Supervised learning of Bayesian network parameters made easy (original) (raw)
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On supervised learning of Bayesian network parameters
2002
Bayesian network models are widely used for supervised prediction tasks such as classification. Usually the parameters of such models are determined using 'unsupervised' methods such as likelihood maximization, as it has not been clear how to find the parameters maximizing the supervised likelihood or posterior globally. In this paper we show how this supervised learning problem can be solved efficiently for a large class of Bayesian network models, including the Naive Bayes (NB) and Tree-augmented NB (TAN) classifiers. We show that there exists an alternative parameterization of these models in which the supervised likelihood becomes concave. From this result it follows that there can be at most one maximum, easily found by local optimization methods.
When discriminative learning of Bayesian network parameters is easy
2003
Bayesian network models are widely used for discriminative prediction tasks such as classification. Usually their parameters are determined using 'unsupervised' methods such as maximization of the joint likelihood. The reason is often that it is unclear how to find the parameters maximizing the conditional (supervised) likelihood. We show how the discriminative learning problem can be solved efficiently for a large class of Bayesian network models, including the Naive Bayes (NB) and treeaugmented Naive Bayes (TAN) models. We do this by showing that under a certain general condition on the network structure, the discriminative learning problem is exactly equivalent to logistic regression with unconstrained convex parameter spaces. Hitherto this was known only for Naive Bayes models. Since logistic regression models have a concave log-likelihood surface, the global maximum can be easily found by local optimization methods.
Efficient parameter learning of Bayesian network classifiers
Machine Learning, 2017
Recent advances have demonstrated substantial benefits from learning with both generative and discriminative parameters. On the one hand, generative approaches address the estimation of the parameters of the joint distribution-P(y, x), which for most network types is very computationally efficient (a notable exception to this are Markov networks) and on the other hand, discriminative approaches address the estimation of the parameters of the posterior distribution-and, are more effective for classification, since they fit P(y|x) directly. However, discriminative approaches are less computationally efficient as the normalization factor in the conditional log-likelihood precludes the derivation of closed-form estimation of parameters. This paper introduces a new discriminative parameter learning method for Bayesian network classifiers that combines in an elegant fashion parameters learned using
Maximum Margin Bayesian Network Classifiers
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2000
We present a maximum margin parameter learning algorithm for Bayesian network classifiers using a conjugate gradient (CG) method for optimization. In contrast to previous approaches, we maintain the normalization constraints on the parameters of the Bayesian network during optimization, i.e., the probabilistic interpretation of the model is not lost. This enables us to handle missing features in discriminatively optimized Bayesian networks. In experiments, we compare the classification performance of maximum margin parameter learning to conditional likelihood and maximum likelihood learning approaches. Discriminative parameter learning significantly outperforms generative maximum likelihood estimation for naive Bayes and tree augmented naive Bayes structures on all considered data sets. Furthermore, maximizing the margin dominates the conditional likelihood approach in terms of classification performance in most cases. We provide results for a recently proposed maximum margin optimization approach based on convex relaxation [1]. While the classification results are highly similar, our CG-based optimization is computationally up to orders of magnitude faster. Margin-optimized Bayesian network classifiers achieve classification performance comparable to support vector machines (SVMs) using fewer parameters. Moreover, we show that unanticipated missing feature values during classification can be easily processed by discriminatively optimized Bayesian network classifiers, a case where discriminative classifiers usually require mechanisms to complete unknown feature values in the data first.
On Discriminative Bayesian Network Classifiers and Logistic Regression
Machine Learning, 2005
Discriminative learning of the parameters in the naive Bayes model is known to be equivalent to a logistic regression problem. Here we show that the same fact holds for much more general Bayesian network models, as long as the corresponding network structure satisfies a certain graph-theoretic property. The property holds for naive Bayes but also for more complex structures such as tree-augmented naive Bayes (TAN) as well as for mixed diagnostic-discriminative structures. Our results imply that for networks satisfying our property, the conditional likelihood cannot have local maxima so that the global maximum can be found by simple local optimization methods. We also show that if this property does not hold, then in general the conditional likelihood can have local, non-global maxima. We illustrate our theoretical results by empirical experiments with local optimization in a conditional naive Bayes model. Furthermore, we provide a heuristic strategy for pruning the number of parameters and relevant features in such models. For many data sets, we obtain good results with heavily pruned submodels containing many fewer parameters than the original naive Bayes model.
Discriminative Parameter Learning of General Bayesian Network Classifiers
structure, and demonstrated that it often produces classifiers that are more accurate than the ones produced using the generative approach (OFE), which finds maximal likelihood parameters. This is especially true when learning parameters for incorrect structures, such as Naïve Bayes (NB). In searching for algorithms to learn better BN classifiers, this paper uses ELR to learn parameters of more nearly correct BN structures -e.g., of a general Bayesian network (GBN) learned from a structure-learning algorithm . While OFE typically produces more accurate classifiers with GBN (vs. NB), we show that ELR does not, when the training data is not sufficient for the GBN structure learner to produce a good model. Our empirical studies also suggest that the better the BN structure is, the less advantages ELR has over OFE, for classification purposes. ELR learning on NB (i.e., with little structural knowledge) still performs about the same as OFE on GBN in classification accuracy, over a large number of standard benchmark datasets.
Efficient Learning of Bayesian Network Classifiers
2007
We introduce a Bayesian network classifier less restrictive than Naive Bayes (NB) and Tree Augmented Naive Bayes (TAN) classifiers. Considering that learning an unrestricted network is unfeasible the proposed classifier is confined to be consistent with the breadth-first search order of an optimal TAN. We propose an efficient algorithm to learn such classifiers for any score that decompose over the network structure, including the well known scores based on information theory and Bayesian scoring functions. We show that the induced classifier always scores better than or the same as the NB and TAN classifiers. Experiments on modeling transcription factor binding sites show that, in many cases, the improved scores translate into increased classification accuracy.
Machine learning, 1997
Recent work in supervised learning has shown that a surprisingly simple Bayesian classifier with strong assumptions of independence among features, called naive Bayes, is competitive with state-of-the-art classifiers such as C4.5. This fact raises the question of whether a classifier with less restrictive assumptions can perform even better. In this paper we evaluate approaches for inducing classifiers from data, based on the theory of learning Bayesian networks. These networks are factored representations of probability distributions that generalize the naive Bayesian classifier and explicitly represent statements about independence. Among these approaches we single out a method we call Tree Augmented Naive Bayes (TAN), which outperforms naive Bayes, yet at the same time maintains the computational simplicity (no search involved) and robustness that characterize naive Bayes. We experimentally tested these approaches, using problems from the University of California at Irvine repository, and compared them to C4.5, naive Bayes, and wrapper methods for feature selection.
Learning Bayesian network classifiers by risk minimization
International Journal of Approximate Reasoning, 2012
Bayesian networks (BNs) provide a powerful graphical model for encoding the probabilistic relationships among a set of variables, and hence can naturally be used for classification. However, Bayesian network classifiers (BNCs) learned in the common way using likelihood scores usually tend to achieve only mediocre classification accuracy because these scores are less specific to classification, but rather suit a general inference problem. We propose risk minimization by cross validation (RMCV) using the 0/1 loss function, which is a classificationoriented score for unrestricted BNCs. RMCV is an extension of classification-oriented scores commonly used in learning restricted BNCs and non-BN classifiers. Using small real and synthetic problems, allowing for learning all possible graphs, we empirically demonstrate RMCV superiority to marginal and class-conditional likelihood-based scores with respect to classification accuracy. Experiments using twenty-two real-world datasets show that BNCs learned using an RMCV-based algorithm significantly outperform the naive Bayesian classifier (NBC), tree augmented NBC (TAN), and other BNCs learned using marginal or conditional likelihood scores and are on par with non-BN state of the art classifiers, such as support vector machine, neural network, and classification tree. These experiments also show that an optimized version of RMCV is faster than all unrestricted BNCs and comparable with the neural network with respect to run-time. The main conclusion from our experiments is that unrestricted BNCs, when learned properly, can be a good alternative to restricted BNCs and traditional machine-learning classifiers with respect to both accuracy and efficiency.
bnclassify: Learning Bayesian Network Classifiers
The R Journal, 2019
The bnclassify package provides state-of-the art algorithms for learning Bayesian network classifiers from data. For structure learning it provides variants of the greedy hill-climbing search, a well-known adaptation of the Chow-Liu algorithm and averaged one-dependence estimators. It provides Bayesian and maximum likelihood parameter estimation, as well as three naive-Bayesspecific methods based on discriminative score optimization and Bayesian model averaging. The implementation is efficient enough to allow for time-consuming discriminative scores on mediumsized data sets. The bnclassify package provides utilities for model evaluation, such as cross-validated accuracy and penalized log-likelihood scores, and analysis of the underlying networks, including network plotting via the Rgraphviz package. It is extensively tested, with over 200 automated tests that give a code coverage of 94%. Here we present the main functionalities, illustrate them with a number of data sets, and comment on related software.