Empirical evaluation of scoring functions for Bayesian network model selection
- Zhifa Liu†^{1, 2},
- Brandon Malone†^{1, 3} and
- Changhe Yuan^{1, 4}Email author
https://doi.org/10.1186/1471-2105-13-S15-S14
© Liu et al.; licensee BioMed Central Ltd. 2012
Published: 11 September 2012
Abstract
In this work, we empirically evaluate the capability of various scoring functions of Bayesian networks for recovering true underlying structures. Similar investigations have been carried out before, but they typically relied on approximate learning algorithms to learn the network structures. The suboptimal structures found by the approximation methods have unknown quality and may affect the reliability of their conclusions. Our study uses an optimal algorithm to learn Bayesian network structures from datasets generated from a set of gold standard Bayesian networks. Because all optimal algorithms always learn equivalent networks, this ensures that only the choice of scoring function affects the learned networks. Another shortcoming of the previous studies stems from their use of random synthetic networks as test cases. There is no guarantee that these networks reflect real-world data. We use real-world data to generate our gold-standard structures, so our experimental design more closely approximates real-world situations. A major finding of our study suggests that, in contrast to results reported by several prior works, the Minimum Description Length (MDL) (or equivalently, Bayesian information criterion (BIC)) consistently outperforms other scoring functions such as Akaike's information criterion (AIC), Bayesian Dirichlet equivalence score (BDeu), and factorized normalized maximum likelihood (fNML) in recovering the underlying Bayesian network structures. We believe this finding is a result of using both datasets generated from real-world applications rather than from random processes used in previous studies and learning algorithms to select high-scoring structures rather than selecting random models. Other findings of our study support existing work, e.g., large sample sizes result in learning structures closer to the true underlying structure; the BDeu score is sensitive to the parameter settings; and the fNML performs pretty well on small datasets. We also tested a greedy hill climbing algorithm and observed similar results as the optimal algorithm.
Introduction
Bayesian networks are compact graphical models for representing uncertain relationships among the random variables in a domain. Often, the relationships are unknown and must be learned from data. A popular approach called score-based learning [1] is to assign a score to each Bayesian network structure according to a scoring function and find the structure that optimizes the score. There are many scoring functions for Bayesian networks, such as minimum description length (MDL) [2] (or equivalently, Bayesian information criterion (BIC) [3]), Akaike's information criterion (AIC) [4], Bayesian Dirichlet equivalence score (BDeu) [5, 6], factorized normalized maximum likelihood (fNML) [7], and others [8, 9].
The score-based approach to learning Bayesian networks has been shown to be NP-hard [10]; both the running time and memory usage of exact learning are exponential in the number of variables in the worst case. Therefore, early research mainly focused on developing approximation methods [1, 11–14]. Recently, however, optimal learning algorithms such as dynamic programming [15–17], branch and bound [18], admissible heuristic search [19–21], and mathematical programming [22, 23] have been developed to learn optimal Bayesian networks with several dozens of variables.
Because of the different theoretical underpinnings of these scoring functions, they typically result in different "optimal" networks. Once a scoring function has been selected, though, all optimal algorithms learn equivalent networks; they only differ in running time and memory usage. A major mystery surrounding Bayesian network learning is which scoring function to use given that there are so many choices. Several empirical investigations have been carried out on the performance of various scoring functions in learning Bayesian networks, e.g. [24–26]. These studies, however, have drawbacks in their evaluations because they used local search methods such as K-2 [1] and Greedy Thick Thinning algorithm [27] to select network structures, or even used randomly generated network structures [26]. These suboptimal structures may affect the reliability of their conclusions regarding the model selection capability of the scoring functions. Furthermore, these studies often generate random synthetic networks as the test cases; experimental data thus generated may not share similar properties as real-world data.
In this study, we use an optimal dynamic programming algorithm [16] to learn Bayesian network structures; any other optimal algorithm would yield the same results, however, because only the choice of scoring function affects the learned networks. We study the capability of four scoring functions, MDL, AIC, BDeu, and fNML, to recover the underlying Bayesian network structures. We generated artificial datasets from a set of gold standard Bayesian networks created based on real-world data, learned optimal Bayesian networks for them using different scoring functions, and compared the learned models with the gold standard models based on various evaluation measures. For comparison, we also included the results of a greedy hill climbing algorithm.
Our results offer new insights into the scoring functions in addition to confirming some other common beliefs. In contrast to the results of existing work, a major finding of our study suggests that the MDL/BIC score consistently outperforms AIC, BDeu, and fNML in recovering the underlying Bayesian network structures across various sample sizes. Other findings of our study support existing work. Our results confirm that the structural Hamming distance gives a more reliable measure of the distance between Bayesian net-work structures. We also observed that a parameter selection greatly affects the BDeu score. Finally, it is confirmed that fNML has good performance when the sample sizes are relatively small. Our results using the greedy hill climbing algorithm are similar to those of the optimal learning algorithm, although with higher variances, so our conclusions also hold for the greedy algorithm.
The remainder of this paper is structured as follows. We first review several prior empirical studies of scoring functions. We then provide an overview of Bayesian network and structure learning. After that, we introduce four scoring functions which we will compare. We follow that with a description of the experimental design of this study. Finally, we present the empirical results and discuss our findings.
Prior work
Several researchers have empirically evaluated the various scoring functions for learning Bayesian networks. In [26], Van Allen and Greiner compared the performance of three different model selection criteria, AIC, BIC, and cross-validation, in finding the right balance between the complexity of the model and the goodness of fit to the training data. First, they randomly generated the gold standard Bayesian network structures as well as the probability parameters. Second, they generated datasets with different sample sizes from the networks. For each dataset, they again randomly constructed a set of hypothesis structures and evaluated their quality based on the scoring functions. They found that AIC and cross-validation perform better in avoiding over-fitting in the model selection. While BIC may still work for large sample sizes, it can perform arbitrarily worse than other functions for small datasets. However, they did not use a learning algorithm to try to find good hypothesis structures; they also randomly generated their gold standard networks. It is unclear whether their results stem from the scoring functions or their random model selection technique, or whether the results can be generalized to real-world datasets.
In Yang and Chang's study [24], they compared the performance of five different scoring functions: uniform prior score metric (UPSM), conditional uniform prior score metrics (CUPSM), Dirichlet prior score metric (DPSM), BDe, and BIC. They restricted their experimental evaluations on random networks with three or five nodes as well as a benchmark network called Alarm. Then they generated random datasets from the networks. They used a K2-like search method [1] to learn Bayesian networks. Their greedy structure learning algorithm assumes an ordering over the variables. Then, it greedily adds parents consistent with that ordering to maximize the likelihood of the structure and data set. Because of the ordering assumption and the greedy approach to adding parents, it does not guarantee finding the globally optimal structure. For evaluation, they use the cross-entropy (KL-Divergence) to measure the difference between the learned networks and the true networks. Their results indicated that UPSM, CUPSM, DPSM and BIC are able to correctly identify the true networks. Meanwhile, BDe and DPSM's performance are very sensitive to the α value. They may fail to find the true network if the α value is not set properly. This study shares the shortcoming of Van Allen and Greiner's study: their gold standard networks are randomly generated, so they may not accurately reflect real-world datasets. Furthermore, their K2-like search method requires an ordering of the variables; in real-world applications, an ordering is often not known a priori. Therefore, it is again unclear how their results generalize to real-world situations.
Another related empirical work by de Jongh and Druzdzel [25] investigates structural evaluation measures for Bayesian networks rather than scoring functions. They generated random datasets with different sizes from four benchmark Bayesian networks. Then for each combination of the network and sample size, they ran a local search algorithm called Greedy Thick Thinning [27] to learn Bayesian network structures and calculated the distances between the learned networks and the gold standard networks based on structural Hamming distance, Hamming distance, and other measures. They concluded that the structural Hamming distance is especially useful when looking for the causal structures.
All of these studies have drawbacks in their empirical evaluations. In particular, the conclusions of Van Allen and Greiner are drawn based on randomly generated network structures. Therefore, it is unclear how reliable their conclusions are regarding the model selection capability of the scoring functions. Additionally, the two studies which evaluate scoring functions rely on randomly generated gold standard networks; these may not accurately reflect real-world datasets. The work of de Jongh and Druzdzel only investigates structural evaluation measures using a single scoring function; other scoring functions may behave differently. The current study is designed to address these concerns.
Bayesian networks
Given a dataset D = {D_{1}, ..., D_{ N } }, where D_{ i } is an instantiation of all the variables in V, Bayesian network structure learning is the problem of learning a network structure from D. Assuming D is complete and discrete, Θ is maximized using frequency counts from the data [7]. Consequently, finding the optimal Bayesian network reduces to finding the optimal structure.
A Bayesian network structure can represent a set of joint probability distributions. Two network structures are said to belong to the same equivalence class if they represent the same set of probability distributions [28]. A scoring function which assigns the same score to networks in the same equivalence class is score equivalent [6].
Unfortunately, the number of possible structures is super-exponential in the number of variables; learning an optimal Bayesian network from D is shown to be NP-hard [10]. Solving the learning problem exactly becomes impractical if the number of variables is too large. Consequently, much early work focused on approximate algorithms, such as greedy hill climbing approaches [1, 11], tabu search with random restarts [13], limiting the number of parents or parameters for each variable [14], searching in the space of equivalence classes of network structures [29], and the optimal reinsertion algorithm (OR) [12]. These algorithms use local search to find "good" networks; however, they offer no guarantee to find the one that optimizes the scoring function. Recently, exact algorithms for learning optimal Bayesian networks have been developed based on dynamic programming [15–17, 30, 31], branch and bound [18], linear and integer programming (LP) [22, 23], and heuristic search [19–21]. These algorithms have enabled us to learn optimal Bayesian networks for datasets with dozens of variables.
Given a scoring function, all optimal learning algorithms learn equivalent networks; hence, the choice of which optimal algorithm is used does not affect the learned network. Consequently, these algorithms make it possible for us to study the behavior of different scoring functions in structure learning without needing to consider the confounding factors resulting from the choice of structure learning algorithms.
Scoring functions
where D_{ ij } is the instantiation of X_{ i } in data point D_{ j }, and PA_{ ij } is the instantiation of X_{ i }'s parents in D_{ j }. Adding an arc to a network never decreases the likelihood of the network. Intuitively, the extra arc is simply ignored if it does not add any more information. The extra arcs pose at least two problems, though. First, they may lead to overfitting of the training data and result in poor performance on testing data. Second, densely connected networks increase the running time when using the networks for downstream analysis, such as inference and prediction.
There are several well-known DPLL scoring functions for learning Bayesian networks. In this study, we consider MDL, AIC, BDeu and fNML. These scoring functions only differ in the penalty terms, so we will focus on discussing the penalty terms in the following discussions. In terms of memory and runtime, all of the scoring functions incur similar overhead [32].
Minimum description length (MDL)
where p_{ i } is the number of parameters for X_{ i }. For MDL, the penalty term reflects that more complex models will require longer encodings. The penalty term for MDL is larger than that of most other scoring functions, so optimal MDL networks tend to be sparser than optimal networks of other scoring functions. As hinted at by its name, an optimal MDL network minimizes rather than maximizes the scoring function. To interpret the penalty as a subtraction, the scores must be multiplied by -1. The Bayesian information criterion (BIC) [3] is a scoring function whose calculation is equivalent to MDL for Bayesian networks, but it is derived based on the asymptotic behavior of the models, that is, BIC is based on having a sufficiently large amount of data. Also, BIC does not require the -1 multiplication.
Akaike's information criterion (AIC)
Because its penalty term is less than that of MDL, AIC tends to favor more complex networks than MDL.
Bayesian Dirichlet with score equivalence and uniform priors (BDeu)
where q_{ i } is the number of possible values of PA_{ i }, r_{ i } is the number of possible values for X_{ i }, D_{ ijk } is the number of times X_{ i } = k and PA_{ i } = j in D, and α_{ ij } is a parameter calculated based on the user-specified α. The original derivations [5, 6] include a more detailed description. The density of the optimal network structure learned with BDeu is correlated with α; low α values typically result in sparser networks than higher α values. Recent studies [35] have shown the behavior of BDeu is very sensitive to α. If the density of the network to be learned is unknown, selecting an appropriate α is difficult.
Factorized normalized maximum likelihood (fNML)
where ${C}_{{N}_{ij}}^{{r}_{i}}$ are the regrets. fNML is not score equivalent.
Methods
Our empirical evaluation of the scoring functions consisted of four phases. First, we created a set of Bayesian networks from real datasets as the gold standard networks. Next, we generated a variety of datasets from each of those gold standard networks by logic sampling. After that, we learned optimal Bayesian networks from the sampled datasets using both an optimal algorithm and a greedy hill climbing algorithm. Finally, we calculated a number of evaluation metrics by comparing the learned networks with the gold standard networks.
Creating gold standard networks
We need a set of gold standard Bayesian networks as the basis for our empirical evaluations. It is possible to use randomly generated Bayesian networks like several existing studies did, but we want to use models that resemble Bayesian networks that are created for real-world applications. There are many benchmark Bayesian networks available, such as Alarm, CPCS, Hepar, etc., but these benchmark models contain too many variables and are intractable for the current optimal learning algorithms. Therefore, we chose to create the gold standard networks by learning optimal Bayesian networks for a set of UCI machine learning datasets [37] with fewer than 25 variables. This section describes our data processing method for the reproducibility of the results.
Summary of gold standard networks
Dataset | Domain | Instances | Nodes | Edges | Average In-degree |
---|---|---|---|---|---|
Statlog (Australian Credit Approval) | Industry | 690 | 15 | 33 | 2.20 |
Breast Cancer | Biology | 699 | 10 | 20 | 2.00 |
Car Evaluation | Industry | 1,728 | 7 | 9 | 1.29 |
Cleveland Heart Disease | Biology | 303 | 14 | 22 | 1.57 |
Credit Approval | Industry | 690 | 16 | 35 | 2.19 |
Diabetes | Biology | 768 | 9 | 13 | 1.44 |
Glass Identification | Industry | 214 | 10 | 17 | 1.70 |
Statlog (Heart) | Biology | 270 | 14 | 21 | 1.50 |
Hepatitis | Biology | 155 | 20 | 36 | 1.80 |
Iris | Biology | 150 | 5 | 8 | 1.60 |
Nursery | Industry | 12,960 | 9 | 14 | 1.56 |
Statlog (Vehicle Silhouettes) | Industry | 846 | 19 | 40 | 2.11 |
Congressional Voting Records | Political | 436 | 17 | 46 | 2.71 |
where S_{1} and S_{2} are the segments of S based on partitioning at T and Ent(·) is the entropy of the single set.
where k_{ i } is the number of distinct values of A in S_{ i }.
Although the MDL discretization technique has the same theoretical basis as the MDL scoring function, it is otherwise unrelated. That is, using the MDL discretization does not favor the MDL scoring function over the others in any way.
We used a k nearest neighbors (kNN) algorithm to impute missing values [39]. The kNN algorithm computes a missing value X_{ p } for record D_{ i } by finding the k closest D_{ j }s (out of those records which are not missing any values) to D_{ i } (using Euclidean distance, for example), excluding X_{ p }. If X_{ p } is a continuous variable, the value of X_{ p } is averaged for each of the D_{ j }s, and that value is assigned to X_{ p } for D_{ i }. If categorical, it is replaced by a majority vote among the k closest neighbors for X_{ p }. We set k = 5.
After processing the datasets, we applied an optimal learning algorithm based on the MDL scoring function [17] to learn optimal Bayesian networks. Again, the use of MDL score here does not affect the conclusions of this study, as other scoring functions yielded similar results. We used the maximum likelihood estimation method to learn the parameters of the networks. We took the learned networks as the gold standard networks and generated datasets from them.
Generating datasets from gold standard networks
After we created the gold standard networks, we generated datasets for each of these Bayesian networks with different numbers of data points ranging from 200 and 1000 (with increments equal to 200) and from 1,000 and 10,000 (with increments equal to 1,000), for a total of 18 sample sizes for each gold standard network. Each data point in a dataset corresponds to one random sample drawn from the joint probability distribution of a Bayesian network using logic sampling [40]. The basic idea is to sample the value for each variable according to the conditional probability distribution of the variable given its parents. The sampling is performed in a topological order of all the variables in order that all the parents already have sampled values before the child variable is sampled.
Learning from the sampled datasets
After generating datasets from the gold standard networks, we learned optimal networks for all the datasets by using the aforementioned scoring metrics. MDL, AIC and fNML are parameterless, so we learned one network for each combination of scoring function and dataset. All optimal learning algorithms would learn an equivalent network, so our choice of optimal learning algorithm does not affect the learned network. We tried the following α values, 0.1, 0.5, 1, 5, 10, 20, 50, 80, 100, for the hyperparameter α of BDeu and learned a network for each combination of α value and dataset. Thus, in total, we learned 12 "optimal" networks for each dataset and sample size. For comparison, we also tested a greedy hill climbing algorithm with random restarts and a tabu list in the same experiments.
Evaluating the learned networks
where a TP is an edge in the correct direction in the learned network, a TN is an edge in neither the learned nor the gold standard network, a FP is an edge in the learned network but not in the gold standard network, and a FN is an edge in the gold standard but not in the learned network. Note that an edge in the wrong direction in the learned network counts as both a FP and a FN.
We also used an evaluation metric called structural Hamming distance (SHD). As mentioned earlier, multiple structures with edges in different directions may belong to the same equivalence class. Intuitively, the distance between Bayesian networks in the same equivalence class should be zero. To accommodate this, SHD first identifies the equivalence class to which a Bayesian network belongs using an algorithm given by Chickering [28]. An equivalence class is represented by a partially directed graph (PDAG) in which some edges are directed and some undirected. The undirected edges can be orientated arbitrary as long as no new V structure in which multiple variables share a child is introduced. SHD then counts the number of directed and undirected edge additions, deletions, reversals and changes in direction to transform one PDAG into the other as the distance between two corresponding Bayesian networks. Tsamardinos et al. [41] provide a more formal algorithm for computing the SHD metric.
Results
In this section, we present the results of our empirical study. We first compared the evaluation metrics in order to select one metric for further analysis. We next looked into the effect of the hyperparameter α on the BDeu score. We then compared the capability of the scoring functions in recovering the Bayesian network structures from the sampled datasets generated from the gold standard Bayesian networks. After that, we compared the effect of sample sizes on the performance of the scoring functions in learning from the datasets when using both an optimal learning algorithm and a greedy hill climbing algorithm.
Comparison of evaluation metrics
BDeu parameterizations
Summary of the effect of different α values on the performance of BDeu
α = 0.1 | α= 0.5 | α= 1 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
GoldNet | Min | Mean | Max | STD | Min | Mean | Max | STD | Min | Mean | Max | STD |
Austra | 0 | 2.44 | 14 | 4.38 | 0 | 2.11 | 14 | 4.32 | 0 | 1.94* | 14 | 4.02 |
Breast | 0 | 0.83 | 8 | 2.18 | 0 | 0.61 | 5 | 1.50 | 0 | 0.61 | 5 | 1.50 |
Car | 0 | 1.44 | 5 | 2.28 | 0 | 0.89 | 5 | 1.91 | 0 | 0.89 | 5 | 1.91 |
Cleve | 1 | 1.83 | 11 | 2.43 | 1 | 1.50 | 7 | 1.54 | 1 | 1.44* | 7 | 1.46 |
Crx | 3 | 5.72* | 18 | 4.56 | 3 | 6.06 | 19 | 5.46 | 3 | 5.72 | 18 | 4.91 |
Diabetes | 1 | 1.72 | 6 | 1.71 | 1 | 1.22* | 4 | 0.73 | 1 | 1.28 | 4 | 0.75 |
Glass | 1 | 1.83 | 7 | 2.01 | 1 | 1.00* | 1 | 0.00 | 1 | 1.00* | 1 | 0.00 |
Heart | 0 | 0.00* | 0 | 0.00 | 0 | 0.00 | 0 | 0.00 | 0 | 0.00 | 0 | 0.00 |
Hepatitis | 3 | 3.72* | 13 | 2.37 | 3 | 4.22 | 12 | 2.62 | 6 | 7.06 | 13 | 2.01 |
Iris | 1 | 1.83 | 7 | 1.72 | 1 | 1.44 | 6 | 1.34 | 1 | 1.33* | 4 | 0.97 |
Nursery | 1 | 4.94 | 8 | 2.75 | 0 | 4.39 | 8 | 2.77 | 0 | 4.06 | 8 | 2.92 |
Vehicle | 0 | 0.67 | 9 | 2.14 | 0 | 0.22 | 4 | 0.94 | 0 | 0.22* | 4 | 0.94 |
Voting | 0 | 1.61 | 23 | 5.38 | 0 | 1.39 | 22 | 5.17 | 0 | 1.28* | 19 | 4.47 |
α = 5 | α = 10 | α = 20 | ||||||||||
GoldNet | Min | Mean | Max | STD | Min | Mean | Max | STD | Min | Mean | Max | STD |
Austra | 0 | 3.61 | 18 | 5.81 | 1 | 13.94 | 21 | 3.81 | 14 | 15.39 | 25 | 3.11 |
Breast | 0 | 0.22* | 4 | 0.94 | 0 | 0.56 | 10 | 2.36 | 0 | 1.33 | 11 | 2.93 |
Car | 0 | 0.28* | 5 | 1.18 | 0 | 0.28 | 5 | 1.18 | 0 | 0.33 | 5 | 1.19 |
Cleve | 4 | 6.61 | 13 | 2.85 | 9 | 12.56 | 19 | 2.18 | 20 | 21.56 | 25 | 1.10 |
Crx | 5 | 13.72 | 21 | 5.49 | 13 | 16.83 | 20 | 1.86 | 18 | 20.17 | 29 | 2.90 |
Diabetes | 3 | 3.22 | 6 | 0.73 | 5 | 5.33 | 10 | 1.19 | 9 | 9.11 | 11 | 0.47 |
Glass | 7 | 7.89 | 8 | 0.32 | 12 | 14.67 | 15 | 0.84 | 18 | 19.83 | 20 | 0.51 |
Heart | 1 | 1.00 | 1 | 0.00 | 1 | 1.44 | 2 | 0.51 | 2 | 2.22 | 4 | 0.55 |
Hepatitis | 25 | 30.22 | 33 | 1.99 | 39 | 41.78 | 44 | 1.31 | 50 | 53.56 | 60 | 2.25 |
Iris | 3 | 3.22 | 5 | 0.55 | 5 | 5.22 | 8 | 0.73 | 9 | 9.61 | 14 | 1.38 |
Nursery | 0 | 3.17 | 8 | 2.64 | 0 | 2.50 | 8 | 2.48 | 0 | 2.39* | 9 | 2.57 |
Vehicle | 0 | 0.44 | 5 | 1.20 | 1 | 2.39 | 10 | 2.03 | 5 | 6.72 | 15 | 2.47 |
Voting | 0 | 1.61 | 22 | 5.14 | 0 | 3.89 | 30 | 7.06 | 0 | 7.06 | 38 | 9.05 |
α = 50 | α = 80 | α = 100 | ||||||||||
GoldNet | Min | Mean | Max | STD | Min | Mean | Max | STD | Min | Mean | Max | STD |
Austra | 16 | 19.67 | 33 | 4.35 | 18 | 22.50 | 42 | 5.99 | 19 | 24.17 | 42 | 5.86 |
Breast | 2 | 7.44 | 14 | 3.35 | 7 | 11.67 | 16 | 3.36 | 7 | 13.72 | 20 | 3.72 |
Car | 0 | 0.67 | 8 | 1.88 | 0 | 1.50 | 8 | 1.76 | 0 | 1.94 | 8 | 1.66 |
Cleve | 26 | 27.50 | 34 | 1.95 | 27 | 29.72 | 41 | 3.27 | 28 | 30.50 | 42 | 3.49 |
Crx | 19 | 25.39 | 37 | 4.41 | 24 | 29.28 | 40 | 4.08 | 27 | 31.78 | 44 | 4.35 |
Diabetes | 13 | 14.89 | 16 | 1.13 | 14 | 16.33 | 18 | 1.37 | 14 | 15.67 | 18 | 1.50 |
Glass | 18 | 18.11 | 20 | 0.47 | 18 | 21.61 | 26 | 2.06 | 20 | 24.44 | 26 | 2.04 |
Heart | 4 | 4.11 | 5 | 0.32 | 4 | 4.61 | 5 | 0.50 | 4 | 4.94 | 5 | 0.24 |
Hepatitis | 59 | 61.50 | 72 | 3.03 | 61 | 64.28 | 75 | 3.92 | 61 | 65.28 | 78 | 4.86 |
Iris | 13 | 15.06 | 18 | 1.35 | 14 | 16.78 | 18 | 1.11 | 14 | 15.83 | 18 | 1.47 |
Nursery | 0 | 2.11 | 11 | 2.78 | 0 | 2.33 | 10 | 2.81 | 0 | 2.61 | 11 | 3.13 |
Vehicle | 14 | 18.44 | 30 | 3.88 | 19 | 23.50 | 36 | 4.08 | 22 | 27.44 | 39 | 4.15 |
Voting | 6 | 22.28 | 43 | 9.78 | 16 | 30.00 | 52 | 8.60 | 23 | 34.50 | 56 | 8.21 |
Gold standard network recovery
A comparison of the performance of four scoring functions in recovering the true underlying Bayesian network structures
AIC | MDL | fNML | BDeu | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
GoldNet | Min | Mean | Max | STD | Min | Mean | Max | STD | Min | Mean | Max | STD | Min | Mean | Max | STD |
Austra | 3 | 10.72 | 21 | 5.02 | 0 | 1.50* | 13 | 3.49 | 11 | 12.44 | 31 | 4.78 | 0 | 1.94 | 14 | 4.02 |
Breast | 0 | 1.28 | 6 | 1.64 | 0 | 0.61 | 5 | 1.50 | 0 | 0.39 | 4 | 1.14 | 0 | 0.22* | 4 | 0.94 |
Car | 0 | 1.00 | 7 | 1.91 | 0 | 1.67 | 5 | 2.43 | 5 | 5.94 | 6 | 0.24 | 0 | 0.28* | 5 | 1.18 |
Cleve | 2 | 10.44 | 22 | 3.94 | 0 | 0.44* | 5 | 1.34 | 0 | 0.94 | 12 | 2.82 | 1 | 1.44 | 7 | 1.46 |
Crx | 9 | 15.28 | 24 | 4.20 | 3 | 4.67* | 18 | 3.79 | 13 | 14.44 | 34 | 4.94 | 3 | 5.72 | 18 | 4.56 |
Diabetes | 0 | 2.00 | 5 | 1.41 | 0 | 0.22* | 3 | 0.73 | 0 | 0.22* | 3 | 0.73 | 1 | 1.22 | 4 | 0.73 |
Glass | 0 | 0.00* | 0 | 0.00 | 0 | 0.00* | 0 | 0.00 | 0 | 0.06 | 1 | 0.24 | 1 | 1.00 | 1 | 0.00 |
Heart | 0 | 0.00* | 0 | 0.00 | 0 | 0.00* | 0 | 0.00 | 0 | 0.00* | 0 | 0.00 | 0 | 0.00* | 0 | 0.00 |
Hepatitis | 17 | 21.83 | 31 | 4.13 | 0 | 0.44* | 6 | 1.46 | 0 | 2.94 | 24 | 5.95 | 3 | 3.72 | 13 | 2.37 |
Iris | 0 | 1.78 | 5 | 1.80 | 0 | 0.33 | 3 | 0.97 | 0 | 0.17* | 3 | 0.71 | 1 | 1.33 | 4 | 0.97 |
Nursery | 0 | 3.61* | 12 | 3.99 | 0 | 4.94 | 8 | 3.28 | 8 | 9.22 | 16 | 2.18 | 0 | 4.39 | 8 | 2.77 |
Vehicle | 0 | 0.72 | 4 | 1.07 | 0 | 1.11 | 16 | 3.77 | 0 | 0.39 | 7 | 1.65 | 0 | 0.22* | 4 | 0.94 |
Voting | 8 | 14.61 | 32 | 6.15 | 0 | 1.11* | 16 | 3.77 | 0 | 2.50 | 31 | 7.42 | 0 | 1.28 | 19 | 4.47 |
In contrast to the results reported by several previous studies, we found that MDL was able to recover the gold standard network more quickly than other scoring functions. We observe these differences both because we use an optimal learning algorithm and because we use gold standard networks representing real-world datasets. Given an appropriate α value, BDeu also converged to the gold standard networks within the sample sizes we tested. In some of the datasets, fNML converged to the gold standard network very quickly, but sometimes it converged to a different network. In contrast, AIC's behavior was much more erratic. It found the gold standard network on 8 of the datasets. But because of its high standard deviation, we infer it never completely converged. Figure 4 supports this conclusion. In light of these results, we conclude that MDL is a good scoring function because it often converges to the gold standard network. BDeu also exhibits good behavior if a suitable α is known before learning.
Convergence behavior
Next, we studied the convergence behavior of each scoring function. We did not consider whether the scoring function converged to the gold standard network; rather, we only focused on whether the scoring function converged to any network. In essence, this part of our study investigates the effect of the size of a dataset on the scoring functions. We again consult Figure 4 and Table 3 but this time look for convergence of the scoring functions; that is, we look to see at what point increasing sampling size does not change SHD anymore. As the figure shows, most of the scoring functions converged. To look for convergence in the table, we consider the mean, minimum, maximum, and standard deviation for the SHD statistics. We expect that if the scoring function converged quickly, its standard deviation will be small. This loose interpretation is robust in that it allows us to conclude that a scoring function converged even if SHD changes slightly from one sample size to the next.
As previously shown [7], fNML converges with fewer samples than the other scoring functions. Because the mean SHD is typically small, we conclude that the network to which it converges is often close to the gold standard network. MDL converged somewhat more slowly, but often converged to the gold standard network. BDeu with an optimal α value tends to converge quickly to a network close to the gold standard networks; however, with a sub-optimal α value, BDeu often neither converges nor comes close to the gold standard networks as shown in Table 2. Because AIC has a very low penalty term, more data encourages it to add more edges. Thus, it tends to overfit the data on large sample sizes and rarely converges. The SHD of AIC does tend to decrease as the sampling size increases, but that trend is somewhat inconsistent. Based on these results, fNML seems to be a good scoring function when data is limited, while MDL is superior when more data is present.
Comparison to greedy hill climbing
Finally, we compared the network recovery and convergence ability of a greedy hill climbing learning algorithm to those from the optimal algorithm. We performed this analysis because, as mentioned, optimal learning algorithms are limited to datasets with several dozens of variables. While some biological datasets (such as the Breast Cancer, Cleveland Heart Database, Diabetes, Statlog (Heart), Hepatitis and Iris datasets included in this study) are within this limit, many others, such as gene expression datasets, include hundreds or thousands of variables. Greedy hill climbing algorithms have been shown to scale to datasets of this size [14]. This part of our study verifies that our conclusions on scoring functions apply to this algorithm, as well.
A Comparison of Structural Error for the suboptimal learning algorithm and the optimal learning algorithm
Greedy Hill Climbing | Optimal | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
GoldNet | Size | Score | Add | Delete | Rev | Mis | Total | Add | Delete | Rev | Mis | Total |
Austr | 200 | AIC | 16 | 14 | 1 | 1 | 32 | 11 | 6 | 2 | 2 | 21 |
200 | MDL | 9 | 17 | 0 | 0 | 26 | 0 | 8 | 1 | 4 | 13 | |
200 | fNML | 11 | 16 | 0 | 1 | 28 | 20 | 7 | 0 | 4 | 31 | |
200 | 0.1 | 7 | 17 | 0 | 1 | 25 | 0 | 10 | 0 | 4 | 14 | |
200 | 0.5 | 9 | 17 | 0 | 0 | 26 | 1 | 9 | 1 | 3 | 14 | |
200 | 1 | 9 | 17 | 0 | 0 | 26 | 1 | 9 | 1 | 3 | 14 | |
200 | 5 | 11 | 12 | 2 | 2 | 27 | 5 | 6 | 1 | 6 | 18 | |
200 | 10 | 14 | 14 | 0 | 2 | 30 | 8 | 7 | 2 | 4 | 21 | |
600 | AIC | 18 | 15 | 1 | 0 | 34 | 7 | 1 | 0 | 0 | 8 | |
600 | MDL | 13 | 15 | 1 | 0 | 29 | 0 | 2 | 0 | 0 | 2 | |
600 | fNML | 13 | 15 | 2 | 0 | 30 | 1 | 3 | 0 | 7 | 11 | |
600 | 0.1 | 11 | 15 | 1 | 1 | 28 | 0 | 4 | 0 | 1 | 5 | |
600 | 0.5 | 12 | 15 | 1 | 1 | 29 | 0 | 3 | 0 | 1 | 4 | |
600 | 1 | 12 | 15 | 1 | 1 | 29 | 0 | 3 | 0 | 1 | 4 | |
600 | 5 | 14 | 14 | 1 | 4 | 33 | 1 | 2 | 0 | 0 | 3 | |
600 | 10 | 15 | 15 | 0 | 3 | 33 | 4 | 3 | 1 | 9 | 17 | |
1000 | AIC | 18 | 13 | 1 | 0 | 32 | 7 | 0 | 1 | 0 | 8 | |
1000 | MDL | 15 | 15 | 1 | 0 | 31 | 0 | 0 | 0 | 0 | 0 | |
1000 | fNML | 16 | 15 | 0 | 3 | 34 | 2 | 1 | 1 | 8 | 12 | |
1000 | 0.1 | 15 | 15 | 1 | 0 | 31 | 0 | 0 | 0 | 0 | 0 | |
1000 | 0.5 | 15 | 15 | 1 | 0 | 31 | 0 | 0 | 0 | 0 | 0 | |
1000 | 1 | 15 | 15 | 1 | 0 | 31 | 0 | 0 | 0 | 0 | 0 | |
1000 | 5 | 17 | 15 | 2 | 1 | 35 | 2 | 0 | 4 | 6 | 12 | |
1000 | 10 | 18 | 15 | 2 | 1 | 36 | 4 | 1 | 1 | 8 | 14 | |
Crx | 200 | AIC | 20 | 14 | 0 | 2 | 36 | 9 | 2 | 4 | 3 | 18 |
200 | MDL | 9 | 16 | 0 | 3 | 28 | 1 | 8 | 0 | 9 | 18 | |
200 | fNML | 16 | 15 | 1 | 1 | 33 | 19 | 5 | 6 | 4 | 34 | |
200 | 0.1 | 6 | 16 | 0 | 3 | 25 | 1 | 11 | 0 | 6 | 18 | |
200 | 0.5 | 10 | 16 | 0 | 3 | 29 | 1 | 8 | 0 | 9 | 18 | |
200 | 1 | 9 | 15 | 0 | 4 | 28 | 1 | 7 | 0 | 10 | 18 | |
200 | 5 | 13 | 14 | 1 | 2 | 30 | 5 | 6 | 3 | 5 | 19 | |
200 | 10 | 19 | 14 | 2 | 0 | 35 | 9 | 4 | 3 | 3 | 19 | |
600 | AIC | 21 | 14 | 0 | 0 | 35 | 8 | 1 | 2 | 0 | 11 | |
600 | MDL | 14 | 16 | 0 | 0 | 30 | 1 | 3 | 1 | 0 | 5 | |
600 | fNML | 14 | 14 | 0 | 4 | 32 | 3 | 3 | 1 | 7 | 14 | |
600 | 0.1 | 11 | 15 | 0 | 1 | 27 | 2 | 6 | 2 | 1 | 11 | |
600 | 0.5 | 13 | 15 | 0 | 0 | 28 | 1 | 3 | 1 | 0 | 5 | |
600 | 1 | 13 | 15 | 0 | 0 | 28 | 1 | 3 | 1 | 0 | 5 | |
600 | 5 | 17 | 13 | 2 | 3 | 35 | 6 | 2 | 2 | 7 | 17 | |
600 | 10 | 18 | 13 | 0 | 3 | 34 | 8 | 3 | 2 | 6 | 19 | |
1000 | AIC | 21 | 15 | 0 | 0 | 36 | 7 | 1 | 1 | 0 | 9 | |
1000 | MDL | 14 | 15 | 1 | 0 | 30 | 1 | 2 | 1 | 1 | 5 | |
1000 | fNML | 17 | 15 | 0 | 4 | 36 | 2 | 2 | 0 | 9 | 13 | |
1000 | 0.1 | 14 | 15 | 0 | 0 | 29 | 1 | 3 | 1 | 1 | 6 | |
1000 | 0.5 | 13 | 15 | 0 | 0 | 28 | 1 | 3 | 1 | 1 | 6 | |
1000 | 1 | 13 | 15 | 0 | 0 | 28 | 1 | 3 | 1 | 1 | 6 | |
1000 | 5 | 17 | 15 | 0 | 0 | 32 | 4 | 2 | 0 | 11 | 17 | |
1000 | 10 | 18 | 14 | 2 | 4 | 38 | 6 | 2 | 1 | 8 | 17 |
Conclusion
In this work, we have empirically investigated the ability of four Bayesian network scoring functions (MDL, AIC, BDeu and fNML) to recover the generating distribution of a dataset; a gold standard Bayesian network represents this distribution. We used an optimal structure learning algorithm to ensure approximation algorithms did not affect the learned networks. All optimal learning algorithms would learn an equivalent network, so our choice of optimal algorithm did not affect our results or conclusions. Then, we controlled scoring function and sample sizes to test their effect on the quality of the learned networks. We also considered four different evaluation metrics: accuracy, sensitivity, AHD and SHD. In addition, we evaluated a greedy hill climbing algorithm to ensure that our conclusions are valid for algorithms which can learn networks with hundreds or thousands of variables.
As a result of our investigation, we discovered that SHD is more well-behaved than the other evaluation metrics because it considers equivalence classes when comparing structures rather than the specific DAGs. Our most surprising result was that MDL was better able to recover gold standard networks than other scoring functions given sufficient data. As expected, BDeu's performance was highly dependent on the selected α parameter, which can be difficult to estimate a priori. We also confirmed that fNML converges even with few samples. Throughout our analysis, we found AIC's behavior erratic and unpredictable. The greedy hill climbing algorithm exhibited similar behavior, so we conclude that our results hold for this algorithm, as well.
We plan to extend this work in several ways. We can use synthetic networks to more carefully control the properties of our gold standard networks. Unlike previous studies, though, we will not rely on random network generation; instead, we will handcraft a variety of networks to reflect a variety of real-world datasets. We will also incorporate other scoring metrics, such as MIT [8], and objectives, such as prediction [9], into our study.
Notes
Declarations
Acknowledgements
This research was supported by the NSF grants IIS-0953723 and EPS-0903787. Some software packages that are used in this study are listed in the Additional File 1 S1.xls (sheet = Software).
This article has been published as part of BMC Bioinformatics Volume 13 Supplement 15, 2012: Proceedings of the Ninth Annual MCBIOS Conference. Dealing with the Omics Data Deluge. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/13/S15
Authors’ Affiliations
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