method of Moments (GMM). Based this insight, we derive an appropriate test statistic and establish its asymptotic distribu- tion correctly reflecting CI by leveraging node- wise regression. Theoretical findings and Empir- ical results across various datasets demonstrate that the superiority and effectiveness of our pro- posed test. Our code implementation is provided in http:://ggitubb..oobbbyaaaggggu
(8)
(b)
Figure I. Illustration of data generative processes using causal graphical models: (a) fork. (b) and (c) chain. discretization process maps latent continuous variables (white nodes) to observ- able discrete variables (gray nodes), denoted with tilde (~>).
overlooked issue is whether the analyzed variables are truly discrete or they are inherently continuous but appear discrete due to measurement limitations.
In many real-world applications, data collection methods impose unavoidable discretization. That is, continuous variables are artificially binned into categories because of constraints in measurement precision. This phenomenon across various fields, including finance (Changsheng & Yongfeng, 2012; Damodaran, 2012), psychology (Mossman et al., 2017; Johnson et al., 2019), and recommendation sys- tems (Sparling & Sen, 2011; I; Dooms et al., 2013). In these stock prices, cognitive ability scores, and user preferences frequently transformed into discrete scales, often leading to biases in statistical inference.
domains, inherently continuous variables-sut
When discretization occurs, traditional CI tests can fail to capture the true CI relationship. As shown in Figurel where illustrate the discretization process with a causal graphic model (Pearl, 2000), for all CI tests unaware of discretiza- tion, the intent is to test the CI of latent continuous variables X1, X3 given X2, what is actually being tested is their dis- cretized counterparts X1, X, given X2. According to the faithfulness assumption (Spirtes et al., 2000), we can in- fer that X, IL Xa|[X2} while X, A Xa1[X2). This mismatch between the latent continuous variables and their discretized counterparts causes traditional CI tests, when applied to discretized observations, to draw incorrect conclu- sions about the true CI relationships of interested variables.
1. Introduction
Conditional independence tests for discrete variables are fundamental in statistical analysis and widely applied across various disciplines. Traditional methods including the chi- squared test (F.R.S., 2009), the G-test (likelihood ratio test) (McDonald, 2009), and measures based on conditional mu- tual information (Kubkowski et al., 2021) are well estab- lished and extensively used. However, a critical yet often
'Mohamed bin Zayed University of Artificial Intelligence "Sydney Al Centre, The University of Sydney 'Carnegie Mellon University 'Peking University. Correspondence to: Yumou Qiu <qiuyumou @ math, wwth.pku..duu...... eku.edu.cn> Kun Zhang <kunzl @cmu.edu>>
Recent work Discretization-Aware CI test (DCT) (Sun et al.,
Proceedings of the nd International Conference Mochine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025
by the author(s).

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A Sample-Efficien Conditional Independence Test in the Presence of Discretization
2024) has attempted to address this issue by establishing the correct relationship between discretized data and latent variables through binarization of the observed data. While this approach facilitates more accurate CI testing by sim- plifying the data structure, it inherently leads to a loss of information. The reduction of data to binary form can sig- nificantly degrade the performance of CI tests, especially in settings with small sample sizes where the preservation of information is crucial for reliable statistical inference.
Their approach can be decomposed into three steps: 1. The calculation of estimated covariance Σ, based the prop- erty that the proportion of both observed variables exceeding their means reflects the underlying covariance, solved using a single equation. 2. The deviation of covariance matrix Σ En follows a multivariate normal distribution utilizing Z-estimator( (aart,, 1998). 3. The deviation of precision matrix 5 S2" also follows multivariate normal distri- bution utilizing node-wise regression (Callot et al., 2019). However, despite having multiple solvable equations from the discretized observations, only one parameter of interest exists per variable pair, leading to an over-identification is- sue. Efficiently utilizing all available information is the key challenge, motivating us to explore the usage of GMM.
Motivated by the limitations of existing methodologies, this paper aims to introduce a samplc-efficien CI test that cir- cumvents the need for binarization, thereby preserving the full richness of the data. Our approach leverages the Gen- eralized Method of Moments (GMM) to address the over- identifying restrictions problem, enabling the estimation of covariance of latent continuous variables without sacrificing information. We then adopt the strategy of DCT to derive test statistics and their asymptotic distribution for CI test- ing, utilizing nodewise regression (Callot et al., 2019). The paper seeks to contribute as follows:
Generalized Method of Moments GMM (Newey, 2007; Hansen, 1982) is statistical estimation technique offer- ing principal solution to the over-identification problem. Suppose 0 denotes a X 1 vector, x² denotes the obser- vation of a data sample where i E (I...., is the index. f()) = f(x', 0) be a m X 1 vector of functions. For the true parameter 0", we will have
We propose Discretization-Awar CI Test with GMM (DCT-GMM), a novel CI test tailored for discretization.
B[f(00]] =0
We provide a theoretical analysis proving that DCT-GMM is consistent and has lower variance than DCT, making it more sample efficient. (Ziegel, 2002).
Using g(0) = 1 E (0) denote the sample average of the fi))). The A is a m x m positive semi-definite matrix. The GMM estimator is given by
We empirically demonstrate DCT-GMM's effectiveness and superiority over state-of-the-ar CI tests, particularly in small-sample regimes.
i arg min
9 Ag(O).
providing a framework for valid inference (Newey, 2007). In our case, by properly designing the moment functions and selecting the parameter of interest, GMM can efficiently fulfill the objective of estimation and addressing the over- identification issue.
2. Related Work
Conditional Independence Test Testing for CI is fun- damental concept in statistics, with linear Gaussian models traditionally dominating due to their simplicity and inter- perability. These models assume linear dependencies and Gaussian noise, providing closed-form solutions for testing through metrics like partial correlation (Yuan & Lin, 2007: Peterson et al., 2015; Mohan al., 2012; Ren et al., 2015). However, the linear Gaussian assumption restricts the gen- erality. Recent CI testing advancements leverage kemnel methods for nonlinear continuous relationships (Fukumizu et al.. 2004). Methods like KCI (Zhang et al., 2012) and RCI (Strobl et al., 2019) analyze partial associations, while KCIP (Doran et al., 2014) employs sample permutations to simulate CI. For discrete variables, G2 (Aliferis et al., 2010) and conditional mutual information (Zhang et al., 2010) are rank
3. DCT-GMM: Discretization-Aware CI Test with GMM
Notation Throughout this work, we use X, to denote the j-th component of the vector of variables X
(X1, (X1,..., Xp) with finite observations {x] { ..),. We denote the sample mean given n samples by [X,]]
I ΣE 2² while its true expectation is E[X;]. Similarly, the empirical probability is represented by P., and the true probability by P. For a parameter a, its true value is a" and its estimation is å. For a matrix X, -T denotes the transpose of its inverse. We use X-; to represent all other columns of X without X,. Similarly, X-j-j is the subma- trix of X without jth column and jth row, and the X-jj is the vector of jth column without jth row. For full notation table, please refer to Appendix A.I.
standard tests. A recent advance on permutation-bast test, MPRT (Dong et al., 2025), can also be used to test van- ishing partial correlation in the presence of discretization.
Prior work DCT (Sun et al., 2024) moves the first step to- wards the CI test specifically for the discretization scenario.
Problem Setting In this paper. we adopt the same non- paranormal model as DCT (Sun et al., 2024). Specifically,
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A Sample-Efficient Conditional Independence Test in the Presence of Discretization
we consider of independent identically distributed (i.i.d.) p-dimensional random discrete variables, denoted For each discrete variable X3 with finite observations ( #77) there exists corresponding latent Gaussian variable Xj. The transformation from X, to X, is governed by an unknown monotone nonlinear function 9j and thresholding function fz. The function f3 o 9] X' + X' maps the continuous domain of X, onto the discrete domain L. Specifically, for each variable Xj. there exists finite constant vector d - (dy Id 11 characterized by strictly increasing elements such that
effectively construct both the estimator Jajj and character- ize the distribution of و - Ti using GMM. Second, we establish that the nodewise regression parameter 83,A serves as an effective surrogate for the precision matrix element Wyk- Finally, we show the asymptotic normal distribution of - -i,x k by analyzing its relationship with the component distributions of منزر "Jisa"
as X = (X1, Xy..............................
3.1. GMM for covariance estimation and inference
Estimating discretization boundaries The discretiza- tion scheme maps the X; onto a finite set of dis- crete values according to the discretization boundaries, maintaining the ordinal relationship of the original con- tinuous variable while reducing its resolution. For ease of notation, we denote the augmented discretiza- tion boundary oj := [0j.0. SA..... 9:MM- C...
1 9j(X,) < <d3,1,
X, - [(,,(0,)))))-
= d2m= 1 < (X,) dz.m: (1)
M (x(X,) > dj.M-1-
m, for 3j (d) < Xj < 9j (dj.m). where m. is an integer ranging from 1 to M. That is, there exists a finite constant vector Cj = 197 (d...)...... 99 (dj,M- acting as the "discretization boundary" that partition the X, into M cate- gories. We refer M as the cardinality of the discrete variable X,. Without loss of generality, we assume X ~ (0, )) with Σ- (T) and وソ = 1. That is, assume the original continuous variables X follow multivariate nor- mal distribution with zero mean and unit variance on the diagonal of Σ. We provide a detailed discussion regarding its rationality and w.l.o.g in Appendix B.
Equivalently, we can conclude X;
[-x., 9j (, (d..)...... 1, (dj. +oo]. We further de- note 2(-) as the cumulative distribution function (cdf) of the standard normal distribution. Our available observation consists of binned discrete values. Since X, ~ (0. 1) ac- cording to the assumption, we conclude that PXX, = m) = P(cj.m-1 <x < (..)). That is, the probability of observ- ing a discrete value corresponds to the probability of the original continuous variable falling into particular region. Although the true probability is not directly accessible, it can be estimated by calculating the sample proportion of the observations within each bin. Specifically, we can obtain the estimation of the discretization boundaries
Objective We aim to develop a CI test to infer the cor- rect CI relationship between latent continuous variables X Xp)) which are the interested ones given their discretized observations X only. By assuming lin- car Gaussian of original continuous variables, our objec- tive directly transfers to deduce the statistical inference of covariance matrix Σ- = (A)ntt) for independent test and precision matrix Ω- = Σー = (w)x)) for CI test (Baba et al., 2004). Specifically, the covariance Tiita = indicates that X,, IL Xjz. and the precision coefficient wiju = indi- cates that X,- 1L other variables in X except X, and Xx- Technically, we are interested in two key tasks:
2j. = +
fg.K),
(2)
where Ty is the empirical probability defined as řj. == P Px,, = k) = -. 1(3) = k)), serving as the esti- mation of the true probability Tg := P(X, = k). The indicator function 1 (condition) is 1 if the condition hold true, 0 otherwise. This formulation provides closed-form solution for estimating the discretization boundaries from observed discrete data.
(2a). where X- (ja) represents all
Estimate covariance through single equation The chal- lenge lies in estimating the latent covariance Oj.z2 with discretized values. For a pair of continuous variables X,, and X22- the discretization scheme essentially creates a "grid". Each cell represents a specific combination of dis- cretized values for both variables. When we count how many samples fall into one cell, the sample proportion within each cell provides an empirical estimate of the joint probability density, which can be expressed as (X,,,, =
Obtain مزز and WزB serving as the estima- tion of the corresponding true parameters with only dis- cretized observations available.
Estimation:
Inference: Derive the distribution مزز - for inde- pendence test and Wjx- wfx for CI test.
m, X₂ j2 = =k), serving as the estimation of the true proba- bility m-1 < X₁, < <h G2 < <X,, < k.
In the subsequent section, we develop our theoretical frame- work through three key steps. First, we demonstrate that for any pair of continuous variables Xj, and Xj2 with their cor- responding discretized observations X3, and X2 we can
According to our assumption that the latent variables fol- low a multivariate normal distribution, the true probability
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A Sample-Efficient Conditional Independence Test in the Presence of Discretization
C1 mi S 12.k-11 Gg. j. 52 which is the edf of bivariate normal distribution with the true covariance 3} integrated over the rectangular region defined by -m m..ll x 1e -- x. For a specific form of the function, please refer to Appendix A.2.
above is given by
first element of Ó. The next question is, how construct the distribution مززز2 "2iia- Given this, propose the following theorem:
Theorem 3.1. Under the null hypothesis that the original continuous variables X II. Xi with the moment function fi( defined as Eq. (4), when number of samples - +00, the estimator Ojúbz is asymptotically normal distributed:
For notational convenience, we define fj.ja.m :- (x,(;, = m, X3 = k) as the empirical joint probability, and ja,m := P(X,; jn = m, X = k) as the true probabil- ity. We use 13 j1 1 ( m, -2 = k) as the indica- tor of sample i. The empirical cell density can be easily com-
+ (CFA))-(GCC)))) (6)
-10))) which will converge in distribution ((, V11).
puted from the observation as fj.z.mm = + >i=1 1 F jk. The estimated covariance م3 can then be obtained by solving following equation:
where V = (G "AGGAG"" (SAG(GTAG)-1 and
V11 is its first entry,
S.m m: j2.k Gjakk ( (jija),
(3)
where can be computed using Eq. (2). We call the equation
ae is the expectation of the Jacobian matrix of the moment function at true parameter 0*,
G = [8| "
0
above "bridge equation", which provides a direct solution
to recovering the underlying covariance by only using the discretized observations.
G = E 1 is sample average of the Jacobian matrix 86
However, this formulation presents an overidentification challenge. For any pair of discrete variables Xj, and X₁2 with cardinalities M and K respectively, we obtain M x K distinct cells, each corresponding to its own equation. This results in an overdetermined system with M K equations but only one parameter of interest jnjje- This overidentifi- cation is a key limitation of DCT (Sun et al., 2024), which utilizes only single equation despite the availability of mul- tiple informative constraints. In the following section, we demonstrate how GMM acts as a principled framework for efficiently leveraging all available information from these multiple equations, thereby offering a preciser solution.
of moment function at estimated parameter 0
S = E is the covariance matrix of the moment function f0
The detailed derivation can be found in Appendix F.2. Since we never have the access the true parameters 0" and the true expectation [[[,, in practice, we can plug in their estimation ET, [ ( and E,, (O)) to calculate the variance of the asymptotic distribution. For the weighting matrix A, theoretically, any positive semi-definite matrix is applicable to the theorem above. A common choice could be the identity matrix.
Move from single equation to multiple equation For pair of variables Xj, and Xj, with cardinality M and K correspondingly, we define the parameters of interest 0- = ja Cni Cj2) E M+M Let (0)) = f( a6... 0) E
Two-step GMM Apparently, the choice of weighting ma- trix A plays an important role in determining the statistical property of the GMM. Specifically, A directly influences the variance of the distribution 0J Jujz The question is how choose A optimally to minimize the asymptotic variance of the estimator. According to the classical theory of GMM (Hansen, 1982), we have the following lemma:
RMK referred as the moment function with the form:
f((0)
(4)
(c 1. 1. Ch.M K K- (2.3)))))))
For the true parameter 0" the population moment condition satisfies [0') O. The detailed derivation of this condition can be found in Appendix F.1. Let the sample analogue of the moment condition be g(0) 1 +> (0). Given a positive semi-definite matrix A E R
Lemma 3.2. Suppose the choice of A D S-1, where S = ) is the covariance matrix of f((0") for the true parameters 0", then
MK x MK as
Valānn -han) - | [TTA))GGAAA (7)
weighting matrix, the GMM estimator given by
θ arg min 8 ((0)TAA(0)
(5)
will converge in distribution to normal distribution with variance (0, Vn1). where V =GTTS-GG) strictly smaller in the positive semi-definite sense compared to the asymptotic covariance matrix in the one-step GMM estima- tor given in Theorem 3.1. Here, G and G have the same definition as in Theorem 3.1.
This formulation leverages all M x K moment functions simultaneously to obtain the estimation 0 efficiently uti- lizing the available information from the discretized obser- vations. The estimated covariance مززز is nothing but the
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A Sample-Efficiet Conditional Independence Test in the Presence of Discretization
The detailed derivation is provided in Appendix F.3. In practice, the procedure begins by estimating the parameter of interest using predefined weighting matrix, such as the identity matrix. Next, the covariance of the moment functions is used to construct the optimal weighting matrix, enabling the final GMM estimator to efficiently re-estimate the parameters. This two-step GMM approach is a well- established technique for achieving asymptotic efficiency, and its superiority is empirically validated in Section 4.
The derivation can be found in Appendix F.4.1. The first row of Eq. (8) indicates that 83. is scaled version of Wike- Since wij will never be zero due to the positive definiteness of Ω. testing if 53 = 0 is exactly the same as testing Wikk = 0. Thus, 8j.k serves as an effective surrogate of wike- Now our focus transfers to calculating By. and deriving the distribution of 93 - f3
We further note that the second row of Eq. (8) constructs consistent relationship between 3, and the covariance matrix Σ. Thus, we can conduct its estimation as B, (3g,)) = ビー -ー
The estimated covariance Jjije serves as an indicator of unconditional independence. Following the framework pro- posed by (Sun et al., 2024), we build such a CI test utilizing nodewise regression.
و where the estimated covariance terms can be obtained through solving Eq. (5).
Statistical Inference for B3, Nodewise regression trans- fers the parameter of interest from Wik to 3r While the estimation of $3.a has been effectively solved, the next ques- tion is how to construct the distribution of B - 8j.N- Fortu- nately, we have already established the asymptotic distribu- tion of juja - - Therefore, if we can express 8, - as linear combination of j, - Ja the problem is readily solved, as 3, 33 will be linear combination of dependent asymptotically normal random variables. The underlying relationship between these variables is as follows:
3.2. Nodewise regression for constructing CI test
In this subsection, we follow (Sun et al., 2024) and leverage the nodewise regression to derive the CI test. For com- pleteness, we present the main results here and refer to the original paper for detailed treatment. The practical imple- mentation is also discussed at the end of this subsection.
By assuming the X follows a multivariate normal distri- bution, our task of constructing CI test is equivalent to (I). computation of likk based on the discretized observations; (2). construction of uik w Targeting both tasks, the nodewise regression leveraged, which shows that:
- = - ( Ez
E Enn))
(9)
The derivation of this result is provided in Appendix F.4.2. For notational convenience, we express the difference be- tween the estimated and true covariances as:
The regression parameter Sy can serve as an effee- tive surrogate for testing the null hypothesis that X, ILI Xx|X- (1883, i.e., Wisk = 0.
1 Σ ni
The formulation of f3 - -8,x can be expressed as linear combination of مززز j..."' which allows to derive its distribution and facilitating the CI test.
0h - -Jiia =
Siiz
(10)
where the specific form of $).u is given in Theorem 3.1 and Lemma 3.2. We further denote E Σーjーj Σ Σーjーj nn Σm = -j- and ورح - Enj = - E ー are matrix form of Eh
The following lemma establishes the key properties of node- wise regression that support this approach.
Lemma 3.3. [Nodewise Regression Properties] (Sun et al., 2024) For a p-dimensional multivariate normal variable X = (X1...., .p)) ~ NOO, with covariance matrix Σ and precision matrix Ω = E-1 = (wik)ss For any j E [I,...,P,, consider the nodewise regression where each Xj is regressed on all other variables:
Conditional Independence Test We apply the follow- ing theorem for the CI test, with proof provided at Ap- pendix F.4.2 for completeness:
Theorem 3.4. (Sun et al., 2024) Under the null hypothesis that X, and Xn are conditional statistically independent
given set of variables X (-sk)) L.e., Bz,k 0. the statistic
X, = Ex33r,++ + Cg.
- E- 11)
(11)
where 3... is the regression coefficient of Xs. in predicting X,. B, = (B),k)kss E RP-1 is the vector of all coefficients,
where [Rc] denotes the element corresponding to the variable Xx in E } 33. has the asymptotic distribution:
and Cj is the residual term. Then the following relationships hold:
vn(β -33,)) S (0, V), where
Wikk
Bi,k= =
j j k.
(8)
B₁ = Σー -ー Σ أز E 1
V a!ti" n
E veel vee(B',) /uec B²)) (a)))
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A Sample-Efficien Conditional Independence Test in the Presence of Discretization
Type ETTE
Type IT crror
Tyno
Type
Number Samples
Cardinality of Conditioning
Number of Samples
Cardinality of Conditioingg Set
Chi-sqtare
DCT-GMM
GMM-
DCT
Fieberz
0.05
Figure 2. Comparison of results of Type I and Type II error (1-power) for discretized observations. DCT-GMM one uses one-step GMM with A setting identity, and DCT-GMM_ uses two-step GMM with A setting the sample covariance of moment functions.
The formal theorem and detailed derivation can be found in Appendix G. Intuitively, if we construct the moment functions the same DCT, we will reach the estimator with the same variance. However, the GMM framework allows the introduction of additional moment functions, thereby reducing the variance. We empirically validate the theorem in Section 4.3.
B' : and By is $3 whose $J O,
)r - k/.
-
alk
and vec is row-wise
vec [5
vectorization of matrix, and ( 1,) denotes the row in ー that corresponds to Xs.
4. Experiment
Practical implementation for DCT-GMM In practical implementation, we estimate the regression parameter B3 substitute for 3j to caleulate variance
We applied the proposed test DCT-GMM to synthetic dataset to evaluate its performance compared with baselines includ- ing DCT (Sun et al., 2024), Fisher-z test (Fisher, 1921), Chi-square test (FR.S., 2009). Specifically, we investigate its Type I and Type II error in different scenarios and its application in causal discovery. The experiments investigat- ing its performance in denser graphs and effectiveness in real- world dataset can be found in Appendix E.
and set f. and conduct the confidence interval test. Specifically, we derive م using the estimation equation in Equation (11), where estimated covariance terms are calculated utilizing GMM. As discussed Section 3.1, we consider two GMM estimators: naive GMM (also called one-step GMM, the one without carefully designing A) and two-step GMM. In this paper, empirically validate the effectiveness of both, whereas A is chosen an identity matrix for the one- step GMM and the E,, (O) for the two-step GMM following Lemma 3.2. The pseudo-code of both approaches is provided in Appendix C.
4.1. On the Effect of the Cardinality of Conditioning Set and the Sample Size
We conducted experimental study to examine the be- havior of Type and Type II error probabilities under two distinct experimental designs. The first design explores the impact of sample size variation, specifically testing sam- ple sizes of n = (200, 500, 1000, 2000) while maintaining single conditioning variable, noted as D = 1. In the second design, we fixed the sample size at n 2000 and systematically varied the number of conditioning variables D = (1, (I.....5 .,)............. 5). We assumed that all variables in the condi- tioning set are influential and affect the confidence intervals of the tested pairs. Each experimental configuration was replicated 2,000 times to ensure robust statistical analysis. We use X and Y to denote the tested pairs and Z to denote the variables being conditioned on.
Under the null hypothesis of conditional independence (3y, = 0), we substitute the calculated 3y into the distribu- tion defined in Theorem 3.4 to obtain the p-value. Statistical inference follows a standard hypothesis testing approach: if the p-value is less than the predefined significance level a (lypically 0.05), we conclude that the tested pairs are con- ditionally dependent. Conversely, if the v-vuuue exceeds a, we fail to reject the null hypothesis and deduce the tested pairs are conditionally independent.
3.3. Comparison with DCT
One of the motivations behind DCT-GMM to address the insufficient data utilization in DCT. This naturally raises the question: DCT-GMM necessarily superior to DCT? Toward this question, propose the following theorem: Theorem 3.5. (Informal) When n > +o,, by constructing the moment functions properly. the DCT-GMM with two- step GMM has a lower variance than DCT.
To assess the accuracy of the derived asymptotic null distri- bution, we evaluated whether the Type eπror probability aligns with the predetermined significance level a 0.05. We first generate Z as an independent multivariate normal distribution whose mean and variance are randomly sampled from a uniform distribution (0, 1). We then generate cor-
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A Cample-Efficient Conditional Independence Test in the Presence of Discretization
score (Skeleton)
1.0
Precision (Skeleton)
1.0
Recall (Skeleton)
1.0
SHD (Skeleton)
7.5
0.8
0.3
2.5
06
0.0
10
Numb of variables
Numb of variables
Numb of variables
Numb of variables
(a) Fixed samples 2000, changing number of nodes p (4,6,811)
F] (Skeleton)
1.0
Precision (Skeleton)
1.0
Recall (Skeleton)
SHD (Skcleton)
0,8-
0.8
6-
0.8
0.6
4
0.6-
1000
1500
2000
500
1500
2000
500
1000
1500
2000
500
1000
1500
2000
Numb of Samples
Numb of Samples
Numb of Samples
Numb of Sumples
(b) Fixed nodes
10, changing samplc size π = (100,500,1000,2000)
Chi-square
DCT-GMM
DCT-GMM
Fisherz
DCT
Fisherz oracle
Figure Experimental result of skeleton discovery on synthetic data for changing number of nodes (a) and changing sample size (b). Fisherz_oracle is the Fisher-z test applied to original continuous data. We evaluate Fi (t), Precision (f), Recall (f) and SHD (L).
responding X and Y using Z, structured E a, a,Z, + E, (for the first scenario, D 1), where a₂ is scalar sampled from standard normal distribution and E; follows a stan- dard normal distribution. This ensures that X Y Z. The data are then discretized into three levels, with random boundaries set based on the support of each variable, pro- ducing the discretized observations X, Y, and Z. The first two columns of Figure show the resulting Type error at significance level of a 0.05.
creases, both the Chi Chi-Square and Fisher-Z tests tend to yield larger Type errors because these methods measure whether X TL Y Z. More samples only reinforce incorrect conclu- sions. Additionally, DCT-GMM demonstrates significantly higher power compared to DCT, particularly for small sam- ple sizes. This advantage arises from DCT-GMM's ability to utilize more information by solving multiple equations, highlighting its superiority and effectiveness.
4.2. Application in Causal Discovery
A robust statistical test should minimize the Type II error, thereby maximizing statistical power. To evaluate the power of the proposed DCT-GMM, we first generate X and Y as independent pairs following a normal distribution, with mean and variance randomly sampled from uniform distri- bution (O, 1). We then generate the conditioning variable Z as Z, = a,X' b,Y E,, where and b; are scalars randomly drawn from a standard normal distribution, and E, follows a standard normal distribution, i.e., X A Y Z. The same discretization approach is applied here. The last two columns of Figure illustrate the Type II error rates for both varying sample sizes and changing cardinalities of the conditioning set scenarios.
CI testing plays pivotal role in causal discovery, which aims to uncover causal relationships from observational data, Two fundamental cessuptiiossffattfulll Markov condition allow causal structures, represented by a Directed Acyclic Graph (DAG) G, to be inferred from statistical independence relations. Based on these principles, constraint- based methods like the PC algorithm (Spirtes et al., 2000) recover graph structures through CI testing, However, discretization compromises the reliability of CI tests. leading incorrect dependence assertions and distort- ing the inferred DAG.
and the causal
To validate the effectiveness of DCT-GMM, we follow the setting of (Sun et al., 2024) and apply the PC algorithm with different CI testing methods on synthetic dataset. Specifi- cally, the true DAG is generated using the Bipartite Pairing (BP) model (Asratian et al., 1998), with weights drawn from uniform distribution U (1, and incorporating noise following standard normal distribution. The number of
According to the first row of Figure 2, DCT-GMM (for both steps) and DCT show superior performance in maintain- ing Type error close to the significance level across all sample sizes and conditioning set cardinalities, while other baselines, which do not account for discretization, exhibit significantly higher Type errors. As the sample size in-

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A Sample-Efficient Conditional Independence Test in the Presence of Discretization
060
Variance
Variance
0.2
041
010
50
150
188
1000
28
530
123 Number of Samples
190
1000
200
300
100
1238
1000
250
50.
70
Number of Samples
Number of Samples
Number of Samples
dxy
Dsy
0.5
DCT-GMM one
DCT-GMM two
DCT
Figure Comparison of Variance and MSE estimated covariance using DCT and DCT-GMM.
edges in the DAG is one fewer than the number of nodes. While this graph is relatively sparse, the main focus DCT- GMM is to correct CIs incorrectly judged as conditional dependence due discretization. As the number of edges increases, such cases of true CI become rarer. We also in- vestigate the performance of DCT-GMM in denser graphs, detailed in Appendix E.I.
lines tend to misinterpret conditional independence as de- pendence, leading to denser inferred graph structures.
4.3. Empirical Comparison of DCT and DCT-GMM
To validate the superiority of DCT-GMM over DCT, we conducted experiments to examine the variance and Mcan Square Error (MSE) of their respective covariance estima- tors. Both methods adopt the same nodewise regression framework to transition from و - Ay. Bi,K* Therefore, evaluating the covariance estimator directly re- flects the performance of the CI test.
The continuous data is then discretized into three levels, with boundaries randomly generated according to each variable's s support. The experiment is divided into two cases: In the first, we fix the sample size at 2000 while varying the number of nodes p = (4, 6, 8, 12). In the second, fix the number of nodes at 10 and explore sample sizes of (100, 500, 1000, 2000).
For a given pair of variables and Y, we denote the covariance estimated by DCT as oxr and that by DCT-GMM as o8 Xx.- We empirically assess the stabil- ity and accuracy of these estimators by comparing their variance and MSE across varying sample sizes (100, 200, 500, 1000, 2000) under the scenario that the true covariance oxy = (0, 0.5) and fixed discretization level M = 3. The Figure 4 shows the results.
We compare DCT-GMM (both steps) against the Fisher- test (Fisher, 1921), the Chi-square test (F.R.S., 2009), and the previous work DCT (Sun et al., 2024) applied to discretized data. Additionally, apply the Fisher-Z test to the original continuous data theoretical upper bound. Since the PC algorithm can only identify the causal graph up to Completed Partially Directed Graph (CPDAG), we apply the same orientation rule from (Dor & Tarsi, 1992), implemented by (Chandler Squires, 2018), to convert the returned CPDAG to a DAG for easier comparison. For each setting, we run 10 graph instances with different seeds and report the mean and standard deviation of Fl-score, precision, recall, and Structural Hamming Distance (SHD) in Figure 3 for skeleton discovery and Figure for DAG comparison Appendix D.
As shown in Figure 4, DCT-GMM consistently outperforms DCT. Specifically, the variance of & is consistently lower than that of à X across all evaluated sample sizes. Addi- tionally, the MSE of & is smaller, indicating more accurate estimation. These experimental results validate the superiority and improved sample efficiency of DCT-GMM, providing support for Theorem 3.5.
5. Conclusion
Experimental results show that DCT-GMM consistently out- performs DCT across all metrics, particularly recall, align- ing with its higher statistical power demonstrated in Sec- tion 4.1. Moreover, DCT-GMM significantly outperforms the Chi-square and Fisher-Z tests, especially at large sam- ple sizes, where the performance of these traditional tests deteriorates as the sample size increases. This phenomenon underscores the importance of discretization-awaal CI test: for all tests not aware of discretization, increasing the sam- ple size only reinforces incorrect conclusions with greater confidence. The lower recall observed in DCT-GMMM and DCT compared to other baselines is expected, as other base-
In this paper, propose DCT-GMM, a novel sample- efficient CI test to address challenges in CI testing with discretized data. By formulating parameter estimation as an overidentification problem and leveraging GMM, DCT- GMM surpasses existing methods, achieving lower estima- tion variance and greater statistical power, particularly in small-sample scenarios. Its proven effectiveness in causal discovery highlights its practical utility, bridging the gap between discretized observations and latent variable rela- tionships in both synthetic and real- world datasets.
8

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A Sample-Efficiet Conditional Independence Test in the Presence of Discretization
Impact Statement
for recommender systems, CrowdRec at RecSys, volume 2013, pp. 43, 2013.
This paper presents work whose goal to advance the field of Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
Dor, D. and Tarsi, M. A simple algorithm to construct a consistent extension of partially oriented graph. 1992. URL https: / api semanticscho org/ Ctr//orpuusaa :1:::::22220 2294914940
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A Sample-Efficient Conditional Independence Test in the Presence of Discretization
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A Sample-Efficiet Conditional Independence Test in the Presence of Discretization
Appendix for
"A Sample-Efficien Conditional Independence Test in the Presence of Discretization"
A. Notation Table and Function Form
A.1. Notation Table
Category
Description
Number and Indices
T
Number of samples
p J.
Number variables
Index of variable J1 J. & .I.....p)
Random Variables
X
A vector of Gaussian variables The discretized counterparts X
X
Σ
Covariance of X
Σ-j-j
Submatrix of Σ with j-th row and j-th column removed j-th column of Σ with j-th row removed Precision matrix of X, equals to Σ-1
Σ-jj
Ω
X
(jk)
All other variables of with X, and X₃ removed
Cj
The discretization boundaries mapping Xj to X, j-th component of the
X
Covariance between X₁, and Xj Precision coefficient wigk i-th sample of X, i-th sample of X
Wyk
Cj.
One component Cj
Tylk
True probability of X, has the value
Taikaank
True probability of X,, has the value m, and Xj, has the value k. Regression coefficient of X₂ in predicting X, Vector of all coefficients regressing X;
By.k
Bj
Influence function component, it represents the influence of the i-th observation on the covariance estimation error Matrix form of g
E
Estimation of Variables
Estimated covariance of X1 Xjz
Σ
Estimated covariance matrix, also the matrix form 5 Estimation of Wik
äsk
Gj.b
Estimation of Cj.,, calculated using Equation (2)
fj.k
Estimation of Ty.k. the sample probability that X, equals to k
Estimation of Tj.i2.mk, the sample probability that X,, equals to m and X,, equals to k
By
Estimation of 3,, calculated as ブーニ -
Functions and Operators
P
True probability
P.
Sample probability
E[Z]
Expectation of random variable Z
[2
Sample mean of random variable Z over n samples
1
Indicator function: is the condition is true, 0 otherwise
(2)
Cumulative distribution function of a standard normal distribution integrated from -00 to

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Sample-Efficient Conditional Independence Test in the Presence of Discretization
Category
Description
Cumulative distribution function of bivariate normal distribution with covariance j
integrated over the rectangular region defined by (a, bj [e,d]
Notations of GMM
f((
Moment function defined Sample mean moment functions given n samples g:(O) - Ei Si(0) Weighting matrix of GMM
())
A
G
The expectation of the Jacobian of the moment function G = |E| -0
00
Covariance matrix of moment function S = [[]((()))((0)))))
A.2. Cumulative Distribution Function of Bivariate Normal Distribution
The probability density function of bivariate normal distribution with random variables X3, Xy,, mean 0, unit variance and covariance jú is given by:
1
x 20 F1id + 2
/1
exp
2(1 ),
(12)
ji
The edf of the bivariate normal distribution with the covariance Jji integrating over the rectangular region defined by m- mm x (c 11 G.
is
{ (Sim f b(zn; J.n d
1 (C m- mi T2 Ma Friz)
(13)
G2ik-11
12

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Sample-Efficiet Conditional Independence Test in the Presence of Discretization
B. Discussion of Assumptions
Rationality the assumption A primary limitation of this work lies in the assumption of multivariate normal distribution for the latent continuous variables, which, to some extent, restricts its generality. However, it important emphasize the inherent challenge of proposing valid conditional independence test discretization scenario without relying on appropriate assumptions.
To accurately infer conditional independence relationships within this framework, three key components are required: 1. A meaningful statistic - capable of capturing the conditional independence among latent variables. 2. A consistent estimator the statistic must be computable solely from discretized observations. 3. Statistical inference - the null distribution of the statistic must derivable. The discretization drastically reduces available information, making these components less straightforward to implement compared to scenarios where all variables are directly observable.
Without parametric assumption, deriving meaningful statistic is already challenging, let alone performing its statistical inference. We adopt the same framework of (Sun et al., 2024), relying on the property that with a parametric assumption, the covariance of the original latent variables is computable, and for Gaussian variables, the covariance matrix corresponds to the independence and conditional independence among variables.
w.l.o.g of the assumption One question that may intrigue readers is why the assumption of zero mean and unit variance made without loss of generality. The answer is straightforward: we can always adjust the discretization function and its boundaries to produce equivalent results for models with non-7ero means and varying variances.
To illustrate, consider an intuitive example. Suppose we observe discrete variable X, with n samples, where half labeled as "ones" and the other half as "twos." " This discrete distribution could correspond to multiple continuous variables. For instance, continuous variable X;- NO. 1) with discretization boundary at 0 and another variable X (11,)) 2) with discretization boundary at 1 would yield exactly the same discretized observations Xy. Thus, the framework presented in this paper supports mapping the same discretized observations to multivariate normal distributions with any mean and variance, i.e., the zero mean and unit variance assumption without loss of generality.
13

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Sample-Efficien Conditional Independence Test in the Presence of Discretization
C. Pseudo Code
Algorithm I one step DCT-GMM
Require:
Observed data matrix X' E Roxd
Tested pair indices j1.j2 with j1 # j2
Conditioning C C (I,...,d) d} (21
Significance level
2: Rearrange Data Matrix
X X(i X (1 cij x
where 2 |C|
3: Initialize Covariance Matrix
$ Ip (identity matrix size p x p)
4: for 6- 1 to p do
5:
for f + to do
6:
Obtain the Cardinality of X' as Q
Obtain Cardinality of X' [; as
Set naive weighting matrix <- Iøx
9:
Compute covariance oe through minimizing Equation (5)
10:
Update covariance matrix:
E(g, kl + 8g
Eik, gl + dG (ensuring symmetry)
11:
end for
12:
end for
13:
Extract Submatrices
and correspond the first and second column of X due the regroup)
Let I-1-1 R2-11xp11 submatrix of Σ without 1st column and row
Let -n11 11 E p be the 1st column of ' with first row removed
Compute Test Statistics
15: Formulate Null Distribution
(=) <- Cumulative distribution function of the Normal Distribution defined in Thm. 3.4
16: Calculate p-value
p-value +- (1-1) (181.2))
17: Make Decision
18: if p-value > then
19: Conclude: Xj, 11. X Xs
20: else
Conclude: X₁, ,1. X12 Xs
22: end if
23: Return The conditional independence decision
14

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A Sample-Efficee Efficien Conditional Independence Test in the Presence of Discretization
Algorithm 2 two step DCT-GMM
I: Require:
Observed data matrix X' E R
Tested pair indices j1.J2 with ji # j2
Conditioning set C _ (1,...,d) (2.
Significance level a
2: Rearrange Data Matrix
X (i), X (2),, ( Cij Rux,,
where p 2 |C]
Initialize Covariance Matrix
$ t Ip (identity matrix of size x p)
4:
for f- to do
5:
for q + do
6:
Ohtain the Cardinality of X' as Q
Ohtain Cardinality of X' [; as
8: Set the naive weighting matrix + IQx
Ohtain estimated parameters 0 through minimizing Equation (5)
10: Set the weighting matrix A E,, [,()((())(( where f0) defined in Equation (4)
Resolve Equation (5) with updated calculate the estimated covariance 6e
12:
Update covariance matrix:
Egg, 8g
Eik, g) 0g (ensuring symmetry)
13:
end
14:
end for
15:
Extract Submatrices (/i and ja correspond the first and second column of X due to the regroup)
Let E-1-1 Rxp1xp-1 the submatrix of Σ without 1st column and 1st row
Let -_ E Re-1 1st column of Σ with first row removed
16: Compute Test Statistics
812 < $11_,
17: Formulate Null Distribution
(2)
Cumulative distribution function of the Normal Distribution defined in Thm. 3.4
18: Calculate p-value
p-value 2 (1- ② (131.a))
19: Make Decision
20: p-value > a then
21: Conclude: Xj, 11. X22 Xs
22: else
23:
Conclude: X,, f1. X2z Xs
24: end if
25: Return The conditional independence decision
15

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Sample-Efficiet Conditional Independence Test in the Presence of Discretization
D. Figure of Main Experiments: Causal Discovery
Fl score (Directed)
1.0
Prectsion (Directed)
1.0
Recall (Directed)
1.0
SHD SHD (Directed)
0.8
0.8
0.6
0.4
04
0.4
0.2
10
12
10
12
10
12
10
12
Numb of variables
Numb of variables
Numb variables
Numb of variables
(a) Fixed samples = 2000, changing number of nodes p = (4,6,8,12)
Procision (Directed)
F1 score (Directed)
Recall (Directed)
SHD (Directed)
0.8
0.8
0.6-
0.6
0.44
1500
2000
1000 Numb of Samples
1500
2000
1000
1500
2000
Numb of Samples
Numb of Samples
Numb of Samples
(b) Fixed nodes p = 10, changing sample size n = (100,50,,0000,2000)
Chi-square
DCT-GMM one
DCT-GMM two
Fisherz
DCT
Fisherz oracle
Figure 5. Experimental result of DAG discovery synthetic data for changing number of nodes (a) and changing sample size(b).
Fisherz oracle is Fisher-z test applied original continuous data. We evaluate Fj (t), Precision (f), Recall (f) and SHD (L).
16

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Sample-Efficien Conditional Independence Test in the Presence of Discretization
E. Additional Experiments
E.1. Denser Graph
DCT-GMM most effective in cases where discretization causes true conditional independencies to incorrectly identified as dependencies. performance is therefore partieularly strong in sparse graph settings, where conditional independence relationships are abundant. However, comprehensively evaluate a test's statistical power-its ability correctly identify true conditional dependencies it is crucial examine its performance in dense graph scenarios. To this end, conduct experiments with p 10 nodes and 2000 samples, varying the edge density 4 The underlying continuous data follows a multivariate Gaussian distribution, with the true DAG G generated using the BP model. We perform 10 independent trials with different random seeds and present both skeleton discovery and DAG reconstruction results in Fig. 6.
8).
Experimental results show that DCT-GMM continues to outperform other baselines in terms of precision and SHD. As the number of edges increases, the advantage of discretization-awaal CI tests (DCT-GMM and DCT) gradually diminishes due to the decreasing prevalence of conditional independence cases. Notably, DCT-GMM maintains superior recall, consistent with the findings from the main causal discovery experiment.
Fl senre (Skoleton)
Precision (Skeleton)
Recall (Skeleton)
SHD (Skeleton)
15
0.8
0.8-
10
0.6-
0.6
F1 score (Directed)
1.0
Precision (Directed)
1.0
Rocall (Directed)
SHD (Directed)
0.8
15
0.6
0.6
0.4
0.2
d
Chi-square
DCT-GMM one
DCT-GMM tWO
Fisherz
DCT
Fisherz_onae
Figure 6. Experimental comparison of causal discovery synthetic datasets for denser graphs with p 10, 2000 and edges varying
p+ 2 4,p 6, + 8. We evaluate Fl (1), Precision (1), Recall (f) and SHD (L) on hoth skeleton and DAG.
17

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Sample-Effiiint Conditional Independence Test in the Presence of Discretization
E.2. Real-world Experiment
Fisher-z test and Chi-square test
DCT
DCT-GMM [ often feel blue]
[I often feel blue]
[I often feel blue]
NIO
NIO
NIO
N3
N4
N3
N4
N3
N4
[I worry about things/
[1 seldom feel bluej
[I worry about things/
seldom feel bluej
[] worry about things)
seldom feel blue)
(a) 1e
Fisher-z test and Chi-square test often feel blue]
DCT
DCT-GMM [I often feet blue]
[1 often feet blue]
NIO
N10
NI0
N4
N3
N4
N3
N4
[1 worry about things]
[I seldom feel blue]
worry about things]
[I seldom feel blue]
[I worry about things]
seldom feel blue]
(b) == 1
Figure 7. PC algorithm applicd on the weal-world datasct with Fisher-z test, Chi-square test, DCT and DCT-GMM for different significance level a. Red edge found by other baselines while DCT-GMM removes.
To validate the effectiveness of DCT-GMM, we conduct experiments on the Big Five Personality dataset, where each variable has discrete values representing agreement levels (1=Disagree to 5=Agree). For example, "N3=1" indicates "I disagree that worry about things.". This setting aligns well with DCT-GMM, as agreement levels are inherently continuous but observed as discrete categories. This dataset has been closely examined by Dong al. (2024a) and Dong et al. (2024b), yet it does not solve the discretization problem. We focus on three variables: [N3: I worry about things], [N10: often feel blue], and [N4: I seldom feel blue]. Using the PC algorithm for causal discovery, compare DCT-GMM with the Chi-square and Fisher-Z tests. Results are shown in Fig. 7.
Experimental results validate both the effectiveness and superiority of DCT-GMM. Notably, both discretization-aware CI tests (DCT and DCT-GMM) successfully remove the edge between N3 and N4, whereas other baselines fail. The inferred graph directly aligns with our motivating causal graph illustrated in Figure 1. Furthermore, DCT-GMM demonstrates stronger ability capture conditional independence relationships. Increasing the significance level generally makes CI tests prone inferring conditional dependence. While DCT fails 10-3, DCT-GMM remains robust, correctly identifying that N3 IL N4 N10.
18

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Sample-Efficiet Conditional Independence Test in the Presence of Discretization
F. Proof and Derivations
F.I. Proof of Moment Condition
In this part, We show the derivation that [[f)( = O. For the moment functions f((0") defined in Equation (4) with the parameters achieving their optimal 0 0", we have the specific form:
Ju ( S1
Ch S2 12,1
f((0)
JKA M S 12 ki.
For any m (I,..., M), (I....,K) the edf term ((G)mm- m-1 of bivariate normal distribution integrated over the region defined by m Cj mmml x k2, S probability terms, this corresponds
momo k Co 22.ki ).12) represents the area of this
-- X3, <c k2 <X,² C22nk..
For its corresponding counterpart of the discrete domain, the relation holds
( m Sy A- J2 (1d2) = P(A, m, Xjz = k).
Recall the definition that 7 m...m 1 ( m, 812 k) is the indicator function of the sample Its expectation is equivalent to the corresponding probability:
Bis jamkk = IXX,, m, X12 k:).
We note that 2 (133)))))) is constant with respect to the sample, can take expectations over fi(0") term-wise:
E BI P - (( Sh G2 S2.orrur Gaii (1j2)
[l) =
M M G.MM S2 Gu.K K ).12).)
Substituting both mn and (c I C 3252) as F(X;, m, &, k)), each term evaluates zero, giving:
B[f:(0")] = 0.
This concludes the proof.
F22 Proof of Theorem 3.1
In this part, we show the detailed derivation of Theorem 3.1. Recall the definition of GMM defined in Equation (5), we are trying to minimize
i ((0))AA(0))
where the (() E MK is the sample mean of the moment functions, and the f E MRM+K-1 the interested parameters. When the interested parameter 0 Ô, we define the Jacobian matrix
G Ōğ(() aĀ
Using the chain rule, we have
a
2 Ag(
00
We note that when the interested parameter 0 = 0, which is the minimum of j(()). its gradient should be zero:
T Ag(O)
(14)
19

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Sample-Efficien Conditional Independence Test in the Presence of Discretization
Leveraging Taylor expansion, have
0(0") 5 G( - 0) +....
where the second-order terms and more are omitted. Rearrange the equation above, we have
9 ((0) ((()) G( 0).
(15)
Substituting back into the first-order condition Equation (14), the Equation (15) becomes
2G (g(0r) - G( 8) = 0.
(16)
Simplify and rearrange terms, have the difference between the estimator and the true parameter:
0 0-0" 0" -G AG) "C" -G" "'AA((0)
Σ
1=1
According the Central Limit Theorem, when n +00, the sample average of the moment functions should be asymptotically normal:
Σ ((0)) 4 N(O, (0,EE(0))00))
Since the mean is zero due to the definition. We note that the Jacobian term
0g(0)
2f:(0))
G
En!
00
00
due to the sample terms are irrelevant with the parameter According the Law of large numbers, when n + +00, the estimated parameter " 0" Thus, the Jacobian
Of:(0))
G G E
00
Let = ( for simplicity of the notation, according to Slutsky's theorem, have
Vn 0*) $ N (0, (G" AG) "G ASAG(G"AG)] AG)
(17)
Since jnja is nothing but the first element of the 0, we conclude that
vū(ōni jnn)) 4 N (0, [(G
(18)
which concludes the proof.
F.3. Proof of Lemma 3.2
In this part, we show the detailed derivation of Lemma 3.2. Our proof is divided in to two parts: we first
show the specific form of variance will follow when A 2, S-1. We then establish it superiority by showing
(GT AG) AGAAGGG AG) > -G
When n + +00, the positive semi-definite weighting matrix A converges to the S the variance of the original asymptotical defined in Theorem 3.1, will be written as:
(G AG) -GG)'GG GG ASAG AG) =0'"s="0---"""=="="""""""""""
(19)
-G)
20

---

A Cample-Efficient Conditional Independence Test in the Presence of Discretization
That is,
0= 4 N (O,,(G's
(20)
Since oud is nothing but the first element of the 0, we conclude that
- - 4 (0. ((G''''"))"))))
(21)
which concludes the first part of the proof. We now dive into the second part.
First, we factor S = CC where C RMKXMK which non-singular. Second, we
H (G" AG) -G- (G's "G''"""""""'''''''''"""""""
Third, we not that
- 0.
Fourth, verify that
(G² AG) ASAGAS" AG) A HH +G'
Since HH is positive semi-definite, the (GT is lower bound of AG) (AA) which concludes the proof.
F.4. Proof of Thm. 3.4
We note that the following proof is direct copy from (Sun et al., 2024). We include it here for completeness.
F.4.1. DERIVATION OF EQUATION 8
Consider our latent continuous variables X = EX.,...,y,, ~ N(O, Σ) and do nodewise regression
X, = X X-33, + +
(22)
where X-j is the submatrix of X with Xj removed. We can divide its covariance Σ and its precision matrix Ω the predictor X-j and outcome variable X, in our regression:
Σ-1 into
Σ jj Σ 1j
Sij Ω2 Ω-5f 5j
Σ
Ω =
Σーj-j)
Σ
(23)
Just like regular linear regression, we can get
n n 00000
31= Σー E-
(24)
From the invertibility of a block matrix
A
B D
(A (A-BD (D-
(25)
c
D
-
D=+
and is invertible, we will have
A
B
(A -D-1C)-1
0
| - -DD-1
1
(26)
C
D
0
DD (D-CA-IB)-1 CC--1
Thus, we can get:
2 = EΣ - Σイ Σー -
(27)
Qz-ン= = -
Σ - Σー -
21

---

Sample-Efficiet Conditional Independence Test in the Presence of Discretization
Move one step forward:
(28)
Take transpose for both sides, as long as S2 is a symmetric matrix and $2 = -3-ー we will have
2 Ω2 = ーー 8z.
(29)
We should note testing Ω = 0 is equivalent to testing 3 = 0 as the S2jj will always be nonzero. The variable Ω captures the CI Cl X, with other variables. As long as the variable $2 is just one scalar, we can get
Wik
A
(30)
Wij
capturing the CI relationship between variable Xj with X₁ conditioning on all other variables.
F.4.2. DETAILED DERIVATION OF INFERENCE FOR By
Nodewise regression allows us to use the regression parameter 3, as the surrogate of f2 The problem now transfers to constructing the inference for 3j. specifically, the derivation of distribution of $3 Bj. The overarching concept is that we are already aware of the distribution of J júúz and we know that there exists a deterministic relationship between 3j with E. Consequently, we can express 33- 33 as composite of jnjz مززز to establish such an inference. Specifically, we have
3, B, = ブー و ورك - Σー Σ-ك
= ー رーك - -ー | -ー
(31)
- (Σーj-18} - E Enj-ißs + -ーjjjaaj - -E-jj)
= (É E- (É- - -E-s))>
where each entry in matrix (Σ-j-j プーーード) and (Σ-j} - Σーjj) denotes the difference between estimated covariance with true covariance. Suppose that we want test the CI of the variable X1 with other variables. j 1. then
δz .22...02p
0 02p
1-11-E---
(32)
022 ... pp
0p 0p
52 ...
E2p
(33)
Sp2
Spp_
where {e), } are i.id random variables with specific form defined Theorem 3.1 for one-step GMM and Lemma 3.2 for two-step GMM correspondingly. Put them together:
81 31,2
S2
B1,2
S2
B1.3 $1,3
Sa
Saa
Sāp
$1.3
Ea
81 31 =
(34)
BI, 31
iml
Sp2
S
As Σ S132 is asymptotically normal, the who vector of 3 B1 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), we should have
Var(Yi) Cou(())
Cov(Y, Y2) [ Var(Yz)
Var(aY + DY2) = [a b]
(35)
b
22

---

Sample-Efficien Conditional Independence Test in the Presence of Discretization
Now, suppose we interested in the distribution of $1.2
$1,2.
we have
22 22
22 S3.p
1
2
&,
S3.
$1,3
Ea,
$1.2 -1,2
(36)
Sp,2 $3.3
Sp.p_
[1.p1
where (E=i-)) is the row of index of X2 of E ([2) denotes the index of the variable, e.g., the first row of Dー since the row of first variable removed. ). For ease of notation, we define
(Σ=1,- 1)) represents
23.3
22.
1
11-1
S32
S3.3
S3
E R2- 18p-1xp-1
111
$3,1
E
GRP-
(37)
52.2 Sp.
Sp.1.
f1,2
31,3
= ( ( Rp-1
GR--1
31
We can rewrite the equation as
$1.2 31, = "Y v²).
We note that Y o² are variables, and u, are constants (just like the example aYı bY2). We further let m p 1 to simplify the notation. We can thus write the equation above vector form:
1 Σ 1=1
Um
_1,2 B1,2
-1,-... Mmn 01101 U1 022 -2...... Um Wm
12
Yi
mm
n ju" vec(uw"))
-v¹ veo(Y
where un represents the k-th element of vector and Yj represents the entry in j-th row and k-th column matrix Y represents the row-wise vectorization of matrix, e.g.
Yn
Y12
vee vec(Y')
Y1s
Y m
Similar as equation 35, the variance calculated
v² vec(Yri)
[
Var (Vn(312-11.)) - -1.22)
= n2
vec(uw
(vee()r)))
u
Now go back the notations of & and Σ. Under the null hypothesis that X1 Il. X2 Xothers: i.e. 31,2 0. We thus use 31 to denote B1 where $1.2 Let
E2 En
E2
Ez
Sp
Bi
8g
Sp2
gi
Spa
4pp
23

---

A Cample-Efficien Conditional Independence Test in the Presence of Discretization
and
-1-1 [2], 34)
|2]
vec
Similarly (35), The variance is calculated as
Var (Vn(012 B1,2)) = =a!2 vec (ec(B",))ec(B",))
Simply replace the index 1, 2 as general index j, k. the distribution of B3NN Bi is
3 - ByNk 4 N(0, UbjT
veoc(BB
In practice, we can plug in the estimates of 3, to estimate the interested distribution and do the CI test by hypothesizing $3.k = 0.
24

---

A Sample-Efficiet Conditional Independence Test in the Presence of Discretization
G. Formal Claim of Theorem 3.5 and Derivation
In this section, we try to demonstrate the theoretical advantage of DCT-GMM over DCT. Specifically, the variance of 33. - -j,x obtained using DCT-GMM is consistently lower than that of DCT. Since DCT-GMM and DCT adopt exactly the same strategy to transition from f1 J13 to ,. x which is simply linear combination of é j:jz - A reduction in the variance of jn مUوز directly translates to reduction in the variance of 83.x -6,, to prove the variance of the estimator of covariance using DCT, denoted Varper(o - Jiiz ), is consistently greater than the one of two-step DCT-GMM, denoted Varcmm T2
j. is. Thus, our task is
The proof is organized into two parts:
Review of Variance Derivation of DCT: We first provide review of the derivation for Varper (ijj T1sz 2.
Moment Function Selection: Next, show that with appropriate moment functions, Varper J ja equals VaramM jud2 )2 We then directly use the property of GMM that incorporating valid moment functions lead to less
variance, which concludes the proof.
G.1. Review of Variance Derivation of DCT
We begin with the derivation of Varoer(0j..j 3232) with particular focus on discrete case. For discretized observed variable pair X,, and X2 DCT implicitly treats it as a pair of binary variables. Recall the definitions in DCT, we have interested parameters (a h5,, hjj)), with the function
$3i *b(h, hzzi T3152)
g(0) = Σ A(00) =
f) ((()))
(38)
52 (11322
For the true parameters 0 = [o]rsr hij, h;g)), we have
$1.30 $bh7 hTp )uis
n f.(0")
g(0") n
f) (h
(39)
F) & (h,
and the function of estimated parameters
क (hhin hipi Tsria)
[i(0))
4) ((hj,)
(40)
71
$2 $ ((hj))
where
S1i 168), > E, [ 51, 1, > ,x,1 serving as the estimation of T/ 1 ( > E[X E(X,,))
o = 1(a), > E,, (X,,)) serving as the estimation of Tj = 1( > I).,,)
(< =) fn 2 (exp-
2201 12 duu is the cumulative distribution function of bivariate
2 21-
normal distribution.
(<x)= (<x), (ax 2) (2,
(1:)
exp
2² du is the cumulative distribution function of a standard normal distribution.
2
Our objective is to construct the distribution of ,,H22 J. equivalently 0 - 0. By leveraging the Taylor expansion, can construct the following equation
( =00))= + 0g(0")
(@ - 0²) +... 00-
(41)
25

---

Sample-Efficient Conditional Independence Test in the Presence of Discretization
where 0el0") is the Jacobian matrix of function g at 0" 96
The second terms and more are omitted here. Rearrange terms,
since g(0) equals to zero, we have
0g(00))
0 0
g(0)),
(42)
80
if the Jacobian is invertible, which will always be true in this framework. Express g in vector form, have
00(
n
h
Φ(h;,
11
0 0
F1 21
((h
(43)
00
((h
When n + +-0, the first term Jacobian matrix 0g(0")
00- will converge E
01:(0') 00-
It's noteworthy that [ff((0-)]
according to the definition. By leveraging the Central limit theorem, we have
Σ f((0)) ~ N NN(,, (EM00)1)))
n + to0,
(44)
n
Thus, we have
1 0 0,
0- 0" ~ N
(
E
(45)
00
00
G.2. Moment Function Selection and Additional Moment Functions
We note that this derivation process is pretty similar to the one using GMM. Intuitively, if the moment functions of GMM are the same as Equation (38), we may have a similar distribution. We now provide the formal statement of the Theorem 3.5: Theorem G.I. For GMM whose a subset of moment functions g(0) are the same as Equation (38), with additional moment functions defined in Equation 4, have strictly less variance than DCT, whose variance is given in (45).
We now provide the proof of the theorem above. With appropriate moment functions, the variance of the م jija gotten using two-step DCT-GMM, is exactly the same as VorGM Mt (رم, )úúz). Specifically, for the interested parameters in hyj ), we define the moment function the same as Equation (38). Specifically, we are solving the following minimization problem:
i
arg min
g Ag)))
(46)
with the moment condition [[){ 0 satisfied for the true parameters 0", According to Lemma 3.2,
θ Ô 0 ~ (0, 1 GG" AG)
(47)
for the two-step estimation where
G = El 0f0) '1 86-
S (() ()
A = S-1
Since G is invertible 1 we can rewrite
GGAG)-I-G-ISCC
(48)
which is the same as the variance in Equation (45). That is, VarGMas(0,..s - ),, Varper -. However, GMM accommodates additional moment functions (solvable equations in Equation 3). Based on the property of
GMM (Newey, 2007), adding valid moment functions (moment functions defined in Equation 4) generally reduces the variance of the parameter estimates, which concludes the proof.
'One may check DCT the analytic form of G. which will a triangular matrix with non-zero diagonal entries.
26


Reviews:
[{"review_id": "G0pYDEdBix", "reviewer": "Reviewer_NWpp", "summary": "The paper proposes a conditional independence (CI) test for testing CI relations in discretized data. It does not rely on binarizing the data to infer the CI relations between latent variables. The paper argues that it does not need to rely on binarization like the previous work to establish correct CI relations between the latent continuous variables. It addresses an issue known as over-identifying restriction problem by using Generalized Method of Moments. The authors derive the test statistic and establish its asymptotic distribution. The key limitation of this work is that it assumes all latent continuous variables follow a multivariate normal distribution.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Some practical scenarios \n\nThank you for your valuable question. We appreciate the chance to highlight the **common and often unavoidable discretization** due to practical measurement constraints.  In principle, **any variables measuring \"degree\" or \"intensity\"**(e.g., happiness, severity) are inherently continuous but often recorded in discrete form. For example\n\n- In medical diagnostics, the **severity of cancer** is commonly categorized into stages like _\"Stage I\"_ or _\"Stage II\"_, not because the disease progresses in discrete jumps, but because there is no equipment that can exactly quantify its severity on a truly continuous scale.\n\n- Similarly, in questionnaires—including the real-world dataset used in our paper—latent continuous variables such as **level of depression** are often measured using Likert scales (e.g., 1 to 5) for fast assessment and clinical usability.\n\nWe hope those practical scenarios can clarifies its practical significance of DCT-GMM.\n\n> Groundtruth of real-world dataset\n\nThanks for your question. You're right—there is no ground truth in the Big Five Personality Dataset. We followed the setup from DCT. Despite the absence of ground truth for reference, the results of the discretization-aware CI tests (DCT and DCT-GMM) appear more reasonable. Specifically, the conclusion that $N4 \\perp N10 \\mid N3$, is intuitively plausible---since N10 represents \"I often feel blue\" is a stronger indicator of mood and should naturally subsume the information in N4 (\"I seldom feel blue\").\n\nAt the same time, we would greatly appreciate it if the reviewer could suggest any datasets that align well with our setting.\n\n> Normal assumption\n\nThank you for this very insightful question. Please kindly refer to our **response~1 to Reviewer 9WLL** for a detailed explanation. Due to the 5000-character limit, we were unable to include the full response here. We sincerely apologize for the inconvenience.\n\n> Citing CI tests that work with discrete data...\n\nThanks for your great suggestion. We have included the suggested reference [1] and other recent CI tests for discrete variables in our revised version [2,3]. \n\n> Sentences are difficult to understand. e.g, the proportion...\n\nThank you for the helpful comment. We have followed your comment and revised it to: “The empirical probability of observed discrete pairs reflects the covariance of the underlying continuous variables.”\n\n> Use heavily from DCT of nodewise regression\n\nThank you for discussing the connection between DCT and ours. Nodewise regression is a standard technique for addressing the inference problem of the precision matrix [4,5]. Both DCT and our method involve the inference of the precision matrix.  By using the nodewise regression in nonparanormal model, the derived CI test in Thoerem 3.4 would have the same form (but the analytical solutions involved are entirely different).  We have properly acknowledged this in the main text and references.\n\nMoreover, it is worth noting that apart from the use of nodewise regression, all other components employ entirely different techniques.\nOur unique contribution lies in making the CI test sample-efficient in the presence of discretization. To achieve this, we cannot follow DCT, which further binarizes observations and conducts estimation and inference based on the binarized data—a limitation explicitly acknowledged in DCT (Appendix G in [6]). Instead, we reformulate the problem as an overparameterized one and leverage GMM to provide a principled solution with theoretical guarantees. The rationale and techniques are fundamentally different.\n\n\n> - DAG generation \n\nKindly refer to lines 380–382, where we state that the DAG is generated using the BP model.\n\n> \"Fisherz\" meaning..\n\nThank you for your insightful question. Yes, \"Fisherz\" refers to directly applying the Fisher z test to discretized data. Our goal is twofold:\n\n1. To compare it with the Fisher z test on the original continuous data and highlight how discretization distorts causal discovery.\n2. To show that even if users know the variables are inherently continuous, directly applying a CI test designed for continuous data—such as the Fisher z test—on discretized values can significantly degrade the resulting causal graph.\n\n\n[1] Li et al., K-nearest-neighbor local sampling based conditional independence testing\n\n[2] Cai et al., A distribution free conditional independence test with applications to causal discovery\n\n[3] Kim et al., Local permutation tests for conditional independence\n\n[4] Qiu et al., Inference on multi-level partial correlations based on multi-subject time series data\n\n[5] Chang et al., Confidence regions for entries of a large precision matrix\n\n[6] Sun et al., A conditional independence test in the presence of discretization\n\n---\n\nWe sincerely thank the reviewer for the thorough and constructive review, which has greatly improved our paper. Please let us know if there are any further questions.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes, the paper uses F1 score, precision, and recall of the skeleton to evaluate the effectiveness of the proposed CI test on learning conditional independence. The authors also show the robustness of the test based on a range of sample sizes and graph sizes. They further test the algorithm on denser graphs and a real-world experiment.", "theoretical_claims": "Yes. I have checked sections F and G. I don’t see any issue.", "experimental_designs_or_analyses": "Yes. Originally, I thought the paper intentionally avoided many recent advanced baselines. However, I later understood that it is not needed because of the neat idea that the conditional independence among the original continuous variables may be wrongly tested as conditional dependence due to discretization. With faithfulness, it makes sense. The experiment demonstrates the contributions of the proposed CI test, which is more sample efficient than the previous work called DCT and outperforms it. \n\nHowever, I do not follow how the authors came up with the ground truth graph to verify their test performance in the real-world experiment. I tried to look up the reference, but I don't see anything describing the ground truth on the subset of variables.", "relation_to_broader_scientific_literature": "The paper aims to solve the overidentification issue (see lines 184-192) found in one recent related work called DCT (Sun et al. 2024) without the need of binarization like DCT. The key contribution of this paper is the content starting from line 197 all the way to proving Theorem 3.1 (con)and Lemma 3.2 along with Theorem 3.5. Theorem 3.1 establishes the asymptotic normality of the covariance estimator derived via GMM.  Lemma 3.2 further claims that choosing an optimal weighting matrix (namely, one converging to the inverse of the covariance of the moment functions) reduces the asymptotic variance compared to a one-step estimator.  Then, the authors follow the same framework of using nodewise regression (Sun et al. 2024) to derive the CI tests. The authors also argue that DCT-GMM, the proposed CI test, achieves lower variance than DCT because it leverages additional valid moment conditions via Theorem 3.5.", "essential_references_not_discussed": "I think the paper should cite the recent work on conditional independence tests though these tests are not primarily designed for the exact same setup. I think at least citing CI tests that work with discrete data is a fair game e.g. [1]\n\nReference:\n\n- [1] Li, Shuai, et al. \"K-nearest-neighbor local sampling based conditional independence testing.\" Advances in Neural Information Processing Systems 36 (2023): 23321-23344.", "other_strengths_and_weaknesses": "Strengths:\n\n- The paper is quite well-written.\n\n- The theoretical claims are logically sound. I appreciate the detailed explanation for the proofs in section F. \n\n- The experiment supports the contributions and claims of the paper. Both synthetic and real-world data are used.\n\nWeaknesses:\n\n- Some sentences are difficult to understand .e.g, \"the proportion of both observed variables exceeding their means reflects the underlying covariance, solved using a single equation.”\n\n- The practical significance of the proposed CI tests is quite limited. Although the previous work (Sun et al. 2024) has been published in ICLR'25, I find it difficult to come up with practical scenarios where one knowingly discretizes some continuous variables, and those continuous variables become unobserved. This is further questioned, especially when it assumes normality for those unobserved continuous variables. \n\n- The paper uses results heavily from (Sun et al. 2024) to derive the CI tests via nodewise regression. The presentation structure e.g., experiments, also closely follows Sun et al. 2024 (see section 3 and the rest in Sun et al. 2024).", "other_comments_or_suggestions": "- inferring \\tilde{X_{1}} being conditionally dependent of  \\tilde{X_{3}} given  \\tilde{X_{2}} is via faithfulness assumption, not causal Markov condition.\n- Minor typo in figure 3b: “Numb”\n- Lines 964-968: DCTG-> DCT-GMM. Also, the algorithm is named DCTG in the appendix, but the main paper refers to it as DCT-GMM.\n- It would be better to describe how the DAG structures are generated in the experiment.\n- I think it will be interesting how robust the proposed CI test is when the unobserved continuous variables do not follow a normal distribution and run the same comparison with the baselines.", "questions_for_authors": "1. In Figure 3b, does ‘Fisherz’ mean that the authors directly apply the Fisher z test to the discretized data? If so, why is that appropriate?\n2. How do the authors obtain the ground truth for the real-world experiment to verify the experimental result, especially for the selected three variables?\n3. Can the authors further give some practical scenarios where some variables are discretized and are known to be both continuous and unobserved?\n4. Can the authors test on an experiment where the unobserved continuous variables do not follow a normal distribution?\n\nI will raise my score if the authors address my questions well.", "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": {"ethical_review_concerns": "N/A"}}, {"review_id": "uCC9RDs98A", "reviewer": "Reviewer_iPdb", "summary": "This work tackles the problem of detecting conditional independence among hidden continuous variables, while the observed variables are discrete. Specifically, the authors rely on a recent work, the Discretization-Aware CI Test (DCT), which establishes a workflow to estimate covariances between continuous variables. This workflow leads to a system of equations that the DCT uses to over-identify the covariance. To address the issue of over-identification, the authors propose using a Generalized Method of Moments (GMM) to leverage all equations constructed with various discretization boundaries to acquire an accurate estimate of the covariance. Subsequently, nodewise regression is used, combining the covariance estimates to determine the conditional independence between variables.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Most of the claims presented are clear, except for Theorem 3.5.\n\nThank you for this construction comments.  We followed your comment and carefully revised Theorem 3.5 to make it more intutive and clear. Note that the formal theorem demonstrating the superiority of DCT-GMM over DCT is provided in Appendix G. We avoid to include it in the main body of the submission because presenting the claim requires a detailed introduction to DCT, which may detract from the overall flow of the main text. \n\n> It would be beneficial if the authors could provide additional details about Theorem 3.3 in the main content \n\nThank you for your thoughtful feedback. We believe you are referring to Lemma 3.3, which is intended to illustrate a property of nodewise regression that supports the derivation of the CI test. Inspired by your comments, we have added sentences to clarify the motivation behind using nodewise regression in this context:\n\n\" While Theorem 3.1 and Lemma 3.2 effectively address the independence test, they do not directly resolve the CI testing problem. A seemingly straightforward approach is to invert the estimated covariance matrix; however, this does not provide a valid solution for inference.\" \n\n> The parameters described in lines 407 to 410 on the left differ from those used in the original DCT work.\n\nThanks for your careful review.  Since DCT is the only method specifically designed to handle discretization scenario, we generally followed their experimental setup for fair comparison. However, our parameter setting for the experiment is **more challenging**, involving **fewer samples** and **more variables** to validate the superiority of DCT-GMM. \nWe summarize the comparison of causal discovery of DCT paper and our own:\n\n| Setting        | DCT                               | DCT-GMM                       |\n| -------------- | --------------------------------- | ----------------------------- |\n| varying sample | $p=8, n=(500, 1000, 5000, 10000)$ | $p=10, n=(100,500,1000,2000)$ |\n| varying nodes  | $n=5000, p=(4,6,8,10)$            | $n=2000, p=(4,6,8,12)$        |\n\n> overlap between lines 91 to 102 and lines 297 to 304 on the right-hand column with the corresponding sections of the DCT work.\n\nWe appreciate the reviewer for the careful review. We have rephrased the related work part (line 91 to 102) with citing additional references [1, 2, 3] (thank again to reviewer NWpp for his great suggestion).\n\nFor the experiment part (line 297 to 304), our intention is to follow the previous work DCT. Given your concern, we have rephrased it as \n\" \nIn the first part, we evaluate the Type I and Type II errors of DCT-GMM across various scenarios, comparing it with baseline methods including DCT (Sun et al., 2024), the Fisher-z test (Fisher, 1921), and the Chi-square test (F.R.S., 2009).  In the second part, we assess the performance of DCT-GMM on causal discovery tasks and compare with same baseline methods.  In the third part, we directly compare DCT-GMM and DCT to empirically validate Theorem 3.5. \n\"\n\n> Benefits of using nodewise regression rather than inverting a covariance matrix? \n\nThanks for your insightful question. The problem with inverting the covariance matrix is that **it does not address the issue of inference**, i.e., **deriving the distribution** of $\\hat \\omega_{jk} - \\omega_{jk}^*$. \n\nWhile the GMM solves the estimation of covariance $\\mathbf{\\hat{\\Sigma}}$ effectively, and we can directly invert it and obtain $\\mathbf{\\hat{\\Omega}}$ , whose entry $\\hat \\omega_{jk}$ captures the relation of $X_j$ with $X_k$ conditioning on all other variables. The problem is, it does not provide a tractable way to infer the distribution $\\hat \\omega_{jk} - \\omega_{jk}^*$. \nTo address this, we adopt a standard technique---nodewise regression [4, 5]. This allows us to:\n\t1. Show $\\beta_{j,k}$ acts as an effective surrogate of $\\omega_{jk}$, thereby the wanted distribution transfers from $\\hat \\omega_{jk} - \\omega_{jk}^*$  to $\\hat \\beta_{j,k} - \\beta_{j,k}^*$;\n\t2. Express $\\hat \\beta_{j,k}-\\beta_{j,k}^*$ as linear combination of the known distribution $\\hat{\\mathbf{\\Sigma}} - \\mathbf{\\Sigma}^*$ as equation~(11).\n\n[1] Li S, et al. K-nearest-neighbor local sampling based conditional independence testing[J]. Advances in Neural Information Processing Systems, 2023.\n\n[2] Cai Z, et al. A distribution free conditional independence test with applications to causal discovery[J]. Journal of Machine Learning Research, 2022.\n\n[3] Kim I, et al. Local permutation tests for conditional independence[J]. The Annals of Statistics, 2022.\n\n[4] Qiu, Yumou, et al. \"Inference on multi-level partial correlations based on multi-subject time series data.\" Journal of the American Statistical Association, 2022.\n\n[5] Chang J, et al. Confidence regions for entries of a large precision matrix[J]. Journal of Econometrics, 2018.", "claims_and_evidence": "Most of the claims presented are clear, except for Theorem 3.5, which is critical to demonstrating why the DCT-GMM approach is superior to the standalone DCT method. It would be beneficial if the authors could provide additional details about Theorem 3.3 in the main content to enhance understanding and support the claims made.", "methods_and_evaluation_criteria": "The evaluation criteria include Type I and Type II errors, F1 Score, precision, recall, etc., which makes sense for evaluating the proposed method.", "theoretical_claims": "I did not check.", "experimental_designs_or_analyses": "I have reviewed the experiments, and they are generally convincing. However, I am a bit concerned about the experimental settings for the application in causal discovery. The parameters described in lines 407 to 410 on the left differ from those used in the original DCT work.", "relation_to_broader_scientific_literature": "I believe the proposed method could prove particularly useful in digital data settings, where data is often discretized for storage in devices.", "essential_references_not_discussed": "I am not aware of.", "other_strengths_and_weaknesses": "I have noticed substantial overlap between lines 91 to 102 and lines 297 to 304 on the right-hand column with the corresponding sections of the DCT work. Please rephrase these sections to ensure your work is original.", "other_comments_or_suggestions": "Please see the weaknesses section.", "questions_for_authors": "What are the benefits of using nodewise regression rather than constructing and inverting a covariance matrix? Is it because the inversion is expensive?", "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": "ErIhICPReE", "reviewer": "Reviewer_bfoQ", "summary": "This paper introduces a Conditional Independence (CI) test designed for scenarios where continuous data is represented at a discretized level due to measurement limitations. In such settings, applying standard CI tests directly can lead to incorrect conclusions. Assuming the continuous data follows a multivariate normal distribution, the paper addresses this issue by proposing a Discretization-Aware CI test that accounts for these limitations and can identify independence relations between continuous latent variables given their discretized versions. This is achieved using the Generalized Method of Moments (GMM), which leverages all available sample information to estimate the covariance matrix of the continuous variables.\n\nBy assuming the original continuous variables follow a Gaussian model, performing the CI test reduces to an inference problem for the precision matrix $\\Omega = \\Sigma^{-1} = (w_{jk})$, since under this model, $w_{jk} = 0$ implies $X_j \\perp\\kern-0.3em\\perp X_k \\mid X_{-jk}$, where $X_{-jk}$ represents all other variables in $X$ except $X_j$ and $X_k$. The paper provides asymptotic error bounds on estimating these parameters and empirically demonstrates the improved performance of the proposed method compared to other CI tests.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> The multivariate normal distribution assumption.\n\nThank you for raising the insightful concern. Please kindly refer to our response~1 to Reviewer 9WLL for a detailed explanation. Due to the 5000-character limit, we were unable to include the full response here. We apologize for the inconvenience.\n\n> The technical contribution is rather limited since the main results use standard asymptotic tools.\n\nThank you for this important comment. We believe this comment arises from a lack of sufficient discussion in our paper. We have included additional discussion to clarify our use of asymptotic tools.\n\n1. **Standard Practice in CI Testing**: Asymptotic approaches are a cornerstone of CI testing. Many well-established tests, including the classical Chi-square test [2], Fisher-z test [3], kernel-based tests [4] and more recent approaches [5], rely on asymptotic theory. While permutation-type tests may be constructed controlling Type I error in some special cases [1], there are currently no non-asymptotic methods available for CI tests involving hidden variables under nonparanormal models.\n\n\n2. **Finite Sample Solution**: Recognizing the limitations inherent in asymptotic analysis, our framework can be readily extended with resampling techniques, such as the bootstrap. These methods offer a viable route to improve finite-sample performance without deviating from the core asymptotic framework.\n\n3. **Empirical Support from Simulation Studies**: Extensive simulation studies in the literature demonstrate that asymptotic approximations yield accurate results even at moderate sample sizes (e.g., n>100 [4, 6]), supporting their practical applicability.\n\n\n> results from DCT\n\nWe are sorry for the confusion. Both DCT and our method involve inferring the precision matrix, for which the standard technique of nodewise regression is employed in both methods [7,8]. This is why the derived CI test in Theorem 3.4 has the same form (though the analytical solutions $\\xi$  involved are entirely different). We have properly acknowledged it. \n\nApart from the use of nodewise regression, all other components employ entirely different techniques.\nOur aim lies in making the CI test sample-efficient in the presence of discretization. To achieve this, we cannot follow DCT, which further binarizes observations and conducts estimation and inference based on the binarized data—a limitation explicitly acknowledged in DCT (Appendix G in [9]). Instead, we reformulate the problem as an overparameterized one and leverage GMM to provide a principled solution with theoretical guarantees. The rationale and techniques are fundamentally different.\n\nWe will carefully discuss the above points in the paper to avoid further confusion. Thanks for the valuable feedback.\n\n> In Equation (1), it should be clarified that m ranges from 2 to M−1,\n\nThanks for your great advice. We have included \"$m$ is an integer ranging from $2$ to $M-1$\" in our revised paper.\n\n> definitions for Σ−jj and Σ−j−j−1 should be stated explicitly\n\nThanks for your great suggestion. Kindly note that we define the notation in right part of line 99-102: \"Similarly, $\\mathbf X_{-j-j}$ is the submatrix of $\\mathbf X$ without $j$th column and $j$th row, and...\". However, your suggestion made us realize that it might be not sufficiently clear. We have included the specific definition of $\\mathbf \\Sigma_{-j-j}$ and $\\mathbf \\Sigma_{-jj}$ in Lemma 3.4 to improve the clarity. \n\n> In the two-step procedure, is there a specific reason for limiting the process to only two steps, rather than iterating further until some form of convergence in the weight matrix is achieved? or the convergence itself is not guaranteed?\n\nThanks for your insightful question. Yes, we can iterate the procedure alternately until convergence, and convergence is indeed guaranteed. we don't adopt the iterative update since the two-step estimator is already consistent and variance-efficient, additional iterations would offer limited benefit while incurring higher computational cost.\n\n---\n\n[1] Berrett et al., The conditional permutation test for independence while controlling for confounders.\n\n[2] Pearson, on the criterion that a given system of deviations from the probable in the case of a correlated system ... \n\n[3] Fisher, On the \"Probable Error\" of a Coefficient of Correlation Deduced from a Small Sample.\n\n[4] Zhang et al., Kernel-based conditional independence test and application in causal discovery\n\n[5] Azadkia & Chatterjee, A Simple Measure of Conditional Dependence\n\n[6] Gretton et al., A kernel two-sample test\n\n[7] Qiu et al., Inference on multi-level partial correlations based on multi-subject time series data\n\n[8] Chang et al., Confidence regions for entries of a large precision matrix\n\n[9] Sun et al., A conditional independence test in the presence of discretization\n\n--- \n\n Please feel free to let us know if any part remains unclear or if you have further questions. Thank you again for your time and valuable feedback.", "claims_and_evidence": "yes, however, the multivariate normal distribution assumption should be stated more explicitly and earlier in the paper, as it is a key part of the analysis. It may be worth mentioning it in the abstract or contributions section since, as it stands, the language suggests a more general claim up until the assumption is introduced in line 130.", "methods_and_evaluation_criteria": "yes", "theoretical_claims": "I went over the proofs but haven't checked line by line.", "experimental_designs_or_analyses": "Yes, results sounds reasonable", "relation_to_broader_scientific_literature": "The paper is related to the literature on Discretization-Aware CI tests (DCT), particularly the work in (1). However, the test in (1) relies on binarizing the observed data. The key contribution of this paper is proposing a sample-efficient CI test that does not require binarization, making it more effective in preserving information from the original data.\n\n(1)  Sun, B., Yao, Y., Hao, H., Qiu, Y., and Zhang, K. (2024). A conditional independence test in the presence of discretization.", "essential_references_not_discussed": "no", "other_strengths_and_weaknesses": "Strengths: \n\n-  Proposing a sample-efficient CI test that does not require binarization by utilizing GMM, making it more effective in preserving information from the original data.\n\n- Promising empirical results\n\n\n\nWeaknesses: \n\n- The the multivariate normal distribution assumption limits the implication of the theoretical results\n- From the technical perspective, the contribution is rather limited since the main results either uses standard asymptotic tools or results from (1)\n- No guarantees for the type-I or type-II errors since the results are asymptotic and under the gaussian assumptions which may not hold in practice\n\n\n(1)  Sun, B., Yao, Y., Hao, H., Qiu, Y., and Zhang, K. (2024). A conditional independence test in the presence of discretization.", "other_comments_or_suggestions": "Minor comments:\n\n- In Equation (1), it should be clarified that $m$ ranges from $2$ to $M-1$, or dots should be added in the brackets to indicate the full range. Initially, it appears as if there are only three cases instead of $M$ cases.  \n\n- In Lemma 3.3, Equation (8), the definitions for $\\Sigma_{-j j}$ and $\\Sigma^{-1}_{-j -j}$ should be stated explicitly in the lemma statement or at least referenced in the appendix (Equation 23). Currently, it is not immediately clear how these terms are defined until reaching Equation (23).", "questions_for_authors": "In the two-step procedure, is there a specific reason for limiting the process to only two steps, rather than iterating further until some form of convergence in the weight matrix is achieved? or the convergence itself is not guaranteed ?", "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": "THOKxRo6Pf", "reviewer": "Reviewer_9WLL", "summary": "This paper addresses the critical challenge of conducting conditional independence tests on discretized data, where traditional methods often fail due to information loss from binning. The authors propose DCT-GMM, a new method leveraging the Generalized Method of Moments to infer latent continuous variable relationships without binarization. Theoretical guarantees for asymptotic normality and reduced estimator variance are provided, and experiments demonstrate superior performance over existing methods in both Type I and Type II error rates and causal discovery tasks.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Multivariate Gaussian distribution.\n\nThank you for raising the concern. We fully agree that the assumption of Gaussianity will limit the generality of the proposed test. At the same time, please allow us to share a few points regarding its reasonableness:\n1. **Challenges in Conditional Independence in the Presence of Discretization**: Inferring the conditional independence of latent variables based on their discretized values is indeed challenging, despite being a common practical issue. _Discretization significantly reduces the available information._ Without introducing mild assumptions, it is particularly difficult—if not overly ambitious—to construct statistics that correctly reflect conditional independence of latent continuous variables, let alone develop valid inference procedures. In this work, we rely on two key properties:  \n\t(1) The Gaussian assumption enables consistent estimation of the latent covariance matrix from discretized observations thanks to its parametric structure. \n\t(2) Under the Gaussian model, conditional independence can be inferred solely from the covariance matrix.\n2.  **Popularity of nonparanormal Model**:  The assumption of latent variables following multivariate Gaussian, also called the nonparanormal model, is well-studied and widely accepted in the community. There is a substantial body of work demonstrating the effectiveness of the nonparanormal model in various scenarios [1,2].\n3. **Empirical Performance:** To alleviate your concern and thanks to the insightul suggestion from Reviewer Nwpp, we conducted experiments investigating the Type I and Type II error of the proposed test where the data generation process violates our assumption. Specifically, the data are generated as either linear or nonlinear non-Gaussian, where the linear parameters follow the same setting of the main experiment and the nonlinear functions are randomly choose from $(a) f(x) = sin(x), (b) f(x) = x^3, (c) f(x) = tanh(x), (d) f(x) = ReLu(x)$.  Figure~1 in the link https://anonymous.4open.science/r/DCT-GMM-0D6D shows the comparison.\n    From the experiment result, DCT-GMM demonstrates comparable or superior Type I error control relative to DCT. In terms of Type II errors, it also outperforms DCT under most distributions. Overall, the discretization-aware tests clearly outperform other baselines.\n\n> Slight size inflation. What might be causing this phenomenon?\n\nWe appreciate the reviewer for the valuable question. The most plausible explanation towards this phenomenon is the neglect of the second order and higher order terms in the deriving distribution of $ \\hat{\\sigma} - \\sigma^*$ (Theorem 3.1 and Lemma 3.2 ). Specifically, our derivation relies on a Taylor expansion of the form (kindly refer to line 1045-1048):\n$$ \\hat{g}(\\mathbf{\\theta}^*) = \\hat{g}(\\hat{\\mathbf{\\theta}}) + \\hat{\\mathbf{G}}(\\mathbf{\\theta}^* - \\hat{\\mathbf{\\theta}}) + \\dots,$$ \nIn our derivation, we  omit higher order terms, which  might influence the accuracy of the derived distribution. To validate the hypothesis, we conducted the experiment where we further increase the sample size, and strengthen the influence from the conditioning set to the observed pairs, thereby reducing the impact of higher-order terms.  As shown in Figure 4 (link: https://anonymous.4open.science/r/DCT-GMM-0D6D), the proposed test effectively controls the Type I error rate under these conditions, supporting the validity of our explanation.\nWe will acknowledge this approximation as a limitation of our approach in the revised version of the paper.\n\n> How does the number of discretization levels impact the results?\n\nThank you for the insightful question. Towards this question, we conducted additional experiments shown in Figure2 and Figure3 in the anonymous link https://anonymous.4open.science/r/DCT-GMM-0D6D. \n\nFigure 2 compares Type I and II errors of DCT-GMM and baselines, using discretization level $M=5$, the cardinality of conditioning set $D=1$ and change the sample size $n=(100,500,1000,2000)$. Similar to the main experiment, both DCT and DCT-GMM control Type I error well, while DCT-GMM achieves higher power.\n\nFigure 3 varies $M=(4,5,6,7,8)$ and fix $D=1$, and $n=2000$ in Figure3. DCT-GMM consistently maintains Type I error control and high power. Notably, the two-step DCT-GMM outperforms the one-step version, supporting our theoretical results.\n\n---\n\n[1] Fan, J., Liu, H., Ning, Y., and Zou, H. High dimensional semiparametric latent graphical model for mixed data.   Journal of the Royal Statistical Society Series B: Statistical Methodology, 79(2):405–421, 2017.\n\n[2] Zhang A, Fang J, Hu W, et al. A latent Gaussian copula model for mixed data analysis in brain imaging genetics[J]. IEEE/ACM transactions on computational biology and bioinformatics, 2019, 18(4): 1350-1360. \n\n---\n\nThank you again for your thoughtful question. We hope the additional results clarify our findings. Please feel free to reach out if you have further questions or feedback.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "No.", "experimental_designs_or_analyses": "Yes, I checked the experiments in Section 4.", "relation_to_broader_scientific_literature": "The key contributions of the paper can advance methodologies in conditional independence testing, causal discovery, and handling discretized data.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths: The integration of GMM to resolve over-identification in discretized data is novel and theoretically sound. The two-step GMM approach optimally weights moment conditions, enhancing efficiency.\n\nWeaknesses: The method assumes latent variables follow a multivariate Gaussian distribution. This greatly limits its applicability to non-Gaussian settings.", "other_comments_or_suggestions": "None.", "questions_for_authors": "1. Given that each experimental configuration was replicated 2,000 times, Figure 2 suggests that the proposed method exhibits slight size inflation. What might be causing this phenomenon?\n\n2. In all experiments, the data are discretized into three levels. How does the number of discretization levels impact the results?", "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:

Meta-review: The paper presents a method for conditional independence testing where the variables are continuous but discretized during measurement, often seen in social science, without binarization. It provides sound asymptotic analysis and experiments on both synthetic datasets indicating improvements over previous methods. However, like many previous works, it assumes that the variables follow a multivariate Gaussian distribution.
Decision: Accept (poster)