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Estimating Continuous Distributions in Bayesian Classifiers (1302.4964v1)

Published 20 Feb 2013 in cs.LG, cs.AI, and stat.ML

Abstract: When modeling a probability distribution with a Bayesian network, we are faced with the problem of how to handle continuous variables. Most previous work has either solved the problem by discretizing, or assumed that the data are generated by a single Gaussian. In this paper we abandon the normality assumption and instead use statistical methods for nonparametric density estimation. For a naive Bayesian classifier, we present experimental results on a variety of natural and artificial domains, comparing two methods of density estimation: assuming normality and modeling each conditional distribution with a single Gaussian; and using nonparametric kernel density estimation. We observe large reductions in error on several natural and artificial data sets, which suggests that kernel estimation is a useful tool for learning Bayesian models.

Citations (3,623)

Summary

  • The paper introduces kernel density estimation as a flexible alternative to the Gaussian assumption in naive Bayesian classifiers.
  • The authors prove strong pointwise consistency for the FLEXIBLE BAYES method and validate its performance on multiple datasets.
  • Experiments show that kernel density estimation yields higher accuracy than traditional methods, especially when data deviates from normality.

Estimating Continuous Distributions in Bayesian Classifiers

George H. John and Pat Langley's paper, "Estimating Continuous Distributions in Bayesian Classifiers," addresses a crucial issue in the field of machine learning: the estimation of probability distributions for continuous variables within the framework of Bayesian classifiers, particularly the naive Bayesian classifier. This paper departs from traditional assumptions, advocating for nonparametric density estimation techniques as a means to enhance the performance and flexibility of such classifiers.

Introduction

The naive Bayesian classifier's popularity stems from its simplicity and competitive performance, even when compared to more sophisticated induction algorithms (Clark & Niblett, 1989; Langley, Iba & Thompson, 1992). However, the traditional naive Bayes method relies on two key assumptions: the conditional independence of predictive attributes given the class and the normality of numeric attribute distributions. These assumptions, while simplifying computational processes, may not hold in all real-world scenarios.

Core Contributions

This paper criticizes the prevalent reliance on a single Gaussian distribution for modeling continuous features. Instead, it introduces the Flexible Bayes (FLEXIBLE BAYES) framework, which employs kernel density estimation to model continuous attributes. This approach is compared against the traditional single Gaussian naive Bayesian classifier on various real-world and synthetic datasets.

Theoretical Underpinnings

Naive Bayes

The naive Bayesian classifier operates under the assumption that each feature is conditionally independent of the others given the class label, leading to an efficient calculation of the posterior probabilities necessary for classification. Continuous features are usually modeled using Gaussian distributions due to their convenient mathematical properties. However, this paper reveals that the Gaussian assumption can be overly restrictive.

Kernel Density Estimation

Kernel density estimation offers a more flexible approach by approximating the density of each continuous variable as a sum of kernel functions centered at observed data points. This method does not assume an underlying parametric distribution, thus accommodating more complex and realistic data distributions. The authors select Gaussian kernels and establish the kernel width parameter as σc=1/nc\sigma_c = 1/\sqrt{n_c}, where ncn_c is the number of training instances in class cc.

Consistency Proofs

The paper provides a rigorous theoretical foundation by proving that the density estimates produced by FLEXIBLE BAYES are strongly pointwise consistent. This consistency means that with sufficient data, the model's estimates converge to the true underlying distributions for both nominal and continuous features. This establishes FLEXIBLE BAYES as a robust and theoretically sound method.

Experimental Evaluation

Natural Data Sets

The authors evaluate FLEXIBLE BAYES through ten-fold cross-validation on 11 real-world datasets, comparing its performance against the traditional naive Bayesian classifier and C4.5, a widely used decision tree algorithm. The results indicate that FLEXIBLE BAYES achieves significantly higher accuracy on several datasets, particularly those where the normality assumption for continuous attributes is violated. For example, on the Glass and Vehicle Silhouette datasets, FLEXIBLE BAYES outperforms the naive Bayesian approach by substantial margins (66.2% vs. 42.9% and 61.5% vs. 44.9%, respectively).

Synthetic Data

Experiments on synthetic data further elucidate the advantages of FLEXIBLE BAYES. In scenarios where the normality assumption holds, both naive and flexible Bayes classifiers converge to similar performance levels as the sample size grows. Conversely, when the data is drawn from a mixture of Gaussians, FLEXIBLE BAYES quickly outperforms the naive Bayes classifier, reaching accuracy levels close to the Bayes optimal error rate. This demonstrates FLEXIBLE BAYES's robustness to different underlying data distributions.

Implications and Future Work

The implications of utilizing kernel density estimation in Bayesian classifiers are both practical and theoretical. Practically, FLEXIBLE BAYES enables better modeling of real-world data distributions, thereby improving the classifier's performance. Theoretically, this work opens avenues for further exploration into adaptive kernel width selection and the integration of model selection methods to optimize the balance between flexibility and overfitting.

Future developments could include incorporating adaptive and data-driven techniques for setting kernel widths, leveraging cross-validation or Bayesian methods for parameter estimation to further enhance performance. Additionally, investigating the computational trade-offs and scalability of FLEXIBLE BAYES in large-scale applications would be beneficial.

Conclusion

George H. John and Pat Langley's paper significantly contributes to the machine learning domain by challenging conventional assumptions in naive Bayesian classifiers and proposing a more flexible and theoretically sound approach through kernel density estimation. The empirical and theoretical findings of this paper underscore the potential of nonparametric methods to enhance probabilistic induction tasks, making FLEXIBLE BAYES a valuable addition to the arsenal of machine learning techniques.