I MPROVING THE IMPUTATION OF MISSING DATA WITH M ARKOV BLANKET DISCOVERY
A BSTRACT
The process of imputation of missing data typically relies on generative and regression models.
These approaches often operate on the unrealistic assumption that all of the data features are directly related with one another, and use all of the available features to impute missing values.
In this paper, we propose a novel
Markov Blanket discovery approach to determine the optimal feature set for a given variable by considering both observed variables and missingness of partially observed variables to account for systematic missingness.
We then incorporate this method to the learning process of the state-of-the-art MissForest imputation algorithm, such that it informs MissForest which features to consider to impute missing values, depending on the variable the missing value belongs to.
Experiments across different case studies and multiple imputation algorithms show that the proposed solution improves imputation accuracy, both under random and systematic missingness.
I NTRODUCTION
Dealing with missing data values represents a common practice across different scientific domains, especially in clinical (Little et al., 2012; Austin et al., 2021), genomic (Petrazzini et al., 2021) and ecological studies (Alsaber et al., 2021; Zhang & Thorburn, 2022).
It represents a problem that can be difficult to address accurately, and this is because missingness can be caused by various known and unknown factors, including machine fault, privacy restriction, data corruption, inconsistencies in the way data are recorded, as well as purely due to human error.
Rubin (1976) categorised the problem of missing data into three classes known as Missing Completely At Random (MCAR), Missing At Random (MAR) and Missing Not At Random (MNAR).
We say data is MCAR when the missingness is purely random, i.e., the missing mechanism is independent of both the observed and unobserved values.
On the other hand, data is MAR when the missingness is dependent on the observed values but independent of the unobserved values given the observed values; implying that MAR data can be effectively imputed by relying on observed data alone.
Lastly, data is said to be MNAR when it is neither MCAR nor MAR and hence, missingness is dependent on both the observed and unobserved values.
While it is tempting to simply remove data rows that contain empty data cells, a process often referred to as list-wise deletion or complete case analysis, past studies have shown that such an approach is ineffective since it tends to lead to poorly trained models (Wilkinson, 1999; Baraldi & Enders, 2010).
On this basis, the problem of missingness is typically handled by imputation approaches which estimate the missing values, often using regression or generative models, and return a complete data set.
The imputation algorithms are often classified as either statistical or machine learning methods (Lin & Tsai, 2020).
Statistical imputation methods include Mean/Mode, which is one of the simplest methods where the imputation is derived by the mean or mode of the observed values found in the same data column.
A more advanced statistical method is the Expectation-Maximization (EM) algorithm (Honaker et al., 2011).
EM computes the expectation of sufficient statistics given the observed data at the E-step (Expectation), and then maximizes likelihood at the M-step (Maximization).
It iterates over these two steps until convergence, at which point the converged parameters are used along with the observed data to impute missing values.
Another statistical al- gorithm is the one proposed by Hastie et al.
(2015), called softImpute, which treats imputation as a matrix completion problem and solves it by finding a rank-restricted singular value decomposition.
Multiple imputation is another popular statistical method for handling missing data, and considers the uncertainty of missing values.
Some classic multiple-imputation algorithms include the Multivariate Normal Imputation (MVNI) (Lee & Carlin, 2010), Multiple Imputation by Chained
Equations (MICE) (Van Buuren & Groothuis-Oudshoorn, 2011), and Extreme Learning Machine (ELM) (Sovilj et al., 2016).
On the other hand, one of the earliest imputation methods that come from the Machine Learning (ML) field include the k-nearest neighbour (k-NN) (Zhang, 2012), which imputes empty cells according to their k-nearest observed data points.
A well-established ML imputation algorithm is MissForest (Stekhoven & B¨uhlmann, 2012), which trains a Random Forest (RF) regression model recursively given the observed data, for every variable containing missing values, and uses the trained RF model to impute missing values.
Recently, deep generative networks have also been used for imputing missing data values.
Yoon et al.
(2018) proposed the Generative Adversarial
Imputation Nets (GAIN) algorithm which trains the generator to impute missing data and the discriminator to distinguish original data and imputed data, and was shown to have higher imputation accuracy compared to previous approaches.
Other ML techniques used for imputation include the optimal transport (Muzellec et al., 2020), a neural network with causal regularizer (Kyono et al., 2021), and automatic model selection (Jarrett et al., 2022).
All of the aforementioned algorithms assume that all the variables in the data correlate with each other, and use all the variables to impute the missing values.
Considering all of the data variables increases the risk of over-fitting, but which can be minimised through L1 and L2 regularization methods often employed by ML algorithms.
However, regularization leads to models that tend to lack interpretability and theoretical guarantees of correctness.
Because this paper focuses on interpretable models, such as those produced by structure learning algorithms, we shall focus on causal feature selection which maintains interpretability, rather than regularization.
This is also partly motivated by Dzulkalnine & Sallehuddin (2019) who showed that using uncorrelated variables to impute missing values not only decreases learning efficiency, but also degrades imputation accuracy.
On this basis, it has recently been suggested to include a feature selection phase that prunes off potentially unrelated variables, for each variable containing missing values, prior to imputation (Bu et al., 2016;
Liu et al., 2020; Hieu Nguyen et al., 2021).
Relevant studies that focus on feature selection for imputation include the work by Doquire & Verleysen (2012) who used Mutual Information (MI) to measure the dependency between variables.
They used a greedy forward search procedure to construct the feature subset, which is an iterative process that constructs feature sets that maximise MI with the dependent variable.
Sefidian &
Daneshpour (2019) also estimate the dependency between variables using MI, and chose to select a set of variables that increase MI above a given threshold, as the features of a given dependent variable.
On the other hand, the algorithm proposed by Dzulkalnine & Sallehuddin (2019) applies a fuzzy Principle Component Analysis (PCA) approach to the complete data cases to remove irrelevant variables from the feature set, followed by a SVM classification feature selection task that returns the set of features that maximise accuracy on the dependent variable.
Lastly, evolutionary optimisation algorithms have also been adopted for feature selection in imputation, and include differential evolution (Tran et al., 2018), genetic algorithms (Awawdeh et al., 2022), and particle swarm optimisation (Jin et al., 2022).
Recently, causal information has also been adopted to feature selection for missing data imputation.
Kyono et al.
(2021) proposed to impute missing values of a variable given its causal parents derived from the weights of the input layer in the neural network.
Similarly, Yu et al.
(2022) proposed the
MimMB framework that learns Markov Blankets (MBs) to be used for feature selection in imputation, which is an iterative process that learns MBs from the imputed data and updates the learned
MB after each iteration.
Note that while MimMB is related to our work, since we also use MB construction for feature selection, an important distinction between the two is that MimMB combines
MBs with imputed data whereas, as we later describe in Section 3, the learning phase of MBs that we propose is separated from imputation, accounts for partially observed variables, and improves computational efficiency.
In this paper, we use the graphical expression of missingness proposed by Mohan et al.
(2013), known as m-graph , which is a graph that captures observed variables in conjuction with the possible causes of missingness as parents of the partially observed variables.
We first show that an original version of the Grow and Shrink (GS) algorithm by Margaritis (2003) is capable of discovering the MBs in m-graphs containing partially observed variables, when applied to test-wise deleted data.
Because this approach relies on CI tests with large conditioning sets, we modify GS such that the number of conditioning sets considered for CI tests is reduced.
We provide proof that the modified GS is capable of discovering the MBs of partially observed variables in m-graphs, under the same assumptions as with the original GS.
We then propose a new imputation algorithm, which we call Markov-Blanket Miss-Forest (MBMF), that combines the modified GS with the state-ofthe-art MissForest (MF) imputation algorithm.
We evaluate the effectiveness of MBMF on both synthetic and real-world data sets.
The empirical experiments show that MBMF outperforms MF, and other relevant state-of-the-art imputation algorithms, under most experiments.
P RELIMINARIES 2. B AYESIAN N ETWORK
A Bayesian network ⟨G , P ⟩ is a probabilistic graphical model represented by a Directed Acyclic
Graph (DAG)
G =
( V , E ) and associated probability distributions P over V .
In DAG G , a path is a sequence of distinct nodes such that every pair of nodes is adjacent in G .
A node V i is called a collider on a path p if at least two of its neighbouring nodes on p are parents of V i in G .
We denote a node V j as a descendant of V i if there is a path from V i to V j such that all arrowheads of the edges on that path are from V i to V j .
A key concept of DAG is d-separation , which defines the conditional independence (CI) between variables in DAG.
D-separation: Two variables X and Y are d-separated conditional on a variable set Z if every path between X and Y has a node W that satisfies one of the following two conditions: i) W is not a collider and W ∈ Z , or ii) W is a collider and none of W or its descendant are in Z (Pearl, 1988).
Conditional independence entailed by a given DAG via d-separation is not always equivalent to the conditional independence of the corresponding probability distribution.
However, we assume the Markov and faithfulness conditions described below, where the DAG and the corresponding distribution express the same set of conditional independence.
Markov condition:
Given a DAG G over V , every variable in V is independent of its nondescendants conditional on its parents.
Faithfulness condition:
Given a DAG G over V , a probability distribution P is faithful to G if and only if the conditional independence relationships in P are exactly the same as the independences entailed by d-separation in G .
Given the faithfulness condition, a variable is conditionally independent of all the other variables given its MB, which contains all its parents, children and parents of its children.
We denote MB of a variable V i as MB ( V i ) .
2. M ISSINGNESS MECHANISM AS AN M - GRAPH
We adopt the graphical representation of the mechanism of missing data known as m-graph , proposed by Mohan et al.
and which makes a slightly stronger assumption on MAR and MNAR scenarios compared to the definition proposed by Rubin (1976).
Given a missing data set, the observed variables V can be partitioned into fully observed variables V o and partially observed variables V m .
In an m-graph, there is an auxiliary indicator variable R i for each partially observed variable V i ∈ V m that specifies the missingness of V
i , such that R i = 1 when V i is missing and
R i = 0 when V i is observed.
We have MCAR if R ⊥⊥ V o ∪ V m , MAR if R ⊥⊥ V m | V o , otherwise MNAR.
We denote R S over a set of variables S as R S =
∪ V i ∈ S { R i | V i ∈ V m } .
We also make the Assumption 1 and Assumption 2 for indicator variables, as described below and based on (Mohan et al., 2013).
Assumption 1.
No missing indicator variable R i can be a parent of observed variables or other indicator variables, i.e., R i could only be a leaf node in m-graph.
Assumption 2.
In an m-graph, no edge can exist from a partially observed variable V i to its corresponding indicator variable R i .
V
V V
V
V 5 V
R
R(a) MCAR
V
V V
V
V 5 V
R
R(b) MAR
V
V V
V
V 5 V
R
R(c) MNAR
Given Assumption 1, if two variables V i and V j are d-separated by a variable set S , they are still d-separated given S ∪ R .
Given Assumption 2, we exclude the situation that there is a causal relation between V i and R i in order to avoid performing CI test between V i and R i .
Figure 1 presents three m-graph examples under different mechanisms of missingness.
M ARKOV B LANKET BASED FEATURE SELECTION FOR IMPUTATION
Given the description of the m-graph and causal faithfulness assumption, the problem of feature selection under incomplete data can be converted into a MB discovery problem over m-graphs that contain partially observed variables and missing indicators 1 .
Because of the possible causal links between partially observed variables V m and indicator variables R (i.e., in the case of MNAR), the
MB of a partially observed variable is likely to contain both observed and indicator variables.
For example, in Figure 1c, the MB of V 1 is { V 2 , V 3 , V 4 , V 5 , V 6 , R 3 } .
We show that the Grow and Shrink (GS) algorithm with test-wise deletion is capable of discovering the m-graph MB of any variable from incomplete data.
Here, unlike list-wise deletion which removes all data rows containing at least one missing value, we use test-wise deletion which removes data cases containing missing values in any of the variables involved in the current CI test.
The pseudo-code of GS (Margaritis, 2003) is presented in Algorithm 1.
Note that we slightly modify the Grow phase, i.e., line 4-6, of GS to eliminate its dependency on the order of the variables in the data (Kitson & Constantinou, 2022).
Given the faithfulness condition, Assumption 1 and Assumption 2, Proposition 1 describes the correctness of GS with test-wise deletion.
Proposition 1.
Given the faithfulness condition and Assumptions 1 and 2, for any observed variable
V i in a m-graph G , the output of GS( V i , V o ∪ V m ∪ R − { V i , R i } , D ) is the MB
( V i ) in G .
The proof of Proposition 1 is provided in Appendix A. Therefore, an intuitive way to determine the relevant features for a given variable is to apply the function GS( V i , V o ∪ V m ∪ R − { V i , R i } , D ) on every V
i ∈ V m .
However, this is impractical since the maximum size of the conditioning sets used for CI testing is | V o | + 2 | V m | − 3 .
In practice, the accuracy of CI tests drops dramatically as we increase the size of the conditional set (Tsamardinos et al., 2003).
To address this, we propose the Markov Blanket Feature Selection (MBFS, Algorithm 2) that aims to restrict the maximum size of the conditional set used by CI tests to | V o | + | V m | − 1 .
MBFS involves two phases, where the first phase involves learning the intrinsic MB of each partially observed variable.
Given a m-graph G , we define the intrinsic MB of a variable V i as the set of variables that are still in the MB of V i after removing all indicator variables from G .
We denote the intrinsic MB of V i by iMB ( V i ) .
Note that iMB ( V i ) is not necessarily equivalent to the set of observed variables in MB ( V i ) .
This is because the missing indicators might be a common effect of two observed variables.
For example, the intrinsic MB of V 1 in Figure 1c is { V 2 , V 3 , V 4 , V 5 } , if any X S i CMB { X
S i
CMB
S
5: add S i p value
6: remove from S 7: end 8: until CMB stays unchanged 9: for Y do ▷
10: if 11: remove procedure whereas the standard MB would have also included V 6 and R 3 .
It is worth noting that, during phase 1, some nodes that do not belong in iMB ( V i ) may still be included in the output CMB .
However, as we show in Appendix B, these nodes are still in the MB ( V i ) in m-graph.
The phase 2 aims to learn all the parents of the missing indicators, in order to complete the feature set of MB ( V i ) .
Propositionstates that MBFS is capable of learning MB ( V i ) from missing data for any V i ∈ V m in a m-graph and thus, it could serve as an effective feature selection approach for imputation algorithms.
Algorithm
Input partially observed variable V i , data D
Output candidate Markov Blanket CMB of V i 1: procedure MBFS( V i , D )
▷ Phase 1 (discover intrinsic MB) 2:
CMB ← GS ( V i , V o ∪ V m − { V i } , D )
▷ Phase 2 (discover other variables in MB caused by indicators) 3: for each R j ∈ R − { R i } do 4:
CPS ← V o ∪ V m − { V j } 5: for each V CPS do 6: if
S 7: remove V k from CPS 8: end if 9: end for 10: if V i ∈ CPS then 11:
CMB ← CMB ∪ { R j } ∪ CPS 12: end if 13: end for 14: return CMB 15: end procedure
Proposition 2.
Given the faithfulness condition, Assumptions 1 and 2, for any observed variable V i in a m-graph G , MBFS( V i , D ) returns MB ( V i ) in G .
The proof of Proposition 2 is provided in Appendix B. Appendix C discusses the implications on the learning performance of MBFS when the Assumptions 1 and 2 are violated.
We then propose a modified version of MissForest that incorporates MBFS as a feature selection process.
The modified version of MissForest, which we call Markov Blanket MissForest (MBMF), takes the feature set
MBFS ( V i , D ) for each partially observed variables V i , as opposed to considering all of the other observed variables as the explanatory features of V i in the Random Forest regression model used in
MissForest.
In other words, MBMF accounts for the possible causal relationships between partially observed variables and the missing indicators, to minimise the risk of considering irrelevant observed variables for imputation by MissForest.
E XPERIMENTS
We test the proposed MBMF algorithm with reference to the standard version of MissForest (MF), the commonly used imputation algorithms Mean and Mode, the K-Nearest Neighbour (KNN), and two state-of-the-art algortihms; the softImpute and GAIN algorithms.
While the evaluation includes experiments on both continuous and categorical data, some of the other algorithms can only process one of the two types of input data and hence, their application is restricted to continuous data (Mean and GAIN) or categorical data (Mode).
We use the scikit-learn python package (Pedregosa et al., 2011) to test the Mean, Mode and KNN algorithms, the MissForest R package (Stekhoven &
Stekhoven, 2013) to test MF, the softImpute R package (Hastie et al., 2015) to test SoftImpute, and the publicly available source code of GAIN.
The implementation of MBMF, described in this paper, is available at: https://github.com/Enderlogic/Markov-Blanket-based-Feature-Selection.
MBMF is applied to continuous data using the Pearson’s correlation test for CI tests, and to categorical data using the G-test statistic, both of which are the default choices for GS.
We also consider the default threshold for independence, which is 0.1 for CI p-value tests.
The other algorithms are also tested with their default hyper-parameters as implemented in their corresponding packages listed above.
4. S YNTHETIC CASE STUDIES BASED ON REAL - WORLD BN S
We first evaluate the algorithms by applying them to synthetic data sampled from three BNs,
ECOLI70, MAGIC-IRRI and ARTH150, taken from the bnlearn repository (Scutari, 2010).
Details about these graphical networks can be found in Appendix D.
We generate complete data sets for each network with sample sizes 500, 1000, 2000 and 3000.
Then, for each complete data set, we create nine incomplete data sets composed of different combinations of missingness rates (i.e., 10%, 30% and 50%) and missingness assumptions (i.e., MCAR, MAR and MNAR).
Appendix E describes the process we followed to generate data sets with different types of missingness.
4. E VALUATION PROCESS
The imputation accuracy is evaluated using two different approaches.
The first approach involves retrieving the Root Mean Squared Error (RMSE) between imputed data and complete data.
Because
RMSE is sensitive to the discrepancy between absolute data values, we normalise the complete data column-wise and re-scale the imputed data with the same normalisation parameters to eliminate bias.
The second approach involves assessing the impact of imputation on structural learning accuracy.
We do this by comparing the Completed Partially Directed Acyclic Graphs (CPDAGs) learned by the state-of-the-art GES causal structure learning algorithm (Chickering, 2002) from imputed data sets produced by the different imputation algorithms.
The second approach is helpful because, while it is reasonable to assume that higher imputation accuracy helps causal machine learning, it is possible that some imputed values are more important than others.
Causal structure learning represents a good approach to test this, and we use the F1 score to measure the accuracy of graphical structures learned by GES, as follows:
[Equation]
where TP , FP and FN represent the number of true positive, false positive and false negative edges in the learned CPDAG, relative to the true CPDAG.
For more information on how to retrieve the
CPDAG of a DAG, please refer to (Chickering, 2002).
Readers are also refered to (Kitson et al., 2021) for a review of structure learning.
[Figure]
4. R ESULTS
Figure 2 depicts the average RMSE of imputed data produced by the different algorithms under different sample sizes.
Note that the results we report on GAIN are, to some degree, inconsistent with the results presented in the original paper (Yoon et al., 2018) , but are consistent with the results presented in follow-up studies (You et al., 2020; Nazabal et al., 2020; Kyono et al., 2021; Jarrett et al., 2022).
In general, the proposed MBMF algorithm is found to outperform the baseline MF under all scenarios of missingness.
Specifically, for MCAR and MAR scenarios with missing rate 10%, MBMF provides a considerable improvement over MF, but this improvement diminishes with the higher missing rates of 30% and 50%.
This is because a higher missing rate tends to decrease the sample size of the test-wise deleted data, which in turn reduces the accuracy of CI tests and the discovered MB set by MBFS.
Importantly, MBMF outperforms MF considerably under all MNAR settings, which better reflect real-world missingness that is generally systematic.
None of the other imputation algorithms provide satisfactory performance in terms of RMSE; at least relative to the
MBMF and MF algorithms.
Figure 3 presents the average F1 scores corresponding to the graphs learned by GES, given the imputed data sets produced by the different imputation algorithms, and across the different sample sizes.
While this evaluation approach decreases the discrepancy in performance scores between the top performing imputation algorithms, the results are consistent with those presented in Figuresince MBMF and MF are found to perform better than the other algorithms in almost all cases, and MBNF performs better than MF in most experiments.
Specifically, MBMF and MF produce similar performance when the rate of missingness is lowest at 10%, with their performance being close to that produced with complete data (dashed line).
These results serve as empirical evidence that both MFBF and MF imputation algorithms perform exceptionally well with relatively low rates of missingness.
When the rate of missingness increases to 30%, MBMF performs better than MF in most cases.
However, when the rate of missingness is highest, at 50%, there is no clear winner between MBMF and MF.
Lastly, we evaluate the computational efficiency of MBMF relative to the original MF.
As shown in
Figure 4, MBMF is generally more efficient than MF.
Note that while MBMF involves an additional phase needed to perform feature selection, the additional time spent by MBMF in that phase is countered by the reduced time MBMF spends to actually impute values.
This is because MBMF
[Figure] [Figure]
will almost always consider less features than MF during the imputation phase.
Specifically, MBMF is slower than MF at the lowest sample size, but becomes increasingly faster than MF with increasing sample size.
Averaging the results across the different mechanisms of missingness and missing rates also shows that MBMF is, in general, more efficient than MF.
Note that MF is slightly more efficient when the rate of missingness is at its highest, 50%, since MF trains its RF regression model on observed data rows only; implying that a higher rate of missingness decreases the training data passed to the RF.
The impact of the rate of missingness is higher on MF than MBMF since the number of independent features considered by MF is, in general, considerably higher than those considered by MBMF.
4. R EAL - WORLD CASE STUDY
We repeat the evaluation by applying the imputation algorithms to six real-world data sets retrieved from the UCI data repository (Dua & Graff, 2017).
A summary of these data sets is given in Tablein Appendix F.
We simulate missingness using the same strategy as described in Section 4.1.
Specif-
[Table]
ically, for each complete real data set, we generate nine incomplete data sets composed of different rates and assumptions of missingness.
We use RMSE to evaluate the imputation accuracy when the algorithms are applied to continuous data, and the Proportion of Falsely Classified entries (PFC) when applied to categorical data.
Note that the second evaluation approach, which involves investigating the impact of imputation accuracy on causal structure learning as described in subsection 4.2, is unsuitable here since we do not have a ground truth causal graph.
Table 1 presents the average scores and standard deviation.
The results show that MBMF and MF continue to outperform the other algorithms
[Table]
and that MBMF continues to outperform MF in most of the experiments.
On this basis, we conclude that the results obtained from the real-world data sets are consistent with the results obtained from the synthetic data sets.
C ONCLUSION
This paper described a novel feature selection algorithm, called MBFS, that recovers the Markov
Blanket of partially observed variables based on the graphical expression of missingness known as the m-graph, which captures observed variables together with missingness indicators and the possible causal links between them.
We then incorporated MBFS into the imputation process of
MissForest, to formulate a new algorithm suitable for imputation under both random and systematic missingness, which we call MBMF.
Empirical experiments based on both synthetic and real-world data sets show that MBMF outperforms the baseline MissForest in most of the experimental settings, and outperforms considerably other well-known or state-of-the-art imputation algorithms under both random and systematic missingness.
Moreover, while MBMF incorporates an additional learning phase needed to perform feature selection for each partially observed variable found in the input data, the results show that
MBMF is generally more efficient than the baseline MissForest, especially at larger sample sizes where efficiency matters the most.
This is because the time saved during imputation due to prior feature selection is higher than the additional time spent determining the best features for each partially observed variable.
Because the feature selection phase can be independent of the imputation phase (i.e., by using the
MBFS algorithm alone), future research works could extend this work to different relevant directions where feature selection is deemed to be important.
For example, MBFS could be combined with other imputation algorithms, including those based on deep learning (Mattei & Frellsen, 2019;
Fortuin et al., 2020; Lin et al., 2022) which are generally powerful but which tend to be time consuming and to overfit the data, since they typically process large numbers of uncorrelated features.
Moreover, since each MB discovered by MBFS could be used to construct the complete m-graph of the input data set, a rather different possible direction for future research would be to investigate the capability of MBFS in recovering the entire m-graph of the input data.
This task would require structural rules to deal with collisions between MBs as well as cycles, and would essentially convert
MBFS into a structure learning algorithm.
A P ROOF OF PROPOSITION
Given Assumption 1 and Assumption 2, R i is not an child of V i nor a parent of any other variables.
Therefore, we do not need to consider R i when learning MB ( V i ) .
We first show that the output Candidate Markov Blanket ( CMB ; i.e., output of Algorithm 1) set learned at the Grow phase contains all the variables of MB ( V i ) .
Assume Y is a parent or child of
V i , given the faithfulness condition, V i ̸⊥⊥ Y | CMB , R { V i ,Y }∪ CMB = 0 for any CMB ⊆
V o ∪
V m ∪ R − { V i , R i } .
Therefore, Y will always be added to CMB during the Grow phase.
Assume
Z → Y and that Y is also a child of V i .
Once Y is added to CMB , V i and Z will not be d-separated by CMB ∪ R { V i ,Y }∪ CMB and thus, Z will also be added to CMB .
Lastly, we show that the Shrink phase preserves the variables in MB ( V i ) only.
Let us assume that T is the first variable in MB ( V i ) when the algorithm enters the Shrink phase and attempts to remove variables in CMB , which have already been added during the Grow phase.
Because
MB ( V i ) − { T } ⊆
CMB , irrespective of T being a neighbour or a parent of a child of V
i , we always have V i ̸⊥⊥ T | CMB −{ T } , R { V i }∪ CMB =
given the faithfulness condition.
Therefore, no variable in MB ( V i ) will be removed during the Shrink phase.
On the other hand, if we assume T ̸∈ MB ( V i ) , since MB ( V i ) ⊆
CMB ∪ R { V i }∪ CMB −{ T } , we have V i ⊥⊥ T | CMB −{ T } , R { V i }∪ CMB =
given the faithfulness condition and thus, T will be removed from CMB .
B P ROOF OF PROPOSITION
For convenience, in this section we denote path p as the path being blocked by a variable set S if there is at least one node on p that satisfies either 1) it is a non-collider and in S or 2) it is a collider and neither it nor any of its descendants are in S .
Thus, if all paths between V i and V j are blocked by S , V i and V j are d-separated by S .
We firstly show that the CMB returned by
GS ( V i , V o ∪ V m − { V i } , D ) (phase 1 of MBFS) contains all but only variables in iMB ( V i ) , in addition to some other observed variables belonging to MB ( V i ) .
For the Grow phase in GS ( V i , V o ∪ V m − { V i } , D ) , assume that Y is a parent or child of V i , and that given the faithfulness condition, V i ̸⊥⊥ Y | CMB , R { V i ,Y }∪ CMB = 0 for any CMB ⊆
V o ∪ V m − { V i } .
Therefore, Y will be added in CMB .
Assume that Z is a parent of an observed variable Y , which is a child of V i .
After Y is added to CMB , V i and Z can no longer be d-separated by CMB ∪ R { V i ,Y }∪ CMB .
Thus, Z will also be added to CMB eventually.
As a result, all variables in iMB ( V i ) will be included in CMB at the end of the Grow phase.
For the Shrink phase, we prove two things: 1) that all the variables in iMB ( V i ) will remain in CMB , and 2) that all the variables not in the MB ( V i ) will be removed from CMB .
Suppose that T is the first variable in iMB ( V i ) that the algorithm attempts to remove from
CMB , and that V i and T are connected by either a direct edge or a path through a collider in iMB ( V i ) .
Therefore, T and V i cannot be d-separated by CMB ∪ R { V i }∪ CMB − { T } , since iMB ( V i ) ⊆
CMB ∪ R { V i }∪ CMB −{ T } .
Then, we have V
i ̸⊥⊥ T | CMB −{ T } , R { V i }∪ CMB = 0 given the faithfulness condition.
Suppose that T is an observed variable in CMB
such that T / ∈ MB ( V i ) .
Given the faithfulness condition, all we need to prove is that all paths between V i and T are blocked by
CMB ∪ R { V i }
∪ CMB − { T } .
• For a path p between V i and T composed by observed variables only, there is at least one non-collider node on p that belongs to iMB ( V i ) , such that p is blocked by CMB ∪ R { V i }∪ CMB − { T } , since iMB ( V i ) ⊆
CMB ∪ R { V i }∪ CMB − { T } .
• For a path p between V i and T contains indicator variables, when an observed node V j is adjacent to V i on p , at least one non-collider (either V j or the parent of V j ) on p will also be in iMB ( V i ) .
This is because no indicator variable can be a parent of V j according to Assumption 1, which implies that p will be blocked by iMB ( V i ) as well as CMB ∪ R { V i }∪ CMB − { T } .
If the adjacent node of V i on p is an indicator variable R j , then according to Assumptions 1 and 2, R j ̸ = R i
and it must have another parent V k ̸
= T on p since T / ∈ MB ( V i ) .
If R j / ∈ R CMB , path p is blocked by CMB ∪ R { V i }∪ CMB − { T } .
If R j ∈ R CMB , V k must be included in CMB during the Grow phase, and cannot be removed during the Shrink phase since V i and V k cannot be d-separated by CMB ∪ R { V i ,V k }∪ CMB .
Therefore, p is still blocked by CMB ∪ R { V i }∪ CMB − { T } since V k is a non-collider on p in CMB ∪ R { V i }∪ CMB − { T } .
In the phase 2 of MBFS, all the parents of each R j should remain in CPS as they cannot be dseparated from R j given any set composed of observed variables.
Besides, all of the other variables should be removed from CPS as they can be d-separated from R j given any set that contains all the parents of R j .
As a result, CMB should return MB ( V i ) at the end of phase 2.
C T HE IMPLICATIONS OF VIOLATING A SSUMPTIONS 1 AND 2 OF MBFS
If the Assumption 1 is violated (e.g., when an indicator variable is also a parent of some other variable), then both phases of MBFS may produce redundant variables in the CMB set.
Let us suppose that the true m-graph is the one shown in Figure 5.
If we use MBFS to learn the MB of V 1 , then V 2 will be included in the CMB set of Phase 1, and this is because V 1 and V 2 are independent only when conditioning on R 4 , whereas V 4 and V 1 are always independent.
This implies that MBFS will never test V 1 and V 2 for CI conditional on R 4 during Phase 1.
In Phase 2, V 6 will be included in the CPS set of R 5 and thus, it will also be added in CMB of V 1 .
If Assumption 2 is violated (e.g., when a partially observed variable V i is also the parent of its own missingness indicator R i ), this will cause MBFS not to include R i in the CMB of V i , since MBFS would not consider V i as a candidate variable in the CPS of R i during Phase 2.
V 1 V
V
V V
V
R R
R
D S UMMARY OF THE REAL - WORLD BN S IN SYNTHETIC EXPERIMENTS
[Table]
E M ISSING VALUE GENERATION PROCESS
We randomly choose 50% of the variables to be made partially observed.
We use α to represent the rate of missingness for each of those variables.
This process differs depending on the underlying assumption of missingness.
Specifically, 1.
For the MCAR condition, a value in V i is removed with probability P
( R i = 1) =
α .
For the MAR condition, a fully observed variable V j ∈ V o is randomly assigned as the parent of R i .
For continuous data, we denote the highest 10th-quantile of V j as q j , and remove values in V i with the following conditional probabilities: P ( R i = 1 | V j > q j ) =
2 α , P
( R i =
| V j < = q j ) = 89 α .
For categorical data, we select a set of states v j of
V j such that α < P ( V j = v j ) < 1 , then we remove values in V i with the following conditional probabilities: P ( R i = 1 | V j ̸ = v j ) =
2 α , P
( R i =
| V j = v j ) =
α − α
P ( V j = v j ) .
For the MNAR condition, a partially observed variable V j ∈ V m − { V i } is randomly assigned as the parent of R i .
Variable values are then removed for each V i using the same strategy as in the case of MAR.
F S UMMARY OF THE REAL - WORLD DATA SETS
[Table]
[Table]