贝叶斯网络在数据挖掘中的应用与优势

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"《网络在数据挖掘中的应用：贝叶斯网络》是一篇经典的论文，发表于1997年的《数据挖掘与知识发现》期刊，作者是David Heckerman。该文章深入探讨了贝叶斯网络在数据挖掘领域的关键作用。贝叶斯网络是一种图形模型，它以概率形式表示感兴趣的变量之间的关系，这在数据建模中具有显著优势。首先，贝叶斯网络的优势在于其能够处理缺失数据。由于模型结构本身就包含了所有变量之间的依赖性，即使部分数据缺失，网络仍能通过概率推断来估计缺失值，这对于实际数据集中的不完整性问题非常有用。其次，贝叶斯网络支持因果关系的学习。通过对数据的分析，网络可以揭示变量间的因果联系，这对于理解问题领域以及预测干预结果至关重要。这种因果理解能力有助于我们对复杂系统进行更深入的洞察。第三，贝叶斯网络结合了因果和概率两种语义，使其成为融合先验知识的理想工具。在许多情况下，我们可能事先了解某些变量间的因果关系，而数据则提供了进一步的支持。这种结合使得模型更具解释性和预测准确性。最后，论文强调了将贝叶斯统计方法与贝叶斯网络相结合的价值。这些方法不仅提供了准确的预测，还能够进行有效的推理和不确定性量化，从而提高数据挖掘任务的精度和可靠性。《Networks for Data Mining》这篇论文不仅介绍了贝叶斯网络的基本原理，还展示了如何将其应用于实际的数据挖掘场景，包括处理缺失数据、学习因果关系、整合先验知识和利用统计技术进行决策分析。这对于理解和实践数据挖掘领域的专业人士来说，是一篇不可或缺的参考文献。"

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Data Mining and Knowledge Discovery KL411-04-Heckerman February 26, 1997 18:6

86 HECKERMAN

Regardless of the functional form, we can learn about the parameters given data using

the Bayesian approach. As we have done in the binomial case, we deﬁne variables corre-

sponding to the unknown parameters, assign priors to these variables, and use Bayes’ rule

to update our beliefs about these parameters given data:

p(θ | D,ξ)=

p(D|θ,ξ) p(θ|ξ)

p(D|ξ)

(9)

We then average over the possible values of 2 to make predictions. For example,

p(x

N+1

| D,ξ)=

p(x

N+1

|θ,ξ) p(θ|D,ξ)dθ (10)

For a class of distributions known as the exponential family, these computations can be

done efﬁciently and in closed form

. Members of this class include the binomial, multi-

nomial, normal, Gamma, Poisson, and multivariate-normal distributions. Each member

of this family has sufﬁcient statistics that are of ﬁxed dimension for any random sample,

and a simple conjugate prior

. Bernardo and Smith (pp. 436–442, 1994) have compiled

the important quantities and Bayesian computations for commonly used members of the

exponential family. Here, we summarize these items for multinomial sampling, which we

use to illustrate many of the ideas in this paper.

In multinomial sampling, the observed variable X is discrete, having r possible states

,...,x

. The likelihood function is given by

p(X = x

|θ,ξ) =θ

, k =1,...,r

where θ ={θ

,...,θ

}are the parameters. (The parameter θ

is given by 1 −

k=2

In this case, as in the case of binomial sampling, the parameters correspond to physical

probabilities. The sufﬁcient statistics for data set D ={X

,...,X

= x

} is

,...,N

}, where N

is the number of times X = x

in D. The simple conjugate prior

used with multinomial sampling is the Dirichlet distribution:

p(θ |ξ) =Dir(θ |α

,...,α

)≡

0(α)

k=1

0(α

)

k=1

−1

(11)

where α =

i=1

, and α

> 0, k = 1,...,r. The posterior distribution p(θ | D,ξ) =

Dir(θ |α

,...,α

). Techniques for assessing the beta distribution, including the

methods of imagined future data and equivalentsamples, can also be used to assess Dirichlet

distributions. Given this conjugate prior and data set D, the probability distribution for the

next observation is given by

p(X

N+1

= x

| D,ξ)=

Dir(θ |α

+ N

,...,α

)dθ =

α+N

(12)

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Data Mining and Knowledge Discovery KL411-04-Heckerman February 26, 1997 18:6

BAYESIAN NETWORKS FOR DATA MINING 87

As we shall see, another important quantity in Bayesian analysis is the marginal likelihood

or evidence p(D |ξ). In this case, we have

p(D |ξ) =

0(α)

0(α + N)

k=1

0(α

+ N

)

0(α

)

(13)

We note that the explicit mention of the state of knowledge ξ is useful, because it reinforces

the notion that probabilities are subjective. Nonetheless, once this concept is ﬁrmly in place,

the notation simply adds clutter. In the remainder of this tutorial, we shall not mention ξ

explicitly.

Inclosing this section, we emphasizethat, althoughthe Bayesianandclassical approaches

may sometimes yield the same prediction, they are fundamentally different methods for

learning from data. As an illustration, let us revisit the thumbtack problem. Here, the

Bayesian “estimate” for the physical probability of heads is obtained in a manner that is

essentially the opposite of the classical approach.

Namely, in the classical approach, θ is ﬁxed (albeit unknown), and we imagine all data

sets of size N that may be generated by sampling from the binomial distribution determined

by θ. Each data set D will occur with some probability p(D |θ) and will produce an

estimate θ

∗

(D). To evaluate an estimator, we compute the expectation and variance of the

estimate with respect to all such data sets:

p(D |θ)

(θ

∗

) =

p(D |θ) θ

∗

(D)

Var

p ( D | θ)

(θ

∗

) =

p(D |θ) (θ

∗

(D) −E

p(D|θ)

(θ

∗

))

(14)

We then choose an estimator that somehow balances the bias (θ −E

p(D |θ)

(θ

∗

)) and variance

of these estimates over the possible values for θ.

Finally, we apply this estimator to the

data set that we actually observe. A commonly-used estimator is the maximum-likelihood

(ML) estimator, which selects the value of θ that maximizes the likelihood p(D |θ).For

binomial sampling, we have

∗

(D) =

k=1

In contrast, in the Bayesian approach, D is ﬁxed, and we imagine all possible values of θ

from which this data set could have been generated. Given θ , the “estimate” of the physical

probability of heads is just θ itself. Nonetheless, we are uncertain about θ, and so our ﬁnal

estimate is the expectation of θ with respect to our posterior beliefs about its value:

p(θ | D,ξ )

(θ) =

θ p(θ | D,ξ)dθ (15)

The expectations in Eqs. (14) and (15) are different and, in many cases, lead to different

“estimates”. One way to frame this difference is to say that the classical and Bayesian

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